Boltzmann and Einstein: Statistics and dynamics –An unsolved ...
Indian Academy of Sciences (India)
The struggle of Boltzmann with the proper description of the behavior of classical macroscopic bodies in equilibrium in terms of the properties of the particles out of which they consist will be sketched. He used both a dynamical and a statistical method. However, Einstein strongly disagreed with Boltzmann's statistical method ...
Boltzmann and Einstein: Statistics and dynamics–An unsolved ...
Indian Academy of Sciences (India)
The struggle of Boltzmann with the proper description of the behavior of classical macroscopic bodies in equilibrium in terms of the properties of the particles out of which they consist will be sketched. He used both a dynamical and a statistical method. However, Einstein strongly disagreed with Boltzmann's statistical method ...
Boltzmann and Einstein: Statistics and dynamics – An unsolved ...
Indian Academy of Sciences (India)
... of watching the ever-shifting battle!” (not to see its outcome). Acknowledgements. The author would like to express my deep appreciation to the IUPAP Commission on Statistical Physics for awarding me the Boltzmann medal 2004. The author is also indebted for financial assistance to the Organizers of STATPHYS 22, T V ...
Mechanistic slumber vs. statistical insomnia: the early history of Boltzmann's H-theorem (1868-1877)
Badino, M.
2011-11-01
An intricate, long, and occasionally heated debate surrounds Boltzmann's H-theorem (1872) and his combinatorial interpretation of the second law (1877). After almost a century of devoted and knowledgeable scholarship, there is still no agreement as to whether Boltzmann changed his view of the second law after Loschmidt's 1876 reversibility argument or whether he had already been holding a probabilistic conception for some years at that point. In this paper, I argue that there was no abrupt statistical turn. In the first part, I discuss the development of Boltzmann's research from 1868 to the formulation of the H-theorem. This reconstruction shows that Boltzmann adopted a pluralistic strategy based on the interplay between a kinetic and a combinatorial approach. Moreover, it shows that the extensive use of asymptotic conditions allowed Boltzmann to bracket the problem of exceptions. In the second part I suggest that both Loschmidt's challenge and Boltzmann's response to it did not concern the H-theorem. The close relation between the theorem and the reversibility argument is a consequence of later investigations on the subject.
He, Ping
2012-01-01
The long-standing puzzle surrounding the statistical mechanics of self-gravitating systems has not yet been solved successfully. We formulate a systematic theoretical framework of entropy-based statistical mechanics for spherically symmetric collisionless self-gravitating systems. We use an approach that is very different from that of the conventional statistical mechanics of short-range interaction systems. We demonstrate that the equilibrium states of self-gravitating systems consist of both mechanical and statistical equilibria, with the former characterized by a series of velocity-moment equations and the latter by statistical equilibrium equations, which should be derived from the entropy principle. The velocity-moment equations of all orders are derived from the steady-state collisionless Boltzmann equation. We point out that the ergodicity is invalid for the whole self-gravitating system, but it can be re-established locally. Based on the local ergodicity, using Fermi-Dirac-like statistics, with the non-degenerate condition and the spatial independence of the local microstates, we rederive the Boltzmann-Gibbs entropy. This is consistent with the validity of the collisionless Boltzmann equation, and should be the correct entropy form for collisionless self-gravitating systems. Apart from the usual constraints of mass and energy conservation, we demonstrate that the series of moment or virialization equations must be included as additional constraints on the entropy functional when performing the variational calculus; this is an extension to the original prescription by White & Narayan. Any possible velocity distribution can be produced by the statistical-mechanical approach that we have developed with the extended Boltzmann-Gibbs/White-Narayan statistics. Finally, we discuss the questions of negative specific heat and ensemble inequivalence for self-gravitating systems.
Fermi-Pasta-Ulam-Tsingou problems: Passage from Boltzmann to q-statistics
Bagchi, Debarshee; Tsallis, Constantino
2018-02-01
The Fermi-Pasta-Ulam (FPU) one-dimensional Hamiltonian includes a quartic term which guarantees ergodicity of the system in the thermodynamic limit. Consistently, the Boltzmann factor P(ε) ∼e-βε describes its equilibrium distribution of one-body energies, and its velocity distribution is Maxwellian, i.e., P(v) ∼e - βv2 /2. We consider here a generalized system where the quartic coupling constant between sites decays as 1 / dijα (α ≥ 0 ;dij = 1 , 2 , …) . Through first-principle molecular dynamics we demonstrate that, for large α (above α ≃ 1), i.e., short-range interactions, Boltzmann statistics (based on the additive entropic functional SB [ P(z) ] = - k ∫ dzP(z) ln P(z)) is verified. However, for small values of α (below α ≃ 1), i.e., long-range interactions, Boltzmann statistics dramatically fails and is replaced by q-statistics (based on the nonadditive entropic functional Sq [ P(z) ] = k(1 - ∫ dz[ P(z) ]q) /(q - 1) , with S1 =SB). Indeed, the one-body energy distribution is q-exponential, P(ε) ∼ eqε-βε ε ≡[ 1 +(qε - 1) βε ε ]-1 /(qε - 1) with qε > 1, and its velocity distribution is given by P(v) ∼ eqv-βvv2 / 2 with qv > 1. Moreover, within small error bars, we verify qε =qv = q, which decreases from an extrapolated value q ≃ 5 / 3 to q = 1 when α increases from zero to α ≃ 1, and remains q = 1 thereafter.
Farrell, Patricio; Koprucki, Thomas; Fuhrmann, Jürgen
2017-10-01
We compare three thermodynamically consistent numerical fluxes known in the literature, appearing in a Voronoï finite volume discretization of the van Roosbroeck system with general charge carrier statistics. Our discussion includes an extension of the Scharfetter-Gummel scheme to non-Boltzmann (e.g. Fermi-Dirac) statistics. It is based on the analytical solution of a two-point boundary value problem obtained by projecting the continuous differential equation onto the interval between neighboring collocation points. Hence, it serves as a reference flux. The exact solution of the boundary value problem can be approximated by computationally cheaper fluxes which modify certain physical quantities. One alternative scheme averages the nonlinear diffusion (caused by the non-Boltzmann nature of the problem), another one modifies the effective density of states. To study the differences between these three schemes, we analyze the Taylor expansions, derive an error estimate, visualize the flux error and show how the schemes perform for a carefully designed p-i-n benchmark simulation. We present strong evidence that the flux discretization based on averaging the nonlinear diffusion has an edge over the scheme based on modifying the effective density of states.
Error statistics of hidden Markov model and hidden Boltzmann model results
Directory of Open Access Journals (Sweden)
Newberg Lee A
2009-07-01
Full Text Available Abstract Background Hidden Markov models and hidden Boltzmann models are employed in computational biology and a variety of other scientific fields for a variety of analyses of sequential data. Whether the associated algorithms are used to compute an actual probability or, more generally, an odds ratio or some other score, a frequent requirement is that the error statistics of a given score be known. What is the chance that random data would achieve that score or better? What is the chance that a real signal would achieve a given score threshold? Results Here we present a novel general approach to estimating these false positive and true positive rates that is significantly more efficient than are existing general approaches. We validate the technique via an implementation within the HMMER 3.0 package, which scans DNA or protein sequence databases for patterns of interest, using a profile-HMM. Conclusion The new approach is faster than general naïve sampling approaches, and more general than other current approaches. It provides an efficient mechanism by which to estimate error statistics for hidden Markov model and hidden Boltzmann model results.
International Nuclear Information System (INIS)
Tripathy, S.; Tiwari, S.K.; Younus, M.; Sahoo, R.
2017-01-01
One of the major goals in heavy-ion physics is to understand the properties of Quark Gluon Plasma (QGP), a deconfined hot and dense state of quarks and gluons existed shortly after the Big Bang. In the present scenario, the high-energy particle accelerators are able to reach energies where this extremely dense nuclear matter can be probed for a short time. Here, we follow our earlier works which use non-extensive statistics in Boltzmann Transport Equation (BTE). We represent the initial distribution of particles with the help of Tsallis power law distribution parameterized by the nonextensive parameter q and the Tsallis temperature T, remembering the fact that their origin is due to hard scatterings. We use the initial distribution (f in ) with Relaxation Time Approximation (RTA) of the BTE and calculate the final distribution (f fin ). Then we calculate ν 2 of the system using the final distribution in the definition of ν2
Thompson, John
2015-04-01
As the Physical Review Focused Collection demonstrates, recent frontiers in physics education research include systematic investigations at the upper division. As part of a collaborative project, we have examined student understanding of several topics in upper-division thermal and statistical physics. A fruitful context for research is the Boltzmann factor in statistical mechanics: the standard derivation involves several physically justified mathematical steps as well as the invocation of a Taylor series expansion. We have investigated student understanding of the physical significance of the Boltzmann factor as well as its utility in various circumstances, and identified various lines of student reasoning related to the use of the Boltzmann factor. Results from written data as well as teaching interviews suggest that many students do not use the Boltzmann factor when answering questions related to probability in applicable physical situations, even after lecture instruction. We designed an inquiry-based tutorial activity to guide students through a derivation of the Boltzmann factor and to encourage deep connections between the physical quantities involved and the mathematics. Observations of students working through the tutorial suggest that many students at this level can recognize and interpret Taylor series expansions, but they often lack fluency in creating and using Taylor series appropriately, despite previous exposure in both calculus and physics courses. Our findings also suggest that tutorial participation not only increases the prevalence of relevant invocation of the Boltzmann factor, but also helps students gain an appreciation of the physical implications and meaning of the mathematical formalism behind the formula. Supported in part by NSF Grants DUE-0817282, DUE-0837214, and DUE-1323426.
Numerical solutions of the semiclassical Boltzmann ellipsoidal-statistical kinetic model equation
Yang, Jaw-Yen; Yan, Chin-Yuan; Huang, Juan-Chen; Li, Zhihui
2014-01-01
Computations of rarefied gas dynamical flows governed by the semiclassical Boltzmann ellipsoidal-statistical (ES) kinetic model equation using an accurate numerical method are presented. The semiclassical ES model was derived through the maximum entropy principle and conserves not only the mass, momentum and energy, but also contains additional higher order moments that differ from the standard quantum distributions. A different decoding procedure to obtain the necessary parameters for determining the ES distribution is also devised. The numerical method in phase space combines the discrete-ordinate method in momentum space and the high-resolution shock capturing method in physical space. Numerical solutions of two-dimensional Riemann problems for two configurations covering various degrees of rarefaction are presented and various contours of the quantities unique to this new model are illustrated. When the relaxation time becomes very small, the main flow features a display similar to that of ideal quantum gas dynamics, and the present solutions are found to be consistent with existing calculations for classical gas. The effect of a parameter that permits an adjustable Prandtl number in the flow is also studied. PMID:25104904
Energy Technology Data Exchange (ETDEWEB)
Zhang Lei; Kashiwakura, Shunsuke; Wagatsuma, Kazuaki, E-mail: wagatuma@imr.tohoku.ac.jp
2011-11-15
A Boltzmann plot for many iron atomic lines having excitation energies of 3.3-6.9 eV was investigated in glow discharge plasmas when argon or neon was employed as the plasma gas. The plot did not show a linear relationship over a wide range of the excitation energy, but showed that the emission lines having higher excitation energies largely deviated from a normal Boltzmann distribution whereas those having low excitation energies (3.3-4.3 eV) well followed it. This result would be derived from an overpopulation among the corresponding energy levels. A probable reason for this is that excitations for the high-lying excited levels would be caused predominantly through a Penning-type collision with the metastable atom of argon or neon, followed by recombination with an electron and then stepwise de-excitations which can populate the excited energy levels just below the ionization limit of iron atom. The non-thermal excitation occurred more actively in the argon plasma rather than the neon plasma, because of a difference in the number density between the argon and the neon metastables. The Boltzmann plots yields important information on the reason why lots of Fe I lines assigned to high-lying excited levels can be emitted from glow discharge plasmas. - Highlights: Black-Right-Pointing-Pointer This paper shows the excitation mechanism of Fe I lines from a glow discharge plasma. Black-Right-Pointing-Pointer A Boltzmann distribution is studied among iron lines of various excitation levels. Black-Right-Pointing-Pointer We find an overpopulation of the high-lying energy levels from the normal distribution. Black-Right-Pointing-Pointer It is caused through Penning-type collision of iron atom with argon metastable atom.
International Nuclear Information System (INIS)
Zhang Lei; Kashiwakura, Shunsuke; Wagatsuma, Kazuaki
2011-01-01
A Boltzmann plot for many iron atomic lines having excitation energies of 3.3–6.9 eV was investigated in glow discharge plasmas when argon or neon was employed as the plasma gas. The plot did not show a linear relationship over a wide range of the excitation energy, but showed that the emission lines having higher excitation energies largely deviated from a normal Boltzmann distribution whereas those having low excitation energies (3.3–4.3 eV) well followed it. This result would be derived from an overpopulation among the corresponding energy levels. A probable reason for this is that excitations for the high-lying excited levels would be caused predominantly through a Penning-type collision with the metastable atom of argon or neon, followed by recombination with an electron and then stepwise de-excitations which can populate the excited energy levels just below the ionization limit of iron atom. The non-thermal excitation occurred more actively in the argon plasma rather than the neon plasma, because of a difference in the number density between the argon and the neon metastables. The Boltzmann plots yields important information on the reason why lots of Fe I lines assigned to high-lying excited levels can be emitted from glow discharge plasmas. - Highlights: ► This paper shows the excitation mechanism of Fe I lines from a glow discharge plasma. ► A Boltzmann distribution is studied among iron lines of various excitation levels. ► We find an overpopulation of the high-lying energy levels from the normal distribution. ► It is caused through Penning-type collision of iron atom with argon metastable atom.
Ludwig Boltzmann, mechanics and vitalism
International Nuclear Information System (INIS)
Broda, E.
1990-01-01
During most of his life Boltzmann considered classical mechanics, based on the ideas of material points and central forces, as the fundament of physics. On this basis he became one of the founders of Statistical Mechanics, through which thermodynamics was interpreted on an atomistic basis. In this work, Boltzmann was opposed by his colleague, Ernst Mach. Boltzmann also devoted much work to attempts to interpret Maxwell's theory of the electromagnetic field, of which he was a main protagonist in Central Europe, through mechanics. However, as a supporter of mechanics Boltzmann was by no means dogmatic. While he was adamant in his rejection of Wilhelm Ostwald's energism, he was openminded in respect to the relationship of mechanics, electromagnetism and atomistics. Personally, Boltzmann wanted to conserve and transmit the enormous achievements of mechanics, especially in connection with the mechanical theory of heat, so that these results should not be lost to future generations, but he encouraged attempts to proceed in new directions. While within the framework of statistical mechanics the atoms were treated like the material points of classical mechanics, Boltzmann resisted the initial, unwarranted, ideas about the structure and the properties of the atoms. When later valid ideas were evolved, Boltzmann warmly welcomed this progress, without however personally taking part in the new developments. In his later years, Boltzmann took an intense interest in biology. He supported Darwin's theories, and he contributed to them. He may be called an 'absolute Darwinist'. In his search for a natural explanation of the phenomena of life, he used the term 'mechanical', without meaning to limit them to the realm of classical mechanics. This terminological laxity is considered as unfortunate. Extending his application of Darwinian principles to advanced species, including man, Boltzmann put forward 'mechanical' explanations of thought
Siegel, Z.; Siegel, Edward Carl-Ludwig
2011-03-01
RANDOMNESS of Numbers cognitive-semantics DEFINITION VIA Cognition QUERY: WHAT???, NOT HOW?) VS. computer-``science" mindLESS number-crunching (Harrel-Sipser-...) algorithmics Goldreich "PSEUDO-randomness"[Not.AMS(02)] mea-culpa is ONLY via MAXWELL-BOLTZMANN CLASSICAL-STATISTICS(NOT FDQS!!!) "hot-plasma" REPULSION VERSUS Newcomb(1881)-Weyl(1914;1916)-Benford(1938) "NeWBe" logarithmic-law digit-CLUMPING/ CLUSTERING NON-Randomness simple Siegel[AMS Joint.Mtg.(02)-Abs. # 973-60-124] algebraic-inversion to THE QUANTUM and ONLY BEQS preferentially SEQUENTIALLY lower-DIGITS CLUMPING/CLUSTERING with d = 0 BEC, is ONLY VIA Siegel-Baez FUZZYICS=CATEGORYICS (SON OF TRIZ)/"Category-Semantics"(C-S), latter intersection/union of Lawvere(1964)-Siegel(1964)] category-theory (matrix: MORPHISMS V FUNCTORS) "+" cognitive-semantics'' (matrix: ANTONYMS V SYNONYMS) yields Siegel-Baez FUZZYICS=CATEGORYICS/C-S tabular list-format matrix truth-table analytics: MBCS RANDOMNESS TRUTH/EMET!!!
Suzuki, Hideyuki; Imura, Jun-ichi; Horio, Yoshihiko; Aihara, Kazuyuki
2013-01-01
The chaotic Boltzmann machine proposed in this paper is a chaotic pseudo-billiard system that works as a Boltzmann machine. Chaotic Boltzmann machines are shown numerically to have computing abilities comparable to conventional (stochastic) Boltzmann machines. Since no randomness is required, efficient hardware implementation is expected. Moreover, the ferromagnetic phase transition of the Ising model is shown to be characterised by the largest Lyapunov exponent of the proposed system. In general, a method to relate probabilistic models to nonlinear dynamics by derandomising Gibbs sampling is presented. PMID:23558425
Tripathy, Sushanta; Kumar Tiwari, Swatantra; Younus, Mohammed; Sahoo, Raghunath
2018-03-01
Elliptic flow in heavy-ion collisions is an important signature of a possible de-confinement transition from hadronic phase to partonic phase. In the present work, we use non-extensive statistics, which has been used for transverse momentum (pT) distribution in proton+proton ( p+p) collisions, as the initial particle distribution function in Boltzmann Transport Equation (BTE). A Boltzmann-Gibbs Blast Wave (BGBW) function is taken as an equilibrium function to get the final distribution to describe the particle production in heavy-ion collisions. In this formalism, we try to estimate the elliptic flow in Pb+Pb collisions at √{s_{NN}} = 2.76 TeV at the LHC for different centralities. The elliptic flow ( v2) of identified particles seems to be described quite well in the available pT range. An approach which combines the non-extensive nature of particle production in p+p collisions through an evolution in kinetic theory using BTE, with BGBW as an equilibrium distribution is successful in describing the spectra and elliptic flow in heavy-ion collisions.
Limitations of Boltzmann's principle
International Nuclear Information System (INIS)
Lavenda, B.H.
1995-01-01
The usual form of Boltzmann's principle assures that maximum entropy, or entropy reduction, occurs with maximum probability, implying a unimodal distribution. Boltzmann's principle cannot be applied to nonunimodal distributions, like the arcsine law, because the entropy may be concave only over a limited portion of the interval. The method of subordination shows that the arcsine distribution corresponds to a process with a single degree of freedom, thereby confirming the invalidation of Boltzmann's principle. The fractalization of time leads to a new distribution in which arcsine and Cauchy distributions can coexist simultaneously for nonintegral degrees of freedom between √2 and 2
Boltzmann factor and Hawking radiation
International Nuclear Information System (INIS)
Ryskin, Gregory
2014-01-01
Hawking radiation has thermal spectrum corresponding to the temperature T H =(8πM) −1 , where M is the mass (energy) of the black hole. Corrections to the Hawking radiation spectrum were discovered by Kraus and Wilczek (1995) and Parikh and Wilczek (2000). Here I show that these corrections follow directly from the basic principles of thermodynamics and statistical mechanics. In essence, it is the Boltzmann factor that ought to be corrected; corrections to the Hawking (or any other) radiation spectrum then follow necessarily
Lindley, David
2002-01-01
Ludwig Boltzmann (1844-1906) è il fisico e matematico austriaco che negli ultimi decenni dell'Ottocento e ancora ai primi del Novecento lottò contro l'opinione dominante tra gli scienziati dell'epoca per affermare la teoria atomica della materia. È noto come con Albert Einstein e fino a oggi la fisica si sia sviluppata e abbia celebrato i propri trionfi lungo le linee anticipate da Boltzmann. La controversia con Mach non riguardava soltanto l'esistenza degli atomi, ma l'intero modo di fare fisica che Boltzmann non riteneva di dover limitare allo studio di quantità misurabili, introducendo invece spiegazioni più elaborate basate su ipotesi più ampie.
Metamaterial characterization using Boltzmann's kinetic equation for electrons
DEFF Research Database (Denmark)
Novitsky, Andrey; Zhukovsky, Sergei; Novitsky, D.
2013-01-01
Statistical properties of electrons in metals are taken into consideration to describe the microscopic motion of electrons. Assuming degenerate electron gas in metal, we introduce the Boltzmann kinetic equation to supplement Maxwell's equations. The solution of these equations clearly shows...
Training Restricted Boltzmann Machines
DEFF Research Database (Denmark)
Fischer, Asja
Restricted Boltzmann machines (RBMs) are probabilistic graphical models that can also be interpreted as stochastic neural networks. Training RBMs is known to be challenging. Computing the likelihood of the model parameters or its gradient is in general computationally intensive. Thus, training...
Indian Academy of Sciences (India)
Home; Journals; Resonance – Journal of Science Education; Volume 6; Issue 9. Entropy à la Boltzmann. Jayanta K Bhattacharjee. General Article Volume 6 Issue 9 September 2001 pp 19-34. Fulltext. Click here to view fulltext PDF. Permanent link: http://www.ias.ac.in/article/fulltext/reso/006/09/0019-0034. Author Affiliations.
Indian Academy of Sciences (India)
Home; Journals; Resonance – Journal of Science Education; Volume 6; Issue 9. Entropy à la Boltzmann. Jayanta K Bhattacharjee. General Article Volume 6 Issue 9 September 2001 pp 19-34. Fulltext. Click here to view fulltext PDF. Permanent link: https://www.ias.ac.in/article/fulltext/reso/006/09/0019-0034 ...
International Nuclear Information System (INIS)
Lim, Gyeong Hui
2008-03-01
This book consists of 15 chapters, which are basic conception and meaning of statistical thermodynamics, Maxwell-Boltzmann's statistics, ensemble, thermodynamics function and fluctuation, statistical dynamics with independent particle system, ideal molecular system, chemical equilibrium and chemical reaction rate in ideal gas mixture, classical statistical thermodynamics, ideal lattice model, lattice statistics and nonideal lattice model, imperfect gas theory on liquid, theory on solution, statistical thermodynamics of interface, statistical thermodynamics of a high molecule system and quantum statistics
Fischer, J.; Fellmuth, B.; Gaiser, C.; Zandt, T.; Pitre, L.; Sparasci, F.; Plimmer, M. D.; de Podesta, M.; Underwood, R.; Sutton, G.; Machin, G.; Gavioso, R. M.; Madonna Ripa, D.; Steur, P. P. M.; Qu, J.; Feng, X. J.; Zhang, J.; Moldover, M. R.; Benz, S. P.; White, D. R.; Gianfrani, L.; Castrillo, A.; Moretti, L.; Darquié, B.; Moufarej, E.; Daussy, C.; Briaudeau, S.; Kozlova, O.; Risegari, L.; Segovia, J. J.; Martín, M. C.; del Campo, D.
2018-04-01
The International Committee for Weights and Measures (CIPM), at its meeting in October 2017, followed the recommendation of the Consultative Committee for Units (CCU) on the redefinition of the kilogram, ampere, kelvin and mole. For the redefinition of the kelvin, the Boltzmann constant will be fixed with the numerical value 1.380 649 × 10-23 J K-1. The relative standard uncertainty to be transferred to the thermodynamic temperature value of the triple point of water will be 3.7 × 10-7, corresponding to an uncertainty in temperature of 0.10 mK, sufficiently low for all practical purposes. With the redefinition of the kelvin, the broad research activities of the temperature community on the determination of the Boltzmann constant have been very successfully completed. In the following, a review of the determinations of the Boltzmann constant k, important for the new definition of the kelvin and performed in the last decade, is given.
The fundamental and universal nature of Boltzmann`s constant
Energy Technology Data Exchange (ETDEWEB)
Biedenharn, L.C. [Univ. of Texas, Austin, TX (United States); Solem, J.C. [Los Alamos National Lab., NM (United States). Theoretical Div.
1996-07-01
The nature of Boltzmann`s constant is very unclear in the physics literature. In the first part of this paper, on general considerations, the authors examine this situation in detail and demonstrate the conclusion that Boltzmann`s constant is indeed both fundamental and universal. As a consequence of their development they find there is an important implication of this work for the problem of the entropy of information. In the second part they discuss, Szilard`s famous construction showing in detail how his result is incompatible with the demonstrations in both parts 1 and 2.
Particle methods for Boltzmann equation
International Nuclear Information System (INIS)
Hermeline, F.
1985-05-01
This work is aimed at showing how to discretize an equation such as Boltzmann equation in its most general form, by particle methods. Then method is applied to some equations of plasma physics which appear as peculiar cases of Boltzmann equation, such as Vlasov equation, Bhatnager-Gross-Krook equation, Fokker-Planck equation and neutron transport equation [fr
The intellectual quadrangle: Mach-Boltzmann-Planck-Einstein
International Nuclear Information System (INIS)
Broda, E.
1981-01-01
These four men were influential in the transition from classical to modern physics. They interacted as scientists, often antagonistically. Thus Boltzmann was the greatest champion of the atom, while Mach remained unconvinced all his life. As a aphysicist, Einstein was greatly influenced by both Mach and Boltzmann, although Mach in the end rejected relativity as well. Because of his work on statistical mechanics, fluctuations, and quantum theory, Einstein has been called the natural successor to Boltzmann. Planck also was influenced by Mach at first. Hence he and Boltzmann were adversaries antil Planck converted to atomistics in 1900 and used the statistical interpretation of entropy to establish his radiation law. Planck accepted relativity early, but in quantum theory he was for a long time partly opposed to Einstein, and vice versa - Einstein considered Planck's derivation of his radiation law as unsound, while Planck could not accept the light quantum. In the case of all four physicists, science was interwoven with philosophy. Boltzmann consistently fought Mach's positivism, while Planck and Einstein moved from positivism to realism. All were also, though in very different ways, actively interested in public affairs. (orig.)
Hayslett, H T
1991-01-01
Statistics covers the basic principles of Statistics. The book starts by tackling the importance and the two kinds of statistics; the presentation of sample data; the definition, illustration and explanation of several measures of location; and the measures of variation. The text then discusses elementary probability, the normal distribution and the normal approximation to the binomial. Testing of statistical hypotheses and tests of hypotheses about the theoretical proportion of successes in a binomial population and about the theoretical mean of a normal population are explained. The text the
Adaptive Non-Boltzmann Monte Carlo
International Nuclear Information System (INIS)
Fitzgerald, M.; Picard, R.R.; Silver, R.N.
1998-01-01
This manuscript generalizes the use of transition probabilities (TPs) between states, which are efficient relative to histogram procedures in deriving system properties. The empirical TPs of the simulation depend on the importance weights and are temperature-specific, so they are not conducive to accumulating statistics as weights change or to extrapolating in temperature. To address these issues, the authors provide a method for inferring Boltzmann-weighted TPs for one temperature from simulations run at other temperatures and/or at different adaptively varying importance weights. They refer to these as canonical transition probabilities (CTPs). System properties are estimated from CTPs. Statistics on CTPs are gathered by inserting a low-cost easily-implemented bookkeeping step into the Metropolis algorithm for non-Boltzmann sampling. The CTP method is inherently adaptive, can take advantage of partitioning of the state space into small regions using either serial or (embarrassingly) parallel architectures, and reduces variance by avoiding histogramming. They also demonstrate how system properties may be extrapolated in temperature from CTPs without the extra memory required by using energy as a microstate label. Nor does it require the solution of non-linear equations used in histogram methods
Entropic lattice Boltzmann representations required to recover Navier-Stokes flows.
Keating, Brian; Vahala, George; Yepez, Jeffrey; Soe, Min; Vahala, Linda
2007-03-01
There are two disparate formulations of the entropic lattice Boltzmann scheme: one of these theories revolves around the analog of the discrete Boltzmann H function of standard extensive statistical mechanics, while the other revolves around the nonextensive Tsallis entropy. It is shown here that it is the nonenforcement of the pressure tensor moment constraints that lead to extremizations of entropy resulting in Tsallis-like forms. However, with the imposition of the pressure tensor moment constraint, as is fundamentally necessary for the recovery of the Navier-Stokes equations, it is proved that the entropy function must be of the discrete Boltzmann form. Three-dimensional simulations are performed which illustrate some of the differences between standard lattice Boltzmann and entropic lattice Boltzmann schemes, as well as the role played by the number of phase-space velocities used in the discretization.
Links to sources of cancer-related statistics, including the Surveillance, Epidemiology and End Results (SEER) Program, SEER-Medicare datasets, cancer survivor prevalence data, and the Cancer Trends Progress Report.
International Nuclear Information System (INIS)
2005-01-01
For the years 2004 and 2005 the figures shown in the tables of Energy Review are partly preliminary. The annual statistics published in Energy Review are presented in more detail in a publication called Energy Statistics that comes out yearly. Energy Statistics also includes historical time-series over a longer period of time (see e.g. Energy Statistics, Statistics Finland, Helsinki 2004.) The applied energy units and conversion coefficients are shown in the back cover of the Review. Explanatory notes to the statistical tables can be found after tables and figures. The figures presents: Changes in GDP, energy consumption and electricity consumption, Carbon dioxide emissions from fossile fuels use, Coal consumption, Consumption of natural gas, Peat consumption, Domestic oil deliveries, Import prices of oil, Consumer prices of principal oil products, Fuel prices in heat production, Fuel prices in electricity production, Price of electricity by type of consumer, Average monthly spot prices at the Nord pool power exchange, Total energy consumption by source and CO 2 -emissions, Supplies and total consumption of electricity GWh, Energy imports by country of origin in January-June 2003, Energy exports by recipient country in January-June 2003, Consumer prices of liquid fuels, Consumer prices of hard coal, natural gas and indigenous fuels, Price of natural gas by type of consumer, Price of electricity by type of consumer, Price of district heating by type of consumer, Excise taxes, value added taxes and fiscal charges and fees included in consumer prices of some energy sources and Energy taxes, precautionary stock fees and oil pollution fees
Classifying images using restricted Boltzmann machines and convolutional neural networks
Zhao, Zhijun; Xu, Tongde; Dai, Chenyu
2017-07-01
To improve the feature recognition ability of deep model transfer learning, we propose a hybrid deep transfer learning method for image classification based on restricted Boltzmann machines (RBM) and convolutional neural networks (CNNs). It integrates learning abilities of two models, which conducts subject classification by exacting structural higher-order statistics features of images. While the method transfers the trained convolutional neural networks to the target datasets, fully-connected layers can be replaced by restricted Boltzmann machine layers; then the restricted Boltzmann machine layers and Softmax classifier are retrained, and BP neural network can be used to fine-tuned the hybrid model. The restricted Boltzmann machine layers has not only fully integrated the whole feature maps, but also learns the statistical features of target datasets in the view of the biggest logarithmic likelihood, thus removing the effects caused by the content differences between datasets. The experimental results show that the proposed method has improved the accuracy of image classification, outperforming other methods on Pascal VOC2007 and Caltech101 datasets.
International Nuclear Information System (INIS)
2000-01-01
For the year 1999 and 2000, part of the figures shown in the tables of the Energy Review are preliminary or estimated. The annual statistics of the Energy Review appear in more detail from the publication Energiatilastot - Energy Statistics issued annually, which also includes historical time series over a longer period (see e.g., Energiatilastot 1998, Statistics Finland, Helsinki 1999, ISSN 0785-3165). The inside of the Review's back cover shows the energy units and the conversion coefficients used for them. Explanatory notes to the statistical tables can be found after tables and figures. The figures presents: Changes in the volume of GNP and energy consumption, Changes in the volume of GNP and electricity, Coal consumption, Natural gas consumption, Peat consumption, Domestic oil deliveries, Import prices of oil, Consumer prices of principal oil products, Fuel prices for heat production, Fuel prices for electricity production, Carbon dioxide emissions, Total energy consumption by source and CO 2 -emissions, Electricity supply, Energy imports by country of origin in January-March 2000, Energy exports by recipient country in January-March 2000, Consumer prices of liquid fuels, Consumer prices of hard coal, natural gas and indigenous fuels, Average electricity price by type of consumer, Price of district heating by type of consumer, Excise taxes, value added taxes and fiscal charges and fees included in consumer prices of some energy sources and Energy taxes and precautionary stock fees on oil products
International Nuclear Information System (INIS)
1999-01-01
For the year 1998 and the year 1999, part of the figures shown in the tables of the Energy Review are preliminary or estimated. The annual statistics of the Energy Review appear in more detail from the publication Energiatilastot - Energy Statistics issued annually, which also includes historical time series over a longer period (see e.g. Energiatilastot 1998, Statistics Finland, Helsinki 1999, ISSN 0785-3165). The inside of the Review's back cover shows the energy units and the conversion coefficients used for them. Explanatory notes to the statistical tables can be found after tables and figures. The figures presents: Changes in the volume of GNP and energy consumption, Changes in the volume of GNP and electricity, Coal consumption, Natural gas consumption, Peat consumption, Domestic oil deliveries, Import prices of oil, Consumer prices of principal oil products, Fuel prices for heat production, Fuel prices for electricity production, Carbon dioxide emissions, Total energy consumption by source and CO 2 -emissions, Electricity supply, Energy imports by country of origin in January-June 1999, Energy exports by recipient country in January-June 1999, Consumer prices of liquid fuels, Consumer prices of hard coal, natural gas and indigenous fuels, Average electricity price by type of consumer, Price of district heating by type of consumer, Excise taxes, value added taxes and fiscal charges and fees included in consumer prices of some energy sources and Energy taxes and precautionary stock fees on oil products
International Nuclear Information System (INIS)
2001-01-01
For the year 2000, part of the figures shown in the tables of the Energy Review are preliminary or estimated. The annual statistics of the Energy Review appear in more detail from the publication Energiatilastot - Energy Statistics issued annually, which also includes historical time series over a longer period (see e.g. Energiatilastot 1999, Statistics Finland, Helsinki 2000, ISSN 0785-3165). The inside of the Review's back cover shows the energy units and the conversion coefficients used for them. Explanatory notes to the statistical tables can be found after tables and figures. The figures presents: Changes in the volume of GNP and energy consumption, Changes in the volume of GNP and electricity, Coal consumption, Natural gas consumption, Peat consumption, Domestic oil deliveries, Import prices of oil, Consumer prices of principal oil products, Fuel prices for heat production, Fuel prices for electricity production, Carbon dioxide emissions from the use of fossil fuels, Total energy consumption by source and CO 2 -emissions, Electricity supply, Energy imports by country of origin in 2000, Energy exports by recipient country in 2000, Consumer prices of liquid fuels, Consumer prices of hard coal, natural gas and indigenous fuels, Average electricity price by type of consumer, Price of district heating by type of consumer, Excise taxes, value added taxes and fiscal charges and fees included in consumer prices of some energy sources and Energy taxes and precautionary stock fees on oil products
International Nuclear Information System (INIS)
2003-01-01
For the year 2002, part of the figures shown in the tables of the Energy Review are partly preliminary. The annual statistics of the Energy Review also includes historical time-series over a longer period (see e.g. Energiatilastot 2001, Statistics Finland, Helsinki 2002). The applied energy units and conversion coefficients are shown in the inside back cover of the Review. Explanatory notes to the statistical tables can be found after tables and figures. The figures presents: Changes in GDP, energy consumption and electricity consumption, Carbon dioxide emissions from fossile fuels use, Coal consumption, Consumption of natural gas, Peat consumption, Domestic oil deliveries, Import prices of oil, Consumer prices of principal oil products, Fuel prices in heat production, Fuel prices in electricity production, Price of electricity by type of consumer, Average monthly spot prices at the Nord pool power exchange, Total energy consumption by source and CO 2 -emissions, Supply and total consumption of electricity GWh, Energy imports by country of origin in January-June 2003, Energy exports by recipient country in January-June 2003, Consumer prices of liquid fuels, Consumer prices of hard coal, natural gas and indigenous fuels, Price of natural gas by type of consumer, Price of electricity by type of consumer, Price of district heating by type of consumer, Excise taxes, value added taxes and fiscal charges and fees included in consumer prices of some energy sources and Excise taxes, precautionary stock fees on oil pollution fees on energy products
International Nuclear Information System (INIS)
2000-01-01
For the year 1999 and 2000, part of the figures shown in the tables of the Energy Review are preliminary or estimated. The annual statistics of the Energy also includes historical time series over a longer period (see e.g., Energiatilastot 1999, Statistics Finland, Helsinki 2000, ISSN 0785-3165). The inside of the Review's back cover shows the energy units and the conversion coefficients used for them. Explanatory notes to the statistical tables can be found after tables and figures. The figures presents: Changes in the volume of GNP and energy consumption, Changes in the volume of GNP and electricity, Coal consumption, Natural gas consumption, Peat consumption, Domestic oil deliveries, Import prices of oil, Consumer prices of principal oil products, Fuel prices for heat production, Fuel prices for electricity production, Carbon dioxide emissions, Total energy consumption by source and CO 2 -emissions, Electricity supply, Energy imports by country of origin in January-June 2000, Energy exports by recipient country in January-June 2000, Consumer prices of liquid fuels, Consumer prices of hard coal, natural gas and indigenous fuels, Average electricity price by type of consumer, Price of district heating by type of consumer, Excise taxes, value added taxes and fiscal charges and fees included in consumer prices of some energy sources and Energy taxes and precautionary stock fees on oil products
International Nuclear Information System (INIS)
2004-01-01
For the year 2003 and 2004, the figures shown in the tables of the Energy Review are partly preliminary. The annual statistics of the Energy Review also includes historical time-series over a longer period (see e.g. Energiatilastot, Statistics Finland, Helsinki 2003, ISSN 0785-3165). The applied energy units and conversion coefficients are shown in the inside back cover of the Review. Explanatory notes to the statistical tables can be found after tables and figures. The figures presents: Changes in GDP, energy consumption and electricity consumption, Carbon dioxide emissions from fossile fuels use, Coal consumption, Consumption of natural gas, Peat consumption, Domestic oil deliveries, Import prices of oil, Consumer prices of principal oil products, Fuel prices in heat production, Fuel prices in electricity production, Price of electricity by type of consumer, Average monthly spot prices at the Nord pool power exchange, Total energy consumption by source and CO 2 -emissions, Supplies and total consumption of electricity GWh, Energy imports by country of origin in January-March 2004, Energy exports by recipient country in January-March 2004, Consumer prices of liquid fuels, Consumer prices of hard coal, natural gas and indigenous fuels, Price of natural gas by type of consumer, Price of electricity by type of consumer, Price of district heating by type of consumer, Excise taxes, value added taxes and fiscal charges and fees included in consumer prices of some energy sources and Excise taxes, precautionary stock fees on oil pollution fees
Lattices for the lattice Boltzmann method.
Chikatamarla, Shyam S; Karlin, Iliya V
2009-04-01
A recently introduced theory of higher-order lattice Boltzmann models [Chikatamarla and Karlin, Phys. Rev. Lett. 97, 190601 (2006)] is elaborated in detail. A general theory of the construction of lattice Boltzmann models as an approximation to the Boltzmann equation is presented. New lattices are found in all three dimensions and are classified according to their accuracy (degree of approximation of the Boltzmann equation). The numerical stability of these lattices is argued based on the entropy principle. The efficiency and accuracy of many new lattices are demonstrated via simulations in all three dimensions.
Return of the Boltzmann brains
Page, Don N.
2008-09-01
Linde in J. Cosmol. Astropart. Phys.1475-7516 01 (2007) 02210.1088/1475-7516/2007/01/022 shows that some (though not all) versions of the global (volume-weighted) description avoid the “Boltzmann brain” problem raised by Page [Phys. Rev. D 78, 063535 (2008)] if the universe does not have a decay time less than 20 Gyr. Here I give an apparently natural version of the volume-weighted description in which the problem persists, highlighting the ambiguity of taking the ratios of infinite volumes that appear to arise from eternal inflation.
On the Boltzmann-Grad Limit for Smooth Hard-Sphere Systems
Tessarotto, Massimo; Cremaschini, Claudio; Mond, Michael; Asci, Claudio; Soranzo, Alessandro; Tironi, Gino
2018-03-01
The problem is posed of the prescription of the so-called Boltzmann-Grad limit operator (L_{BG}) for the N-body system of smooth hard-spheres which undergo unary, binary as well as multiple elastic instantaneous collisions. It is proved, that, despite the non-commutative property of the operator L_{BG}, the Boltzmann equation can nevertheless be uniquely determined. In particular, consistent with the claim of Uffink and Valente (Found Phys 45:404, 2015) that there is "no time-asymmetric ingredient" in its derivation, the Boltzmann equation is shown to be time-reversal symmetric. The proof is couched on the "ab initio" axiomatic approach to the classical statistical mechanics recently developed (Tessarotto et al. in Eur Phys J Plus 128:32, 2013). Implications relevant for the physical interpretation of the Boltzmann H-theorem and the phenomenon of decay to kinetic equilibrium are pointed out.
Lattice Boltzmann scheme for relativistic fluids
Mendoza, M.; Boghosian, B.; Herrmann, H. J.; Succi, S.
2009-01-01
A Lattice Boltzmann formulation for relativistic fluids is presented and numerically verified through quantitative comparison with recent hydrodynamic simulations of relativistic shock-wave propagation in viscous quark-gluon plasmas. This formulation opens up the possibility of exporting the main advantages of Lattice Boltzmann methods to the relativistic context, which seems particularly useful for the simulation of relativistic fluids in complicated geometries.
Pruning Boltzmann networks and hidden Markov models
DEFF Research Database (Denmark)
Pedersen, Morten With; Stork, D.
1996-01-01
Boltzmann chains and hidden Markov models (HMMs), we argue that our method can be applied to HMMs as well. We illustrate pruning on Boltzmann zippers, which are equivalent to two HMMs with cross-connection links. We verify that our second-order approximation preserves the rank ordering of weight saliencies...
Global existence proof for relativistic Boltzmann equation
International Nuclear Information System (INIS)
Dudynski, M.; Ekiel-Jezewska, M.L.
1992-01-01
The existence and causality of solutions to the relativistic Boltzmann equation in L 1 and in L loc 1 are proved. The solutions are shown to satisfy physically natural a priori bounds, time-independent in L 1 . The results rely upon new techniques developed for the nonrelativistic Boltzmann equation by DiPerna and Lions
Boltzmann, Einstein, Natural Law and Evolution
International Nuclear Information System (INIS)
Broda, E.
1980-01-01
Like Boltzmann, Einstein was a protagonist of atomistics. As a physicist, he has been called Boltzmann's true successor. Also in epistemology, after overcoming the positivist influence of Mach, Einstein approached Boltzmann. Any difference between Boltzmann's realism, or even materialism, and Einstein's pantheism may be merely a matter of emphasis. Yet a real difference exists in another respect. Boltzmann explained man's power of thinking and feeling, his morality and his esthetic sense, on an evolutionary, Darwinian, basis. In contrast, evolution had no role in Einstein's thought, though Darwin was accepted by him. This lack of appreciation of the importance of evolution is now attributed to socio-political factors. (author)
Ludwig Boltzmann - The Man and His Work
International Nuclear Information System (INIS)
Broda, E.
1982-01-01
It is argued that Ludwig Boltzmann was, along with Newton and Maxwell, one of the three greatest theoretical physicists of classical times. It is less generally known that he was also a powerful realist-materialist philosopher and a keen opponent of Ernst Mach's positivism and of the philosophical idealism of Berkeley, Hegel and Schopenhauer. Boltzmann was also opposed to Kant. Moreover, he had a lively interest in biology and especially in Darwinian evolution, and he should be taken as one of the founders of biophysics. Boltzmann discussed the origin of life and of the mind. Finally, he also was a most vigorous, colourful and attractive person. (author)
Comparing Boltzmann and Gibbs definitions of entropy in small systems
Ferrari, Loris
2017-11-01
The long-standing contrast between Boltzmann's and Gibbs' approach to statistical thermodynamics has been recently rekindled by Dunkel and Hilbert, who criticize the notion of negative absolute temperature (NAT) as a misleading consequence of Boltzmann's definition of entropy. A different definition, due to Gibbs, has been proposed, which forbids NAT and makes the energy equipartition rigorous in arbitrarily sized systems. The two approaches, however, are shown to converge to the same results in the thermodynamical limit. A vigorous debate followed Dunkel and Hilbert's work, with arguments against and in favor of Gibbs' entropy. In an attempt to leave the speculative level and give the discussion some deal of concreteness, we analyze the practical consequences of Gibbs' definition in two finite-size systems: a non-interacting gas of N atoms with two-level internal spectrum, and an Ising model of N interacting spins. It is shown that, for certain measurable quantities, the difference resulting from Boltzmann's and Gibbs' approach vanishes as N -1/2 , much less rapidly than the 1/ N slope expected. As shown by numerical estimates, this makes the experimental solution of the controversy a feasible task.
Entropic multirelaxation lattice Boltzmann models for turbulent flows.
Bösch, Fabian; Chikatamarla, Shyam S; Karlin, Ilya V
2015-10-01
We present three-dimensional realizations of a class of lattice Boltzmann models introduced recently by the authors [I. V. Karlin, F. Bösch, and S. S. Chikatamarla, Phys. Rev. E 90, 031302(R) (2014)] and review the role of the entropic stabilizer. Both coarse- and fine-grid simulations are addressed for the Kida vortex flow benchmark. We show that the outstanding numerical stability and performance is independent of a particular choice of the moment representation for high-Reynolds-number flows. We report accurate results for low-order moments for homogeneous isotropic decaying turbulence and second-order grid convergence for most assessed statistical quantities. It is demonstrated that all the three-dimensional lattice Boltzmann realizations considered herein converge to the familiar lattice Bhatnagar-Gross-Krook model when the resolution is increased. Moreover, thanks to the dynamic nature of the entropic stabilizer, the present model features less compressibility effects and maintains correct energy and enstrophy dissipation. The explicit and efficient nature of the present lattice Boltzmann method renders it a promising candidate for both engineering and scientific purposes for highly turbulent flows.
The Statistical Interpretation of Entropy: An Activity
Timmberlake, Todd
2010-01-01
The second law of thermodynamics, which states that the entropy of an isolated macroscopic system can increase but will not decrease, is a cornerstone of modern physics. Ludwig Boltzmann argued that the second law arises from the motion of the atoms that compose the system. Boltzmann's statistical mechanics provides deep insight into the…
Income distribution: Boltzmann analysis and its extension
Yuqing, He
2007-04-01
The paper aims at describing income distribution in moderate income regions. Starting with dividing income behaviors into the two parts: random and deterministic, and by introducing “instantaneous model” for theoretical derivations and “cumulative model” for positive tests, this paper applies the equilibrium approach of statistical mechanics in the study of nonconserved individual income course. The random income follows a stationary distribution similar to the Maxwell-Boltzmann distribution in the instantaneous model. Combining this result with marginal analysis, the probability distribution of individual income process that is composed of the random and deterministic income courses approximately obeys a distribution law mixing exponential function with a logarithmic prefactor. Using the census or income survey data of USA, UK, Japan, and New Zealand, the distribution law has been tested. The results show that it agrees very well with most of the empirical data. The discussion suggests that there might be essentially different income processes to happen in moderate and high income regions.
An introduction to the theory of the Boltzmann equation
Harris, Stewart
2011-01-01
Boltzmann's equation (or Boltzmann-like equations) appears extensively in such disparate fields as laser scattering, solid-state physics, nuclear transport, and beyond the conventional boundaries of physics and engineering, in the fields of cellular proliferation and automobile traffic flow. This introductory graduate-level course for students of physics and engineering offers detailed presentations of the basic modern theory of Boltzmann's equation, including representative applications using both Boltzmann's equation and the model Boltzmann equations developed within the text. It emphasizes
Relativistic Boltzmann theory for a plasma
International Nuclear Information System (INIS)
Erkelens, H. van.
1984-01-01
This thesis gives a self-contained treatment of the relativistic Boltzmann theory for a plasma. Here plasma means any mixture containing electrically charged particles. The relativistic Boltzmann equation is linearized for the case of a plasma. The Chapman-Enskog method is elaborated further for transport phenomena. Linear laws for viscous phenomena are derived. Then the collision term in the Boltzmann theory is dealt with. Using the transport equation, a kinetic theory of wave phenomena is developed and the dissipation of hydromagnetic waves in a relativistic plasma is investigated. In the final chapter, it is demonstrated how the relativistic Boltzmann theory can be applied in cosmology. In doing so, expressions are derived for the electric conductivity of the cosmological plasma in the lepton era, the plasma era and the annihilation era. (Auth.)
Lattice Boltzmann approach for complex nonequilibrium flows.
Montessori, A; Prestininzi, P; La Rocca, M; Succi, S
2015-10-01
We present a lattice Boltzmann realization of Grad's extended hydrodynamic approach to nonequilibrium flows. This is achieved by using higher-order isotropic lattices coupled with a higher-order regularization procedure. The method is assessed for flow across parallel plates and three-dimensional flows in porous media, showing excellent agreement of the mass flow with analytical and numerical solutions of the Boltzmann equation across the full range of Knudsen numbers, from the hydrodynamic regime to ballistic motion.
Essentially Entropic Lattice Boltzmann Model
Atif, Mohammad; Kolluru, Praveen Kumar; Thantanapally, Chakradhar; Ansumali, Santosh
2017-12-01
The entropic lattice Boltzmann model (ELBM), a discrete space-time kinetic theory for hydrodynamics, ensures nonlinear stability via the discrete time version of the second law of thermodynamics (the H theorem). Compliance with the H theorem is numerically enforced in this methodology and involves a search for the maximal discrete path length corresponding to the zero dissipation state by iteratively solving a nonlinear equation. We demonstrate that an exact solution for the path length can be obtained by assuming a natural criterion of negative entropy change, thereby reducing the problem to solving an inequality. This inequality is solved by creating a new framework for construction of Padé approximants via quadrature on appropriate convex function. This exact solution also resolves the issue of indeterminacy in case of nonexistence of the entropic involution step. Since our formulation is devoid of complex mathematical library functions, the computational cost is drastically reduced. To illustrate this, we have simulated a model setup of flow over the NACA-0012 airfoil at a Reynolds number of 2.88 ×106.
Operational derivation of Boltzmann distribution with Maxwell's demon model.
Hosoya, Akio; Maruyama, Koji; Shikano, Yutaka
2015-11-24
The resolution of the Maxwell's demon paradox linked thermodynamics with information theory through information erasure principle. By considering a demon endowed with a Turing-machine consisting of a memory tape and a processor, we attempt to explore the link towards the foundations of statistical mechanics and to derive results therein in an operational manner. Here, we present a derivation of the Boltzmann distribution in equilibrium as an example, without hypothesizing the principle of maximum entropy. Further, since the model can be applied to non-equilibrium processes, in principle, we demonstrate the dissipation-fluctuation relation to show the possibility in this direction.
Atoms, mechanics, and probability Ludwig Boltzmann's statistico-mechanical writings : an exegesis
Darrigol, Olivier
2018-01-01
One of the pillars of modern science, statistical mechanics, owes much to one man, the Austrian physicist Ludwig Boltzmann (1844-1906). As a result of his unusual working and writing styles, his enormous contribution remains little read and poorly understood. The purpose of this book is to make the Boltzmann corpus more accessible to physicists, philosophers, and historians, and so give it new life. The means are introductory biographical and historical materials, detailed and lucid summaries of every relevant publication, and a final chapter of critical synthesis. Special attention is given to Boltzmann's theoretical tool-box and to his patient construction of lofty formal systems even before their full conceptual import could be known. This constructive tendency largely accounts for his lengthy style, for the abundance of new constructions, for the relative vagueness of their object--and for the puzzlement of commentators. This book will help the reader cross the stylistic barrier and see how ingeniously B...
Boltzmann-Electron Model in Aleph.
Energy Technology Data Exchange (ETDEWEB)
Hughes, Thomas Patrick; Hooper, Russell
2014-11-01
We apply the Boltzmann-electron model in the electrostatic, particle-in-cell, finite- element code Aleph to a plasma sheath. By assuming a Boltzmann energy distribution for the electrons, the model eliminates the need to resolve the electron plasma fre- quency, and avoids the numerical "grid instability" that can cause unphysical heating of electrons. This allows much larger timesteps to be used than with kinetic electrons. Ions are treated with the standard PIC algorithm. The Boltzmann-electron model re- quires solution of a nonlinear Poisson equation, for which we use an iterative Newton solver (NOX) from the Trilinos Project. Results for the spatial variation of density and voltage in the plasma sheath agree well with an analytic model
The Acoustic Limit for the Boltzmann Equation
Bardos, Claude; Golse, François; Levermore, C. David
The acoustic equations are the linearization of the compressible Euler equations about a spatially homogeneous fluid state. We first derive them directly from the Boltzmann equation as the formal limit of moment equations for an appropriately scaled family of Boltzmann solutions. We then establish this limit for the Boltzmann equation considered over a periodic spatial domain for bounded collision kernels. Appropriately scaled families of DiPerna-Lions renormalized solutions are shown to have fluctuations that converge entropically (and hence strongly in L1) to a unique limit governed by a solution of the acoustic equations for all time, provided that its initial fluctuations converge entropically to an appropriate limit associated to any given L2 initial data of the acoustic equations. The associated local conservation laws are recovered in the limit.
Fast lattice Boltzmann solver for relativistic hydrodynamics.
Mendoza, M; Boghosian, B M; Herrmann, H J; Succi, S
2010-07-02
A lattice Boltzmann formulation for relativistic fluids is presented and numerically validated through quantitative comparison with recent hydrodynamic simulations of relativistic fluids. In order to illustrate its capability to handle complex geometries, the scheme is also applied to the case of a three-dimensional relativistic shock wave, generated by a supernova explosion, impacting on a massive interstellar cloud. This formulation opens up the possibility of exporting the proven advantages of lattice Boltzmann methods, namely, computational efficiency and easy handling of complex geometries, to the context of (mildly) relativistic fluid dynamics at large, from quark-gluon plasmas up to supernovae with relativistic outflows.
Kinetic Boltzmann, Vlasov and Related Equations
Sinitsyn, Alexander; Vedenyapin, Victor
2011-01-01
Boltzmann and Vlasov equations played a great role in the past and still play an important role in modern natural sciences, technique and even philosophy of science. Classical Boltzmann equation derived in 1872 became a cornerstone for the molecular-kinetic theory, the second law of thermodynamics (increasing entropy) and derivation of the basic hydrodynamic equations. After modifications, the fields and numbers of its applications have increased to include diluted gas, radiation, neutral particles transportation, atmosphere optics and nuclear reactor modelling. Vlasov equation was obtained in
Celebrating Cercignani's conjecture for the Boltzmann equation
Villani, Cédric
2011-01-01
Cercignani\\'s conjecture assumes a linear inequality between the entropy and entropy production functionals for Boltzmann\\'s nonlinear integral operator in rarefied gas dynamics. Related to the field of logarithmic Sobolev inequalities and spectral gap inequalities, this issue has been at the core of the renewal of the mathematical theory of convergence to thermodynamical equilibrium for rarefied gases over the past decade. In this review paper, we survey the various positive and negative results which were obtained since the conjecture was proposed in the 1980s. © American Institute of Mathematical Sciences.
Boltzmann and Einstein: Statistics and dynamics – An unsolved ...
Indian Academy of Sciences (India)
mechanical meaning of the second law of the theory of heat [1] in which he claimed to prove the second law of thermodynamics purely from dynamics, i.e., the second law was just a consequence of Newton's equations of motion. At that time he was not even aware of the necessity of introducing probability concepts in order ...
Multispeed models in off-lattice Boltzmann simulations
Bardow, A.; Karlin, I.V.; Gusev, A.A.
2008-01-01
The lattice Boltzmann method is a highly promising approach to the simulation of complex flows. Here, we realize recently proposed multispeed lattice Boltzmann models [S. Chikatamarla et al., Phys. Rev. Lett. 97 190601 (2006)] by exploiting the flexibility offered by off-lattice Boltzmann methods.
Singularities in the nonisotropic Boltzmann equation
International Nuclear Information System (INIS)
Garibotti, C.R.; Martiarena, M.L.; Zanette, D.
1987-09-01
We consider solutions of the nonlinear Boltzmann equation (NLBE) with anisotropic singular initial conditions, which give a simplified model for the penetration of a monochromatic beam on a rarified target. The NLBE is transformed into an integral equation which is solved iteratively and the evolution of the initial singularities is discussed. (author). 5 refs
Quantum Heat Engine and Negative Boltzmann Temperature
Xi, Jing-Yi; Quan, Hai-Tao
2017-09-01
To clarify the ambiguity on negative Boltzmann temperature in literature, we study the Carnot and the Otto cycle with one of the heat reservoirs at the negative Boltzmann temperature based on a canonical ensemble description. The work extraction, entropy production and the efficiency of these cycles are explored. Conditions for constructing and properties of these thermodynamic cycles are elucidated. We find that the apparent “violation” of the second law of thermodynamics in these cycles are due to the fact that the traditional definition of thermodynamic efficiency is inappropriate in this situation. When properly understanding the efficiency and the adiabatic processes, in which the system crosses over “absolute ZERO” in a limit sense, the Carnot cycle with one of the heat reservoirs at a negative Boltzmann temperature can be understood straightforwardly, and it contradicts neither the second nor the third law of thermodynamics. Hence, negative Boltzmann temperature is a consistent concept in thermodynamics. We use a two-level system and an Ising spin system to illustrate our central results. Support from the National Science Foundation of China under Grants Nos. 11375012, 11534002, and The Recruitment Program of Global Youth Experts of China
Quantum Heat Engine and Negative Boltzmann Temperature
International Nuclear Information System (INIS)
Xi Jing-Yi; Quan Hai-Tao
2017-01-01
To clarify the ambiguity on negative Boltzmann temperature in literature, we study the Carnot and the Otto cycle with one of the heat reservoirs at the negative Boltzmann temperature based on a canonical ensemble description. The work extraction, entropy production and the efficiency of these cycles are explored. Conditions for constructing and properties of these thermodynamic cycles are elucidated. We find that the apparent “violation” of the second law of thermodynamics in these cycles are due to the fact that the traditional definition of thermodynamic efficiency is inappropriate in this situation. When properly understanding the efficiency and the adiabatic processes, in which the system crosses over “absolute ZERO” in a limit sense, the Carnot cycle with one of the heat reservoirs at a negative Boltzmann temperature can be understood straightforwardly, and it contradicts neither the second nor the third law of thermodynamics. Hence, negative Boltzmann temperature is a consistent concept in thermodynamics. We use a two-level system and an Ising spin system to illustrate our central results. (paper)
A Study of the Boltzmann Sequence-Structure Channel.
Magner, Abram; Kihara, Daisuke; Szpankowski, Wojciech
2017-02-01
We rigorously study a channel that maps sequences from a finite alphabet to self-avoiding walks in the two-dimensional grid, inspired by a model of protein folding from statistical physics and studied empirically by biophysicists. This channel, which we call the Boltzmann sequence-structure channel, is characterized by a Boltzmann/Gibbs distribution with a free parameter corresponding to temperature. In our previous work, we verified empirically that the channel capacity appears to have a phase transition for small temperature and decays to zero for high temperature. In this paper, we make some progress toward theoretically explaining these phenomena. We first estimate the conditional entropy between the input sequence and the output fold, giving an upper bound which exhibits a phase transition with respect to temperature. Next, we formulate a class of parameter settings under which the dependence between walk energies is governed by their number of shared contacts. In this setting, we derive a lower bound on the conditional entropy. This lower bound allows us to conclude that the mutual information tends to zero in a nontrivial regime of high temperature, giving some support to the empirical fact regarding capacity. Finally, we construct an example setting of the parameters of the model for which the conditional entropy is exactly calculable and which does not exhibit a phase transition.
Beyond Gibbs-Boltzmann-Shannon: General Entropies -- The Gibbs-Lorentzian Example
Directory of Open Access Journals (Sweden)
Rudolf A. Treumann
2014-08-01
Full Text Available We propose a generalisation of Gibbs' statistical mechanics into the domain of non-negligible phase space correlations. Derived are the probability distribution and entropy as a generalised ensemble average, replacing Gibbs-Boltzmann-Shannon's entropy definition enabling construction of new forms of statistical mechanics. The general entropy may also be of importance in information theory and data analysis. Application to generalised Lorentzian phase space elements yields the Gibbs-Lorentzian power law probability distribution and statistical mechanics. The corresponding Boltzmann, Fermi and Bose-Einstein distributions are found. They apply only to finite temperature states including correlations. As a by-product any negative absolute temperatures are categorically excluded, supporting a recent ``no-negative $T$ claim.
Lattice-Boltzmann simulations of droplet evaporation
Ledesma-Aguilar, Rodrigo
2014-09-04
© the Partner Organisations 2014. We study the utility and validity of lattice-Boltzmann (LB) simulations to explore droplet evaporation driven by a concentration gradient. Using a binary-fluid lattice-Boltzmann algorithm based on Cahn-Hilliard dynamics, we study the evaporation of planar films and 3D sessile droplets from smooth solid surfaces. Our results show that LB simulations accurately reproduce the classical regime of quasi-static dynamics. Beyond this limit, we show that the algorithm can be used to explore regimes where the evaporative and diffusive timescales are not widely separated, and to include the effect of boundaries of prescribed driving concentration. We illustrate the method by considering the evaporation of a droplet from a solid surface that is chemically patterned with hydrophilic and hydrophobic stripes. This journal is
Nonequilibrium thermodynamics of restricted Boltzmann machines
Salazar, Domingos S. P.
2017-08-01
In this work, we analyze the nonequilibrium thermodynamics of a class of neural networks known as restricted Boltzmann machines (RBMs) in the context of unsupervised learning. We show how the network is described as a discrete Markov process and how the detailed balance condition and the Maxwell-Boltzmann equilibrium distribution are sufficient conditions for a complete thermodynamics description, including nonequilibrium fluctuation theorems. Numerical simulations in a fully trained RBM are performed and the heat exchange fluctuation theorem is verified with excellent agreement to the theory. We observe how the contrastive divergence functional, mostly used in unsupervised learning of RBMs, is closely related to nonequilibrium thermodynamic quantities. We also use the framework to interpret the estimation of the partition function of RBMs with the annealed importance sampling method from a thermodynamics standpoint. Finally, we argue that unsupervised learning of RBMs is equivalent to a work protocol in a system driven by the laws of thermodynamics in the absence of labeled data.
Velocity-Field Theory, Boltzmann's Transport Equation and Geometry
Ichinose, Shoichi
Boltzmann equation describes the time development of the velocity distribution in the continuum fluid matter. We formulate the equation using the field theory where the velocity-field plays the central role. The matter (constituent particles) fields appear as the density and the viscosity. Fluctuation is examined, and is clearly discriminated from the quantum effect. The time variable is emergently introduced through the computational process step. The collision term, for the (velocity)**4 potential (4-body interaction), is explicitly obtained and the (statistical) fluctuation is closely explained. The present field theory model does not conserve energy and is an open-system model. (One dimensional) Navier-Stokes equation or Burger's equation, appears. In the latter part, we present a way to directly define the distribution function by use of the geometry, appearing in the mechanical dynamics, and Feynman's path-integral.
Boltzmann, Darwin and Directionality theory
Demetrius, Lloyd A.
2013-09-01
Boltzmann’s statistical thermodynamics is a mathematical theory which relates the macroscopic properties of aggregates of interacting molecules with the laws of their interaction. The theory is based on the concept thermodynamic entropy, a statistical measure of the extent to which energy is spread throughout macroscopic matter. Macroscopic evolution of material aggregates is quantitatively explained in terms of the principle: Thermodynamic entropy increases as the composition of the aggregate changes under molecular collision. Darwin’s theory of evolution is a qualitative theory of the origin of species and the adaptation of populations to their environment. A central concept in the theory is fitness, a qualitative measure of the capacity of an organism to contribute to the ancestry of future generations. Macroscopic evolution of populations of living organisms can be qualitatively explained in terms of a neo-Darwinian principle: Fitness increases as the composition of the population changes under variation and natural selection. Directionality theory is a quantitative model of the Darwinian argument of evolution by variation and selection. This mathematical theory is based on the concept evolutionary entropy, a statistical measure which describes the rate at which an organism appropriates energy from the environment and reinvests this energy into survivorship and reproduction. According to directionality theory, microevolutionary dynamics, that is evolution by mutation and natural selection, can be quantitatively explained in terms of a directionality principle: Evolutionary entropy increases when the resources are diverse and of constant abundance; but decreases when the resource is singular and of variable abundance. This report reviews the analytical and empirical support for directionality theory, and invokes the microevolutionary dynamics of variation and selection to delineate the principles which govern macroevolutionary dynamics of speciation and
Energy Dependent Streaming in Lattice Boltzmann Simulations
Czech Academy of Sciences Publication Activity Database
Pavlo, Pavol; Vahala, G.; Vahala, L.
2001-01-01
Roč. 46, č. 8 (2001), s. 241 ISSN 0003-0503. [Annual Meeting of the Division of Plasma Physics of the American Physical Society/43rd./. Long Beach, CA, 29.10.2001-02.11.2001] R&D Projects: GA ČR GA202/00/1216 Institutional research plan: CEZ:AV0Z2043910 Keywords : Lattice Boltzmann Simulations Subject RIV: BL - Plasma and Gas Discharge Physics
Lattice Boltzmann Approach to Resistive MHD
Czech Academy of Sciences Publication Activity Database
Macnab, A.; Vahala, G.; Vahala, L.; Pavlo, Pavol; Soe, M.
2002-01-01
Roč. 47, č. 9 (2002), s. 51 ISSN 0003-0503. [Annual Meeting of the Division of Plasma Physics of the American Physical Society/44th./. Orlando , Florida, 11.11.2001-15.11.2001] R&D Projects: GA ČR GA202/00/1216 Institutional research plan: CEZ:AV0Z2043910 Keywords : Lattice Boltzmann, magnetic fields Subject RIV: BL - Plasma and Gas Discharge Physics
The Boltzmann equation in the difference formulation
Energy Technology Data Exchange (ETDEWEB)
Szoke, Abraham [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Brooks III, Eugene D. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
2015-05-06
First we recall the assumptions that are needed for the validity of the Boltzmann equation and for the validity of the compressible Euler equations. We then present the difference formulation of these equations and make a connection with the time-honored Chapman - Enskog expansion. We discuss the hydrodynamic limit and calculate the thermal conductivity of a monatomic gas, using a simplified approximation for the collision term. Our formulation is more consistent and simpler than the traditional derivation.
Contact Angle Measurement in Lattice Boltzmann Method
Wen, Binghai; Huang, Bingfang; Qin, Zhangrong; Wang, Chunlei; Zhang, Chaoying
2017-01-01
Contact angle is an essential characteristic in wetting, capillarity and moving contact line; however, although contact angle phenomena are effectively simulated, an accurate and real-time measurement for contact angle has not been well studied in computational fluid dynamics, especially in dynamic environments. Here, we design a geometry-based mesoscopic scheme to onthesport measure the contact angle in the lattice Boltzmann method. The computational results without gravity effect are in exc...
Learning Algorithm of Boltzmann Machine Based on Spatial Monte Carlo Integration Method
Directory of Open Access Journals (Sweden)
Muneki Yasuda
2018-04-01
Full Text Available The machine learning techniques for Markov random fields are fundamental in various fields involving pattern recognition, image processing, sparse modeling, and earth science, and a Boltzmann machine is one of the most important models in Markov random fields. However, the inference and learning problems in the Boltzmann machine are NP-hard. The investigation of an effective learning algorithm for the Boltzmann machine is one of the most important challenges in the field of statistical machine learning. In this paper, we study Boltzmann machine learning based on the (first-order spatial Monte Carlo integration method, referred to as the 1-SMCI learning method, which was proposed in the author’s previous paper. In the first part of this paper, we compare the method with the maximum pseudo-likelihood estimation (MPLE method using a theoretical and a numerical approaches, and show the 1-SMCI learning method is more effective than the MPLE. In the latter part, we compare the 1-SMCI learning method with other effective methods, ratio matching and minimum probability flow, using a numerical experiment, and show the 1-SMCI learning method outperforms them.
Boltzmann, Darwin and Directionality theory
International Nuclear Information System (INIS)
Demetrius, Lloyd A.
2013-01-01
Boltzmann’s statistical thermodynamics is a mathematical theory which relates the macroscopic properties of aggregates of interacting molecules with the laws of their interaction. The theory is based on the concept thermodynamic entropy, a statistical measure of the extent to which energy is spread throughout macroscopic matter. Macroscopic evolution of material aggregates is quantitatively explained in terms of the principle: Thermodynamic entropy increases as the composition of the aggregate changes under molecular collision. Darwin’s theory of evolution is a qualitative theory of the origin of species and the adaptation of populations to their environment. A central concept in the theory is fitness, a qualitative measure of the capacity of an organism to contribute to the ancestry of future generations. Macroscopic evolution of populations of living organisms can be qualitatively explained in terms of a neo-Darwinian principle: Fitness increases as the composition of the population changes under variation and natural selection. Directionality theory is a quantitative model of the Darwinian argument of evolution by variation and selection. This mathematical theory is based on the concept evolutionary entropy, a statistical measure which describes the rate at which an organism appropriates energy from the environment and reinvests this energy into survivorship and reproduction. According to directionality theory, microevolutionary dynamics, that is evolution by mutation and natural selection, can be quantitatively explained in terms of a directionality principle: Evolutionary entropy increases when the resources are diverse and of constant abundance; but decreases when the resource is singular and of variable abundance. This report reviews the analytical and empirical support for directionality theory, and invokes the microevolutionary dynamics of variation and selection to delineate the principles which govern macroevolutionary dynamics of speciation and
Boltzmann, Darwin and Directionality theory
Energy Technology Data Exchange (ETDEWEB)
Demetrius, Lloyd A., E-mail: ldemetr@oeb.harvard.edu
2013-09-01
Boltzmann’s statistical thermodynamics is a mathematical theory which relates the macroscopic properties of aggregates of interacting molecules with the laws of their interaction. The theory is based on the concept thermodynamic entropy, a statistical measure of the extent to which energy is spread throughout macroscopic matter. Macroscopic evolution of material aggregates is quantitatively explained in terms of the principle: Thermodynamic entropy increases as the composition of the aggregate changes under molecular collision. Darwin’s theory of evolution is a qualitative theory of the origin of species and the adaptation of populations to their environment. A central concept in the theory is fitness, a qualitative measure of the capacity of an organism to contribute to the ancestry of future generations. Macroscopic evolution of populations of living organisms can be qualitatively explained in terms of a neo-Darwinian principle: Fitness increases as the composition of the population changes under variation and natural selection. Directionality theory is a quantitative model of the Darwinian argument of evolution by variation and selection. This mathematical theory is based on the concept evolutionary entropy, a statistical measure which describes the rate at which an organism appropriates energy from the environment and reinvests this energy into survivorship and reproduction. According to directionality theory, microevolutionary dynamics, that is evolution by mutation and natural selection, can be quantitatively explained in terms of a directionality principle: Evolutionary entropy increases when the resources are diverse and of constant abundance; but decreases when the resource is singular and of variable abundance. This report reviews the analytical and empirical support for directionality theory, and invokes the microevolutionary dynamics of variation and selection to delineate the principles which govern macroevolutionary dynamics of speciation and
Statistical physics; Physique statistique
Energy Technology Data Exchange (ETDEWEB)
Couture, L.; Zitoun, R. [Universite Pierre et Marie Curie, 75 - Paris (France)
1992-12-31
The basis of statistical physics is exposed. The statistical models of Maxwell-Boltzmann, of Bose-Einstein and of Fermi-Dirac and their particular application fields are presented. The statistical theory is applied in different ranges of physics: gas characteristics, paramagnetism, crystal thermal properties and solid electronic properties. A whole chapter is dedicated to helium and its characteristics such as superfluidity, another deals with superconductivity. Superconductivity is presented both experimentally and theoretically. Meissner effect and Josephson effect are described and the framework of BCS theory is drawn. (A.C.)
Sapteka, A. A. N. G.; Narottama, A. A. N. M.; Winarta, A.; Amerta Yasa, K.; Priambodo, P. S.; Putra, N.
2018-01-01
Solar energy utilized with solar panel is a renewable energy that needs to be studied further. The site nearest to the equator, it is not surprising, receives the highest solar energy. In this paper, a modelling of electrical characteristics of 150-Watt peak solar panels using Boltzmann sigmoid function under various temperature and irradiance is reported. Current, voltage, temperature and irradiance data in Denpasar, a city located at just south of equator, was collected. Solar power meter is used to measure irradiance level, meanwhile digital thermometer is used to measure temperature of front and back panels. Short circuit current and open circuit voltage data was also collected at different temperature and irradiance level. Statistically, the electrical characteristics of 150-Watt peak solar panel can be modelled using Boltzmann sigmoid function with good fit. Therefore, it can be concluded that Boltzmann sigmoid function might be used to determine current and voltage characteristics of 150-Watt peak solar panel under various temperature and irradiance.
Identifying product order with restricted Boltzmann machines
Rao, Wen-Jia; Li, Zhenyu; Zhu, Qiong; Luo, Mingxing; Wan, Xin
2018-03-01
Unsupervised machine learning via a restricted Boltzmann machine is a useful tool in distinguishing an ordered phase from a disordered phase. Here we study its application on the two-dimensional Ashkin-Teller model, which features a partially ordered product phase. We train the neural network with spin configuration data generated by Monte Carlo simulations and show that distinct features of the product phase can be learned from nonergodic samples resulting from symmetry breaking. Careful analysis of the weight matrices inspires us to define a nontrivial machine-learning motivated quantity of the product form, which resembles the conventional product order parameter.
Scattering theory of the linear Boltzmann operator
International Nuclear Information System (INIS)
Hejtmanek, J.
1975-01-01
In time dependent scattering theory we know three important examples: the wave equation around an obstacle, the Schroedinger and the Dirac equation with a scattering potential. In this paper another example from time dependent linear transport theory is added and considered in full detail. First the linear Boltzmann operator in certain Banach spaces is rigorously defined, and then the existence of the Moeller operators is proved by use of the theorem of Cook-Jauch-Kuroda, that is generalized to the case of a Banach space. (orig.) [de
On some asymptotic relations in the Boltzmann-Enskog model
International Nuclear Information System (INIS)
Sadovnikov, B.I.; Inozemtseva, N.G.
1977-04-01
The coefficients in the tsup(-3/2) asymptotics of the time autocorrelation functions are successively determined in the framework of the non-linear Boltzmann-Enskog model. The left and right eigenfunction systems are constructed for the Boltzmann-Enskog operator
Soluble Boltzmann equations for internal state and Maxwell models
Futcher, E.; Hoare, M.R.; Hendriks, E.M.; Ernst, M.H.
We consider a class of scalar nonlinear Boltzmann equations describing the evolution of a microcanonical ensemble in which sub-systems exchange internal energy ‘randomly’ in binary interactions. In the continuous variable version these models can equally be interpreted as Boltzmann equations for
Immiscible multicomponent lattice Boltzmann model for fluids with ...
Indian Academy of Sciences (India)
Abstract. An immiscible multicomponent lattice Boltzmann model is developed for fluids with high relaxation time ratios, which is based on the model proposed by Shan and Chen (SC). In the SC model, an interaction potential between particles is incorporated into the discrete lattice. Boltzmann equation through the ...
Partial entropic stabilization of lattice Boltzmann magnetohydrodynamics
Flint, Christopher; Vahala, George
2018-01-01
The entropic lattice Boltzmann algorithm of Karlin et al. [Phys. Rev. E 90, 031302 (2014), 10.1103/PhysRevE.90.031302] is partially extended to magnetohydrodynamics, based on the Dellar model of introducing a vector distribution for the magnetic field. This entropic ansatz is now applied only to the scalar particle distribution function so as to permit the many problems entailing magnetic field reversal. A 9-bit lattice is employed for both particle and magnetic distributions for our two-dimensional simulations. The entropic ansatz is benchmarked against our earlier multiple relaxation lattice-Boltzmann model for the Kelvin-Helmholtz instability in a magnetized jet. Other two-dimensional simulations are performed and compared to results determined by more standard direct algorithms: in particular the switch over between the Kelvin-Helmholtz or tearing mode instability of Chen et al. [J. Geophys. Res.: Space Phys. 102, 151 (1997), 10.1029/96JA03144], and the generalized Orszag-Tang vortex model of Biskamp-Welter [Phys. Fluids B 1, 1964 (1989), 10.1063/1.859060]. Very good results are achieved.
U.S. stock market interaction network as learned by the Boltzmann machine
Borysov, Stanislav S.; Roudi, Yasser; Balatsky, Alexander V.
2015-12-01
We study historical dynamics of joint equilibrium distribution of stock returns in the U.S. stock market using the Boltzmann distribution model being parametrized by external fields and pairwise couplings. Within Boltzmann learning framework for statistical inference, we analyze historical behavior of the parameters inferred using exact and approximate learning algorithms. Since the model and inference methods require use of binary variables, effect of this mapping of continuous returns to the discrete domain is studied. The presented results show that binarization preserves the correlation structure of the market. Properties of distributions of external fields and couplings as well as the market interaction network and industry sector clustering structure are studied for different historical dates and moving window sizes. We demonstrate that the observed positive heavy tail in distribution of couplings is related to the sparse clustering structure of the market. We also show that discrepancies between the model's parameters might be used as a precursor of financial instabilities.
Beyond standard Poisson-Boltzmann theory: ion-specific interactions in aqueous solutions
International Nuclear Information System (INIS)
Ben-Yaakov, Dan; Andelman, David; Harries, Daniel; Podgornik, Rudi
2009-01-01
The Poisson-Boltzmann mean-field description of ionic solutions has been successfully used in predicting charge distributions and interactions between charged macromolecules. While the electrostatic model of charged fluids, on which the Poisson-Boltzmann description rests, and its statistical mechanical consequences have been scrutinized in great detail, much less is understood about its probable shortcomings when dealing with various aspects of real physical, chemical and biological systems. These shortcomings are not only a consequence of the limitations of the mean-field approximation per se, but perhaps are primarily due to the fact that the purely Coulombic model Hamiltonian does not take into account various additional interactions that are not electrostatic in their origin. We explore several possible non-electrostatic contributions to the free energy of ions in confined aqueous solutions and investigate their ramifications and consequences on ionic profiles and interactions between charged surfaces and macromolecules.
Matin, Rastin; Hernandez, Anier; Misztal, Marek; Mathiesen, Joachim
2015-04-01
Many hydrodynamic phenomena ranging from flows at micron scale in porous media, large Reynolds numbers flows, non-Newtonian and multiphase flows have been simulated on computers using the lattice Boltzmann (LB) method. By solving the Lattice Boltzmann Equation on unstructured meshes in three dimensions, we have developed methods to efficiently model the fluid flow in real rock samples. We use this model to study the spatio-temporal statistics of the velocity field inside three-dimensional real geometries and investigate its relation to the, in general, anomalous transport of passive tracers for a wide range of Peclet and Reynolds numbers. We extend this model by free-energy based method, which allows us to simulate binary systems with large-density ratios in a thermodynamically consistent way and track the interface explicitly. In this presentation we will present our recent results on both anomalous transport and multiphase segregation.
Boltzmann equations for a binary one-dimensional ideal gas.
Boozer, A D
2011-09-01
We consider a time-reversal invariant dynamical model of a binary ideal gas of N molecules in one spatial dimension. By making time-asymmetric assumptions about the behavior of the gas, we derive Boltzmann and anti-Boltzmann equations that describe the evolution of the single-molecule velocity distribution functions for an ensemble of such systems. We show that for a special class of initial states of the ensemble one can obtain an exact expression for the N-molecule velocity distribution function, and we use this expression to rigorously prove that the time-asymmetric assumptions needed to derive the Boltzmann and anti-Boltzmann equations hold in the limit of large N. Our results clarify some subtle issues regarding the origin of the time asymmetry of Boltzmann's H theorem.
Exploring cluster Monte Carlo updates with Boltzmann machines.
Wang, Lei
2017-11-01
Boltzmann machines are physics informed generative models with broad applications in machine learning. They model the probability distribution of an input data set with latent variables and generate new samples accordingly. Applying the Boltzmann machines back to physics, they are ideal recommender systems to accelerate the Monte Carlo simulation of physical systems due to their flexibility and effectiveness. More intriguingly, we show that the generative sampling of the Boltzmann machines can even give different cluster Monte Carlo algorithms. The latent representation of the Boltzmann machines can be designed to mediate complex interactions and identify clusters of the physical system. We demonstrate these findings with concrete examples of the classical Ising model with and without four-spin plaquette interactions. In the future, automatic searches in the algorithm space parametrized by Boltzmann machines may discover more innovative Monte Carlo updates.
Exploring cluster Monte Carlo updates with Boltzmann machines
Wang, Lei
2017-11-01
Boltzmann machines are physics informed generative models with broad applications in machine learning. They model the probability distribution of an input data set with latent variables and generate new samples accordingly. Applying the Boltzmann machines back to physics, they are ideal recommender systems to accelerate the Monte Carlo simulation of physical systems due to their flexibility and effectiveness. More intriguingly, we show that the generative sampling of the Boltzmann machines can even give different cluster Monte Carlo algorithms. The latent representation of the Boltzmann machines can be designed to mediate complex interactions and identify clusters of the physical system. We demonstrate these findings with concrete examples of the classical Ising model with and without four-spin plaquette interactions. In the future, automatic searches in the algorithm space parametrized by Boltzmann machines may discover more innovative Monte Carlo updates.
New Monte Carlo approach to the adjoint Boltzmann equation
International Nuclear Information System (INIS)
De Matteis, A.; Simonini, R.
1978-01-01
A class of stochastic models for the Monte Carlo integration of the adjoint neutron transport equation is described. Some current general methods are brought within this class, thus preparing the ground for subsequent comparisons. Monte Carlo integration of the adjoint Boltzmann equation can be seen as a simulation of the transport of mathematical particles with reaction kernels not normalized to unity. This last feature is a source of difficulty: It can influence the variance of the result negatively and also often leads to preparation of special ''libraries'' consisting of tables of normalization factors as functions of energy, presently used by several methods. These are the two main points that are discussed and that are taken into account to devise a nonmultigroup method of solution for a certain class of problems. Reactions considered in detail are radiative capture, elastic scattering, discrete levels and continuum inelastic scattering, for which the need for tables has been almost completely eliminated. The basic policy pursued to avoid a source of statistical fluctuations is to try to make the statistical weight of the traveling particle dependent only on its starting and current energies, at least in simple cases. The effectiveness of the sampling schemes proposed is supported by numerical comparison with other more general adjoint Monte Carlo methods. Computation of neutron flux at a point by means of an adjoint formulation is the problem taken as a test for numerical experiments. Very good results have been obtained in the difficult case of resonant cross sections
Poisson-Boltzmann versus Size-Modified Poisson-Boltzmann Electrostatics Applied to Lipid Bilayers.
Wang, Nuo; Zhou, Shenggao; Kekenes-Huskey, Peter M; Li, Bo; McCammon, J Andrew
2014-12-26
Mean-field methods, such as the Poisson-Boltzmann equation (PBE), are often used to calculate the electrostatic properties of molecular systems. In the past two decades, an enhancement of the PBE, the size-modified Poisson-Boltzmann equation (SMPBE), has been reported. Here, the PBE and the SMPBE are reevaluated for realistic molecular systems, namely, lipid bilayers, under eight different sets of input parameters. The SMPBE appears to reproduce the molecular dynamics simulation results better than the PBE only under specific parameter sets, but in general, it performs no better than the Stern layer correction of the PBE. These results emphasize the need for careful discussions of the accuracy of mean-field calculations on realistic systems with respect to the choice of parameters and call for reconsideration of the cost-efficiency and the significance of the current SMPBE formulation.
Boltzmann babies in the proper time measure
Energy Technology Data Exchange (ETDEWEB)
Bousso, Raphael; Bousso, Raphael; Freivogel, Ben; Yang, I-Sheng
2007-12-20
After commenting briefly on the role of the typicality assumption in science, we advocate a phenomenological approach to the cosmological measure problem. Like any other theory, a measure should be simple, general, well defined, and consistent with observation. This allows us to proceed by elimination. As an example, we consider the proper time cutoff on a geodesic congruence. It predicts that typical observers are quantum fluctuations in the early universe, or Boltzmann babies. We sharpen this well-known youngness problem by taking into account the expansion and open spatial geometry of pocket universes. Moreover, we relate the youngness problem directly to the probability distribution for observables, such as the temperature of the cosmic background radiation. We consider a number of modifications of the proper time measure, but find none that would make it compatible with observation.
The Lattice Boltzmann method principles and practice
Krüger, Timm; Kuzmin, Alexandr; Shardt, Orest; Silva, Goncalo; Viggen, Erlend Magnus
2017-01-01
This book is an introduction to the theory, practice, and implementation of the Lattice Boltzmann (LB) method, a powerful computational fluid dynamics method that is steadily gaining attention due to its simplicity, scalability, extensibility, and simple handling of complex geometries. The book contains chapters on the method's background, fundamental theory, advanced extensions, and implementation. To aid beginners, the most essential paragraphs in each chapter are highlighted, and the introductory chapters on various LB topics are front-loaded with special "in a nutshell" sections that condense the chapter's most important practical results. Together, these sections can be used to quickly get up and running with the method. Exercises are integrated throughout the text, and frequently asked questions about the method are dealt with in a special section at the beginning. In the book itself and through its web page, readers can find example codes showing how the LB method can be implemented efficiently on a va...
Flux Limiter Lattice Boltzmann for Compressible Flows
International Nuclear Information System (INIS)
Chen Feng; Li Yingjun; Xu Aiguo; Zhang Guangcai
2011-01-01
In this paper, a new flux limiter scheme with the splitting technique is successfully incorporated into a multiple-relaxation-time lattice Boltzmann (LB) model for shacked compressible flows. The proposed flux limiter scheme is efficient in decreasing the artificial oscillations and numerical diffusion around the interface. Due to the kinetic nature, some interface problems being difficult to handle at the macroscopic level can be modeled more naturally through the LB method. Numerical simulations for the Richtmyer-Meshkov instability show that with the new model the computed interfaces are smoother and more consistent with physical analysis. The growth rates of bubble and spike present a satisfying agreement with the theoretical predictions and other numerical simulations. (electromagnetism, optics, acoustics, heat transfer, classical mechanics, and fluid dynamics)
Lattice Boltzmann model for numerical relativity.
Ilseven, E; Mendoza, M
2016-02-01
In the Z4 formulation, Einstein equations are written as a set of flux conservative first-order hyperbolic equations that resemble fluid dynamics equations. Based on this formulation, we construct a lattice Boltzmann model for numerical relativity and validate it with well-established tests, also known as "apples with apples." Furthermore, we find that by increasing the relaxation time, we gain stability at the cost of losing accuracy, and by decreasing the lattice spacings while keeping a constant numerical diffusivity, the accuracy and stability of our simulations improve. Finally, in order to show the potential of our approach, a linear scaling law for parallelization with respect to number of CPU cores is demonstrated. Our model represents the first step in using lattice kinetic theory to solve gravitational problems.
Lattice-Boltzmann Simulation of Tablet Disintegration
Jiang, Jiaolong; Sun, Ning; Gersappe, Dilip
Using the lattice-Boltzmann method, we developed a 2D model to study the tablet disintegration involving the swelling and wicking mechanisms. The surface area and disintegration profile of each component were obtained by tracking the tablet structure in the simulation. Compared to pure wicking, the total surface area is larger for swelling and wicking, which indicates that the swelling force breaks the neighboring bonds. The disintegration profiles show that the tablet disintegrates faster than pure wicking, and there are more wetted active pharmaceutical ingredient particles distributed on smaller clusters. Our results indicate how the porosity would affect the disintegration process by changing the wetting area of the tablet as well as by changing the swelling force propagation.
Nonextensive statistical mechanics of ionic solutions
Energy Technology Data Exchange (ETDEWEB)
Varela, L.M. [Grupo de Nanomateriales y Materia Blanda, Departamento de Fisica de la Materia Condensada, Universidad de Santiago de Compostela, E-15782 Santiago de Compostela (Spain)], E-mail: fmluis@usc.es; Carrete, J. [Grupo de Nanomateriales y Materia Blanda, Departamento de Fisica de la Materia Condensada, Universidad de Santiago de Compostela, E-15782 Santiago de Compostela (Spain); Munoz-Sola, R. [Departamento de Matematica Aplicada, Universidad de Santiago de Compostela, E-15782 Santiago de Compostela (Spain); Rodriguez, J.R.; Gallego, J. [Grupo de Nanomateriales y Materia Blanda, Departamento de Fisica de la Materia Condensada, Universidad de Santiago de Compostela, E-15782 Santiago de Compostela (Spain)
2007-10-29
Classical mean-field Poisson-Boltzmann theory of ionic solutions is revisited in the theoretical framework of nonextensive Tsallis statistics. The nonextensive equivalent of Poisson-Boltzmann equation is formulated revisiting the statistical mechanics of liquids and the Debye-Hueckel framework is shown to be valid for highly diluted solutions even under circumstances where nonextensive thermostatistics must be applied. The lowest order corrections associated to nonadditive effects are identified for both symmetric and asymmetric electrolytes and the behavior of the average electrostatic potential in a homogeneous system is analytically and numerically analyzed for various values of the complexity measurement nonextensive parameter q.
Coakley, Kevin J.; Qu, Jifeng
2017-04-01
In the electronic measurement of the Boltzmann constant based on Johnson noise thermometry, the ratio of the power spectral densities of thermal noise across a resistor at the triple point of water, and pseudo-random noise synthetically generated by a quantum-accurate voltage-noise source is constant to within 1 part in a billion for frequencies up to 1 GHz. Given knowledge of this ratio, and the values of other parameters that are known or measured, one can determine the Boltzmann constant. Due, in part, to mismatch between transmission lines, the experimental ratio spectrum varies with frequency. We model this spectrum as an even polynomial function of frequency where the constant term in the polynomial determines the Boltzmann constant. When determining this constant (offset) from experimental data, the assumed complexity of the ratio spectrum model and the maximum frequency analyzed (fitting bandwidth) dramatically affects results. Here, we select the complexity of the model by cross-validation—a data-driven statistical learning method. For each of many fitting bandwidths, we determine the component of uncertainty of the offset term that accounts for random and systematic effects associated with imperfect knowledge of model complexity. We select the fitting bandwidth that minimizes this uncertainty. In the most recent measurement of the Boltzmann constant, results were determined, in part, by application of an earlier version of the method described here. Here, we extend the earlier analysis by considering a broader range of fitting bandwidths and quantify an additional component of uncertainty that accounts for imperfect performance of our fitting bandwidth selection method. For idealized simulated data with additive noise similar to experimental data, our method correctly selects the true complexity of the ratio spectrum model for all cases considered. A new analysis of data from the recent experiment yields evidence for a temporal trend in the offset
A Boltzmann constant determination based on Johnson noise thermometry
Flowers-Jacobs, N. E.; Pollarolo, A.; Coakley, K. J.; Fox, A. E.; Rogalla, H.; Tew, W. L.; Benz, S. P.
2017-10-01
A value for the Boltzmann constant was measured electronically using an improved version of the Johnson Noise Thermometry (JNT) system at the National Institute of Standards and Technology (NIST), USA. This system is different from prior ones, including those from the 2011 determination at NIST and both 2015 and 2017 determinations at the National Institute of Metrology (NIM), China. As in all three previous determinations, the main contribution to the combined uncertainty is the statistical uncertainty in the noise measurement, which is mitigated by accumulating and integrating many weeks of cross-correlated measured data. The second major uncertainty contribution also still results from variations in the frequency response of the ratio of the measured spectral noise of the two noise sources, the sense resistor at the triple-point of water and the superconducting quantum voltage noise source. In this paper, we briefly describe the major differences between our JNT system and previous systems, in particular the input circuit and approach we used to match the frequency responses of the two noise sources. After analyzing and integrating 50 d of accumulated data, we determined a value: k~=1.380 642 9(69)× {{10}-23} J K-1 with a relative standard uncertainty of 5.0× {{10}-6} and relative offset -4.05× {{10}-6} from the CODATA 2014 recommended value.
Lattice Boltzmann method fundamentals and engineering applications with computer codes
Mohamad, A A
2014-01-01
Introducing the Lattice Boltzmann Method in a readable manner, this book provides detailed examples with complete computer codes. It avoids the most complicated mathematics and physics without scarifying the basic fundamentals of the method.
Computational Aeroacoustics Using the Generalized Lattice Boltzmann Equation, Phase I
National Aeronautics and Space Administration — The overall objective of the proposed project is to develop a generalized lattice Boltzmann (GLB) approach as a potential computational aeroacoustics (CAA) tool for...
Analysis of spectral methods for the homogeneous Boltzmann equation
Filbet, Francis
2011-04-01
The development of accurate and fast algorithms for the Boltzmann collision integral and their analysis represent a challenging problem in scientific computing and numerical analysis. Recently, several works were devoted to the derivation of spectrally accurate schemes for the Boltzmann equation, but very few of them were concerned with the stability analysis of the method. In particular there was no result of stability except when the method was modified in order to enforce the positivity preservation, which destroys the spectral accuracy. In this paper we propose a new method to study the stability of homogeneous Boltzmann equations perturbed by smoothed balanced operators which do not preserve positivity of the distribution. This method takes advantage of the "spreading" property of the collision, together with estimates on regularity and entropy production. As an application we prove stability and convergence of spectral methods for the Boltzmann equation, when the discretization parameter is large enough (with explicit bound). © 2010 American Mathematical Society.
Boltzmann, Gibbs and Darwin-Fowler approaches in parastatistics
International Nuclear Information System (INIS)
Ponczek, R.L.; Yan, C.C.
1976-01-01
Derivations of the equilibrium values of occupation numbers are made using three approaches, namely, the Boltzmann 'elementary' one, the ensemble method of Gibbs, and that of Darwin and Fowler as well [pt
Maxwell iteration for the lattice Boltzmann method with diffusive scaling
Zhao, Weifeng; Yong, Wen-An
2017-03-01
In this work, we present an alternative derivation of the Navier-Stokes equations from Bhatnagar-Gross-Krook models of the lattice Boltzmann method with diffusive scaling. This derivation is based on the Maxwell iteration and can expose certain important features of the lattice Boltzmann solutions. Moreover, it will be seen to be much more straightforward and logically clearer than the existing approaches including the Chapman-Enskog expansion.
Implicitly charge-conserving solver for Boltzmann electrons
International Nuclear Information System (INIS)
Carlsson, Johan; Manente, Marco; Pavarin, Daniele
2009-01-01
An implicitly charge-conserving algorithm has been developed for solving the nonlinear Poisson equation that results from the use of Boltzmann electrons. The new algorithm solves for the Boltzmann density parameter and, in the case of a Neumann boundary condition, the surface-charge density, simultaneously as it solves for the discretized electrostatic potential. Numerical stability is demonstrated for time steps exceeding the electron plasma period and spatial resolutions much coarser than the Debye length.
Hot electrons in superlattices: quantum transport versus Boltzmann equation
DEFF Research Database (Denmark)
Wacker, Andreas; Jauho, Antti-Pekka; Rott, S.
1999-01-01
A self-consistent solution of the transport equation is presented for semiconductor superlattices within different approaches: (i) a full quantum transport model based on nonequilibrium Green functions, (ii) the semiclassical Boltzmann equation for electrons in a miniband, and (iii) Boltzmann...... equation for electrons in Wannier-Stark states. We find good quantitative agreement of the approximations (ii) and (iii) with (i) in their respective ranges of validity. (C) 1999 Elsevier Science B.V. All rights reserved....
Müller-Kirsten, Harald J W
2013-01-01
Statistics links microscopic and macroscopic phenomena, and requires for this reason a large number of microscopic elements like atoms. The results are values of maximum probability or of averaging. This introduction to statistical physics concentrates on the basic principles, and attempts to explain these in simple terms supplemented by numerous examples. These basic principles include the difference between classical and quantum statistics, a priori probabilities as related to degeneracies, the vital aspect of indistinguishability as compared with distinguishability in classical physics, the differences between conserved and non-conserved elements, the different ways of counting arrangements in the three statistics (Maxwell-Boltzmann, Fermi-Dirac, Bose-Einstein), the difference between maximization of the number of arrangements of elements, and averaging in the Darwin-Fowler method. Significant applications to solids, radiation and electrons in metals are treated in separate chapters, as well as Bose-Eins...
Ludwig Boltzmann, Albert Einstein and Franz Joseph
International Nuclear Information System (INIS)
Broda, E.
1983-01-01
Under the Emperor Francis Joseph (1848-1916) the natural sciences were less weIl supported in Austria than in other countries of Europe. This is explained by the fact that the German speaking middle classes accepted the preeminence of the feudal forces with their antiscientific attitude. The reason for this readiness to subordination was that those middle classes feIt threatened in their relatively favourable situation by Slavs and Latins. Francis Joseph was the typical representative of the aristocracy. Personally, he did his duty conscientiously and was not corrupt, but progressive ideas and scientific thought were alien to him. From his desk he treated Boltzmann benevolently, but he had no wish to meet personally the greatest mind of the Empire or in any respect to ask his views. Another famous subject of the Emperor, Albert Einstein, was apparently ignored altogether. The structural weakness of Austria, due to the national problems, led to immobilism in her scientific life, but also, up to a point, to tolerance. The impression of Victor Adler on Einstein is considered in this historical context. (author) [de
Extended lattice Boltzmann scheme for droplet combustion.
Ashna, Mostafa; Rahimian, Mohammad Hassan; Fakhari, Abbas
2017-05-01
The available lattice Boltzmann (LB) models for combustion or phase change are focused on either single-phase flow combustion or two-phase flow with evaporation assuming a constant density for both liquid and gas phases. To pave the way towards simulation of spray combustion, we propose a two-phase LB method for modeling combustion of liquid fuel droplets. We develop an LB scheme to model phase change and combustion by taking into account the density variation in the gas phase and accounting for the chemical reaction based on the Cahn-Hilliard free-energy approach. Evaporation of liquid fuel is modeled by adding a source term, which is due to the divergence of the velocity field being nontrivial, in the continuity equation. The low-Mach-number approximation in the governing Navier-Stokes and energy equations is used to incorporate source terms due to heat release from chemical reactions, density variation, and nonluminous radiative heat loss. Additionally, the conservation equation for chemical species is formulated by including a source term due to chemical reaction. To validate the model, we consider the combustion of n-heptane and n-butanol droplets in stagnant air using overall single-step reactions. The diameter history and flame standoff ratio obtained from the proposed LB method are found to be in good agreement with available numerical and experimental data. The present LB scheme is believed to be a promising approach for modeling spray combustion.
Exploiting Restricted Boltzmann Machines and Deep Belief Networks in Compressed Sensing
Polania, Luisa F.; Barner, Kenneth E.
2017-09-01
This paper proposes a CS scheme that exploits the representational power of restricted Boltzmann machines and deep learning architectures to model the prior distribution of the sparsity pattern of signals belonging to the same class. The determined probability distribution is then used in a maximum a posteriori (MAP) approach for the reconstruction. The parameters of the prior distribution are learned from training data. The motivation behind this approach is to model the higher-order statistical dependencies between the coefficients of the sparse representation, with the final goal of improving the reconstruction. The performance of the proposed method is validated on the Berkeley Segmentation Dataset and the MNIST Database of handwritten digits.
Statistical thermodynamics of nonequilibrium processes
Keizer, Joel
1987-01-01
The structure of the theory ofthermodynamics has changed enormously since its inception in the middle of the nineteenth century. Shortly after Thomson and Clausius enunciated their versions of the Second Law, Clausius, Maxwell, and Boltzmann began actively pursuing the molecular basis of thermo dynamics, work that culminated in the Boltzmann equation and the theory of transport processes in dilute gases. Much later, Onsager undertook the elucidation of the symmetry oftransport coefficients and, thereby, established himself as the father of the theory of nonequilibrium thermodynamics. Com bining the statistical ideas of Gibbs and Langevin with the phenomenological transport equations, Onsager and others went on to develop a consistent statistical theory of irreversible processes. The power of that theory is in its ability to relate measurable quantities, such as transport coefficients and thermodynamic derivatives, to the results of experimental measurements. As powerful as that theory is, it is linear and...
Tomography and generative training with quantum Boltzmann machines
Kieferová, Mária; Wiebe, Nathan
2017-12-01
The promise of quantum neural nets, which utilize quantum effects to model complex data sets, has made their development an aspirational goal for quantum machine learning and quantum computing in general. Here we provide methods of training quantum Boltzmann machines. Our work generalizes existing methods and provides additional approaches for training quantum neural networks that compare favorably to existing methods. We further demonstrate that quantum Boltzmann machines enable a form of partial quantum state tomography that further provides a generative model for the input quantum state. Classical Boltzmann machines are incapable of this. This verifies the long-conjectured connection between tomography and quantum machine learning. Finally, we prove that classical computers cannot simulate our training process in general unless BQP=BPP , provide lower bounds on the complexity of the training procedures and numerically investigate training for small nonstoquastic Hamiltonians.
Navier-Stokes Dynamics by a Discrete Boltzmann Model
Rubinstein, Robet
2010-01-01
This work investigates the possibility of particle-based algorithms for the Navier-Stokes equations and higher order continuum approximations of the Boltzmann equation; such algorithms would generalize the well-known Pullin scheme for the Euler equations. One such method is proposed in the context of a discrete velocity model of the Boltzmann equation. Preliminary results on shock structure are consistent with the expectation that the shock should be much broader than the near discontinuity predicted by the Pullin scheme, yet narrower than the prediction of the Boltzmann equation. We discuss the extension of this essentially deterministic method to a stochastic particle method that, like DSMC, samples the distribution function rather than resolving it completely.
Stabilizing the thermal lattice Boltzmann method by spatial filtering.
Gillissen, J J J
2016-10-01
We propose to stabilize the thermal lattice Boltzmann method by filtering the second- and third-order moments of the collision operator. By means of the Chapman-Enskog expansion, we show that the additional numerical diffusivity diminishes in the low-wavnumber limit. To demonstrate the enhanced stability, we consider a three-dimensional thermal lattice Boltzmann system involving 33 discrete velocities. Filtering extends the linear stability of this thermal lattice Boltzmann method to 10-fold smaller transport coefficients. We further demonstrate that the filtering does not compromise the accuracy of the hydrodynamics by comparing simulation results to reference solutions for a number of standardized test cases, including natural convection in two dimensions.
Large Time Behavior of the Vlasov-Poisson-Boltzmann System
Directory of Open Access Journals (Sweden)
Li Li
2013-01-01
Full Text Available The motion of dilute charged particles can be modeled by Vlasov-Poisson-Boltzmann system. We study the large time stability of the VPB system. To be precise, we prove that when time goes to infinity, the solution of VPB system tends to global Maxwellian state in a rate Ot−∞, by using a method developed for Boltzmann equation without force in the work of Desvillettes and Villani (2005. The improvement of the present paper is the removal of condition on parameter λ as in the work of Li (2008.
Lattice Boltzmann method with the cell-population equilibrium
International Nuclear Information System (INIS)
Zhou Xiaoyang; Cheng Bing; Shi Baochang
2008-01-01
The central problem of the lattice Boltzmann method (LBM) is to construct a discrete equilibrium. In this paper, a multi-speed 1D cell-model of Boltzmann equation is proposed, in which the cell-population equilibrium, a direct non-negative approximation to the continuous Maxwellian distribution, plays an important part. By applying the explicit one-order Chapman–Enskog distribution, the model reduces the transportation and collision, two basic evolution steps in LBM, to the transportation of the non-equilibrium distribution. Furthermore, 1D dam-break problem is performed and the numerical results agree well with the analytic solutions
Boltzmann learning of parameters in cellular neural networks
DEFF Research Database (Denmark)
Hansen, Lars Kai
1992-01-01
The use of Bayesian methods to design cellular neural networks for signal processing tasks and the Boltzmann machine learning rule for parameter estimation is discussed. The learning rule can be used for models with hidden units, or for completely unsupervised learning. The latter is exemplified ...... by unsupervised adaptation of an image segmentation cellular network. The learning rule is applied to adaptive segmentation of satellite imagery......The use of Bayesian methods to design cellular neural networks for signal processing tasks and the Boltzmann machine learning rule for parameter estimation is discussed. The learning rule can be used for models with hidden units, or for completely unsupervised learning. The latter is exemplified...
On a Boltzmann-type price formation model
Burger, Martin
2013-06-26
In this paper, we present a Boltzmann-type price formation model, which is motivated by a parabolic free boundary model for the evolution of price presented by Lasry and Lions in 2007. We discuss the mathematical analysis of the Boltzmann-type model and show that its solutions converge to solutions of the model by Lasry and Lions as the transaction rate tends to infinity. Furthermore, we analyse the behaviour of the initial layer on the fast time scale and illustrate the price dynamics with various numerical experiments. © 2013 The Author(s) Published by the Royal Society. All rights reserved.
Ethic and Evolution in Boltzmann's and Einstein's Thought
International Nuclear Information System (INIS)
Broda, E.
1980-01-01
In physics and to a large extent in epistomology, Einstein was the natural successor to Boltzmann. But while Boltzmann was an ardent evolutionist, Einstein cared little for biology. Boltzmann applied Darwinian principles also to ethics, but remained aloof from politics. In contrast, Einstein's morality, though expressed in magnificent and selfless activity, lacked a firm theoretical basis. (author)
Axiomatic nonextensive statistics at NICA energies
Energy Technology Data Exchange (ETDEWEB)
Nasser Tawfik, Abdel [Modern University for Technology and Information (MTI), Egyptian Center for Theoretical Physics (ECTP), Cairo (Egypt); World Laboratory for Cosmology And Particle Physics (WLCAPP), Cairo (Egypt); Network for Nuclear Sciences (NNS), Academy for Scientific Research and Technology (ASRT), Cairo (Egypt)
2016-08-15
We discuss the possibility of implementing axiomatic nonextensive statistics, where it is conjectured that the phase-space volume determines the (non)extensive entropy, on the particle production at NICA energies. Both Boltzmann-Gibbs and Tsallis statistics are very special cases of this generic (non)extensivity. We conclude that the lattice thermodynamics is ab initio extensive and additive and thus the nonextensive approaches including Tsallis statistics categorically are not matching with them, while the particle production, for instance the particle ratios at various center-of-mass energies, is likely a nonextensive process but certainly not of Tsallis type. The resulting freezeout parameters, the temperature and the chemical potentials, are approximately compatible with the ones deduced from Boltzmann-Gibbs statistics. (orig.)
On real statistics of relaxation in gases
Kuzovlev, Yu. E.
2016-02-01
By example of a particle interacting with ideal gas, it is shown that the statistics of collisions in statistical mechanics at any value of the gas rarefaction parameter qualitatively differ from that conjugated with Boltzmann's hypothetical molecular chaos and kinetic equation. In reality, the probability of collisions of the particle in itself is random. Because of that, the relaxation of particle velocity acquires a power-law asymptotic behavior. An estimate of its exponent is suggested on the basis of simple kinematic reasons.
A Parallel Lattice Boltzmann Model of a Carotid Artery
Boyd, J.; Ryan, S. J.; Buick, J. M.
2008-11-01
A parallel implementation of the lattice Boltzmann model is considered for a three dimensional model of the carotid artery. The computational method and its parallel implementation are described. The performance of the parallel implementation on a Beowulf cluster is presented, as are preliminary hemodynamic results.
Nonequilibrium phenomena in QCD and BEC. Boltzmann and beyond
Energy Technology Data Exchange (ETDEWEB)
Stockamp, T.
2006-12-22
In chapter 2 we chose the real time formalism to discuss some basic principles in quantum field theory at finite temperature. This enables us to derive the quantum Boltzmann equation from the Schwinger-Dyson series. We then shortly introduce the basic concepts of QCD which are needed to understand the physics of QGP formation. After a detailed account on the bottom-up scenario we show the consistency of this approach by a diagramatical analysis of the relevant Boltzmann collision integrals. Chapter 3 deals with BEC dynamics out of equilibrium. After an introduction to the fundamental theoretical tool - namely the Gross-Pitaevskii equation - we focus on a generalization to finite temperature developed by Zaremba, Nikuni and Griffin (ZNG). These authors use a Boltzmann equation to describe the interactions between condensed and excited atoms and manage in this way to describe condensate growth. We then turn to a discussion on the 2PI effective action and derive equations of motion for a relativistic scalar field theory. In the nonrelativistic limit these equations are shown to coincide with the ZNG theory when a quasiparticle approximation is applied. Finally, we perform a numerical analysis of the full 2PI equations. These remain valid even at strong coupling and far from equilibrium, and thus go far beyond Boltzmann's approach. For simplicity, we limit ourselves to a homogeneous system and present the first 3+1 dimensional study of condensate melting. (orig.)
Revisiting Boltzmann learning: parameter estimation in Markov random fields
DEFF Research Database (Denmark)
Hansen, Lars Kai; Andersen, Lars Nonboe; Kjems, Ulrik
1996-01-01
and generalization in the context of Boltzmann machines. We provide an illustrative example concerning parameter estimation in an inhomogeneous Markov field. The regularized adaptation produces a parameter set that closely resembles the “teacher” parameters, hence, will produce segmentations that closely reproduce...
Some properties of the Boltzmann elastic collision operator
International Nuclear Information System (INIS)
Delcroix, J. L.; Salmon, J.
1959-01-01
The authors point out some properties (an important one is a variational property) of the Boltzmann elastic collision operator, valid in a more general framework than that of the Lorentz gas. Reprint of a paper published in 'Le journal de physique et le radium', tome 20, Jun 1959, p. 594-596 [fr
Coupling Boltzmann and Navier-Stokes Equations by Friction
Bourgat, Jean-François; Le Tallec, Patrick; Tidriri, Moulay D.
1995-01-01
Projet MENUSIN; The aim of this paper is to introduce and validate a coupled Navier-Stokes Boltzmann approach for the calculation of hypersonic rarefied flows around manoeuvering vehicles. The proposed strategy uses locally a kinetic model in the boundary layer coupled through wall friction forces to a global Navier-Stokes solver. Different numerical experiments illustrate the potentialities of the method.
Simulation of 2D rarefied gas flows based on the numerical solution of the Boltzmann equation
Poleshkin, Sergey O.; Malkov, Ewgenij A.; Kudryavtsev, Alexey N.; Shershnev, Anton A.; Bondar, Yevgeniy A.; Kohanchik, A. A.
2017-10-01
There are various methods for calculating rarefied gas flows, in particular, statistical methods and deterministic methods based on the finite-difference solutions of the Boltzmann nonlinear kinetic equation and on the solutions of model kinetic equations. There is no universal method; each has its disadvantages in terms of efficiency or accuracy. The choice of the method depends on the problem to be solved and on parameters of calculated flows. Qualitative theoretical arguments help to determine the range of parameters of effectively solved problems for each method; however, it is advisable to perform comparative tests of calculations of the classical problems performed by different methods and with different parameters to have quantitative confirmation of this reasoning. The paper provides the results of the calculations performed by the authors with the help of the Direct Simulation Monte Carlo method and finite-difference methods of solving the Boltzmann equation and model kinetic equations. Based on this comparison, conclusions are made on selecting a particular method for flow simulations in various ranges of flow parameters.
A fast iterative scheme for the linearized Boltzmann equation
Wu, Lei; Zhang, Jun; Liu, Haihu; Zhang, Yonghao; Reese, Jason M.
2017-06-01
Iterative schemes to find steady-state solutions to the Boltzmann equation are efficient for highly rarefied gas flows, but can be very slow to converge in the near-continuum flow regime. In this paper, a synthetic iterative scheme is developed to speed up the solution of the linearized Boltzmann equation by penalizing the collision operator L into the form L = (L + Nδh) - Nδh, where δ is the gas rarefaction parameter, h is the velocity distribution function, and N is a tuning parameter controlling the convergence rate. The velocity distribution function is first solved by the conventional iterative scheme, then it is corrected such that the macroscopic flow velocity is governed by a diffusion-type equation that is asymptotic-preserving into the Navier-Stokes limit. The efficiency of this new scheme is assessed by calculating the eigenvalue of the iteration, as well as solving for Poiseuille and thermal transpiration flows. We find that the fastest convergence of our synthetic scheme for the linearized Boltzmann equation is achieved when Nδ is close to the average collision frequency. The synthetic iterative scheme is significantly faster than the conventional iterative scheme in both the transition and the near-continuum gas flow regimes. Moreover, due to its asymptotic-preserving properties, the synthetic iterative scheme does not need high spatial resolution in the near-continuum flow regime, which makes it even faster than the conventional iterative scheme. Using this synthetic scheme, with the fast spectral approximation of the linearized Boltzmann collision operator, Poiseuille and thermal transpiration flows between two parallel plates, through channels of circular/rectangular cross sections and various porous media are calculated over the whole range of gas rarefaction. Finally, the flow of a Ne-Ar gas mixture is solved based on the linearized Boltzmann equation with the Lennard-Jones intermolecular potential for the first time, and the difference
Geva, A B; Shtram, L; Policker, S
2000-07-01
One of the most fascinating aspects of brain research is the subject of language. As in many other cases, the malfunctions that occur in different persons for various reasons give us insight on the mechanisms that support our ability to talk, read and listen. Following the work of Plaut and associates, we deal with the dyslexia disorder, which is the overall name for a large number of reading disorders. A Boltzmann machine neural network scheme was trained to implement the nonlinear mapping task of graphic representation into semantic representation, which may model the brain sections responsible for the translation of a written word into meanings and syllables. After training, various types of lesions were applied and the performance of the network was tested in order to measure the effect of each lesion on the error rate and type distribution that were detected. The system's errors were classified into several categories and the distribution of errors between the categories was studied. Using the simulations, it is demonstrated that a finite scheduling process in the Boltzmann machine causes the distribution of the network's errors to be unique and different from its expected error distribution. The phenomenon is given a mathematical explanation rooted in the statistical mechanics basics of the Boltzmann machine. Test results suggest the localization of certain reading functions within the network. Comparison is made to relevant types of dyslexia and shows resemblance in major symptoms as well as in certain known side effects.
Multilevel Methods for the Poisson-Boltzmann Equation
Holst, Michael Jay
We consider the numerical solution of the Poisson -Boltzmann equation (PBE), a three-dimensional second order nonlinear elliptic partial differential equation arising in biophysics. This problem has several interesting features impacting numerical algorithms, including discontinuous coefficients representing material interfaces, rapid nonlinearities, and three spatial dimensions. Similar equations occur in various applications, including nuclear physics, semiconductor physics, population genetics, astrophysics, and combustion. In this thesis, we study the PBE, discretizations, and develop multilevel-based methods for approximating the solutions of these types of equations. We first outline the physical model and derive the PBE, which describes the electrostatic potential of a large complex biomolecule lying in a solvent. We next study the theoretical properties of the linearized and nonlinear PBE using standard function space methods; since this equation has not been previously studied theoretically, we provide existence and uniqueness proofs in both the linearized and nonlinear cases. We also analyze box-method discretizations of the PBE, establishing several properties of the discrete equations which are produced. In particular, we show that the discrete nonlinear problem is well-posed. We study and develop linear multilevel methods for interface problems, based on algebraic enforcement of Galerkin or variational conditions, and on coefficient averaging procedures. Using a stencil calculus, we show that in certain simplified cases the two approaches are equivalent, with different averaging procedures corresponding to different prolongation operators. We also develop methods for nonlinear problems based on a nonlinear multilevel method, and on linear multilevel methods combined with a globally convergent damped-inexact-Newton method. We derive a necessary and sufficient descent condition for the inexact-Newton direction, enabling the development of extremely
Statistical models for nuclear decay from evaporation to vaporization
Cole, A J
2000-01-01
Elements of equilibrium statistical mechanics: Introduction. Microstates and macrostates. Sub-systems and convolution. The Boltzmann distribution. Statistical mechanics and thermodynamics. The grand canonical ensemble. Equations of state for ideal and real gases. Pseudo-equilibrium. Statistical models of nuclear decay. Nuclear physics background: Introduction. Elements of the theory of nuclear reactions. Quantum mechanical description of scattering from a potential. Decay rates and widths. Level and state densities in atomic nuclei. Angular momentum in quantum mechanics. History of statistical
Understanding Statistics - Cancer Statistics
Annual reports of U.S. cancer statistics including new cases, deaths, trends, survival, prevalence, lifetime risk, and progress toward Healthy People targets, plus statistical summaries for a number of common cancer types.
Dechant, Andreas; Shafier, Shalom Tzvi; Kessler, David A; Barkai, Eli
2016-08-01
The Boltzmann-Gibbs density, a central result of equilibrium statistical mechanics, relates the energy of a system in contact with a thermal bath to its equilibrium statistics. This relation is lost for nonthermal systems such as cold atoms in optical lattices, where the heat bath is replaced with the laser beams of the lattice. We investigate in detail the stationary phase-space probability for Sisyphus cooling under harmonic confinement. In particular, we elucidate whether the total energy of the system still describes its stationary state statistics. We find that this is true for the center part of the phase-space density for deep lattices, where the Boltzmann-Gibbs density provides an approximate description. The relation between energy and statistics also persists for strong confinement and in the limit of high energies, where the system becomes underdamped. However, the phase-space density now exhibits heavy power-law tails. In all three cases we find expressions for the leading-order phase-space density and corrections which break the equivalence of probability and energy and violate energy equipartition. The nonequilibrium nature of the steady state is corroborated by explicit violations of detailed balance. We complement these analytical results with numerical simulations to map out the intricate structure of the phase-space density.
Comparison of Einstein-Boltzmann solvers for testing general relativity
Bellini, E.; Barreira, A.; Frusciante, N.; Hu, B.; Peirone, S.; Raveri, M.; Zumalacárregui, M.; Avilez-Lopez, A.; Ballardini, M.; Battye, R. A.; Bolliet, B.; Calabrese, E.; Dirian, Y.; Ferreira, P. G.; Finelli, F.; Huang, Z.; Ivanov, M. M.; Lesgourgues, J.; Li, B.; Lima, N. A.; Pace, F.; Paoletti, D.; Sawicki, I.; Silvestri, A.; Skordis, C.; Umiltà, C.; Vernizzi, F.
2018-01-01
We compare Einstein-Boltzmann solvers that include modifications to general relativity and find that, for a wide range of models and parameters, they agree to a high level of precision. We look at three general purpose codes that primarily model general scalar-tensor theories, three codes that model Jordan-Brans-Dicke (JBD) gravity, a code that models f (R ) gravity, a code that models covariant Galileons, a code that models Hořava-Lifschitz gravity, and two codes that model nonlocal models of gravity. Comparing predictions of the angular power spectrum of the cosmic microwave background and the power spectrum of dark matter for a suite of different models, we find agreement at the subpercent level. This means that this suite of Einstein-Boltzmann solvers is now sufficiently accurate for precision constraints on cosmological and gravitational parameters.
Polar Coordinate Lattice Boltzmann Kinetic Modeling of Detonation Phenomena
International Nuclear Information System (INIS)
Lin Chuan-Dong; Li Ying-Jun; Xu Ai-Guo; Zhang Guang-Cai
2014-01-01
A novel polar coordinate lattice Boltzmann kinetic model for detonation phenomena is presented and applied to investigate typical implosion and explosion processes. In this model, the change of discrete distribution function due to local chemical reaction is dynamically coupled into the modified lattice Boltzmann equation which could recover the Navier—Stokes equations, including contribution of chemical reaction, via the Chapman—Enskog expansion. For the numerical investigations, the main focuses are the nonequilibrium behaviors in these processes. The system at the disc center is always in its thermodynamic equilibrium in the highly symmetric case. The internal kinetic energies in different degrees of freedom around the detonation front do not coincide. The dependence of the reaction rate on the pressure, influences of the shock strength and reaction rate on the departure amplitude of the system from its local thermodynamic equilibrium are probed. (electromagnetism, optics, acoustics, heat transfer, classical mechanics, and fluid dynamics)
Riemann-Theta Boltzmann Machine arXiv
Krefl, Daniel; Haghighat, Babak; Kahlen, Jens
A general Boltzmann machine with continuous visible and discrete integer valued hidden states is introduced. Under mild assumptions about the connection matrices, the probability density function of the visible units can be solved for analytically, yielding a novel parametric density function involving a ratio of Riemann-Theta functions. The conditional expectation of a hidden state for given visible states can also be calculated analytically, yielding a derivative of the logarithmic Riemann-Theta function. The conditional expectation can be used as activation function in a feedforward neural network, thereby increasing the modelling capacity of the network. Both the Boltzmann machine and the derived feedforward neural network can be successfully trained via standard gradient- and non-gradient-based optimization techniques.
Lattice Boltzmann model for three-phase viscoelastic fluid flow
Xie, Chiyu; Lei, Wenhai; Wang, Moran
2018-02-01
A lattice Boltzmann (LB) framework is developed for simulation of three-phase viscoelastic fluid flows in complex geometries. This model is based on a Rothman-Keller type model for immiscible multiphase flows which ensures mass conservation of each component in porous media even for a high density ratio. To account for the viscoelastic effects, the Maxwell constitutive relation is correctly introduced into the momentum equation, which leads to a modified lattice Boltzmann evolution equation for Maxwell fluids by removing the normal but excess viscous term. Our simulation tests indicate that this excess viscous term may induce significant errors. After three benchmark cases, the displacement processes of oil by dispersed polymer are studied as a typical example of three-phase viscoelastic fluid flow. The results show that increasing either the polymer intrinsic viscosity or the elastic modulus will enhance the oil recovery.
International Nuclear Information System (INIS)
Hwang, Jeong Ui; Jang, Jong Jae; Jee, Jong Gi
1987-01-01
The contents of this book are thermodynamics on the law of thermodynamics, classical thermodynamics and molecule thermodynamics, basics of molecule thermodynamics, molecule and assembly partition function, molecule partition function, classical molecule partition function, thermodynamics function for ideal assembly in fixed system, thermodynamics function for ideal assembly in running system, Maxwell-Boltzmann's law of distribution, chemical equilibrium like calculation of equilibrium constant and theory of absolute reaction rate.
Comment on ''Boltzmann equation and the conservation of particle number''
International Nuclear Information System (INIS)
Zanette, D.
1990-09-01
In a recent paper (Z. Banggu, Phys. Rev. A 42, 761 (1990)) it is argued that some solutions of the Boltzmann equation do not satisfy particle conservation as a consequence of the independence of velocity on position. In this comment, the arguments and conclusions of that paper are discussed. In particular, it is stressed that the temporal series used for solving the kinetic equation are generally divergent. A discussion about the particle conservation in its solutions is also provided. (author). 4 refs
International Nuclear Information System (INIS)
Liu, Minghua; Shi, Yong; Yan, Jiashu; Yan, Yuying
2017-01-01
Highlights: • A numerical capability combining the lattice Boltzmann method with simulated annealing algorithm is developed. • Digitized representations of random porous media are constructed using limited but meaningful statistical descriptors. • Pore-scale flow and heat transfer information in random porous media is obtained by the lattice Boltzmann simulation. • The effective properties at the representative elementary volume scale are well specified using appropriate upscale averaging. - Abstract: In this article, the lattice Boltzmann (LB) method for transport phenomena is combined with the simulated annealing (SA) algorithm for digitized porous-medium construction to study flow and heat transfer in random porous media. Importantly, in contrast to previous studies which simplify porous media as arrays of regularly shaped objects or effective pore networks, the LB + SA method in this article can model statistically meaningful random porous structures in irregular morphology, and simulate pore-scale transport processes inside them. Pore-scale isothermal flow and heat conduction in a set of constructed random porous media characterized by statistical descriptors were then simulated through use of the LB + SA method. The corresponding averages over the computational volumes and the related effective transport properties were also computed based on these pore scale numerical results. Good agreement between the numerical results and theoretical predictions or experimental data on the representative elementary volume scale was found. The numerical simulations in this article demonstrate combination of the LB method with the SA algorithm is a viable and powerful numerical strategy for simulating transport phenomena in random porous media in complex geometries.
Finite Element Based Formulation of Lattice Boltzmann Equation
International Nuclear Information System (INIS)
Jo, Jong Chull; Roh, Kyung Wan; Kwon, Young W.; Kwon, Young W.
2008-01-01
The Lattice Boltzmann Method (LBM) has been developed for application to thermal-fluid problems. Recently, the technique was also applied to fluid-structure interaction problems. Most of those studies considered a regular shape of lattice or mesh like square and cubic grids. In order to apply the LBM to more practical cases, it is necessary to be able to solve complex or irregular shapes of problem domains. There have been different kinds of approaches to address the problems. The most common technique was using the finite volume formulation of the lattice Boltzmann equation. Another approach was a point-wise interpolation technique for irregular grids. Other techniques were based on the finite element method. Generally, the finite element method is very powerful for solving two or three-dimensional complex or irregular shapes of domains using the isoparametric element formulation which is based on a mathematical mapping from a regular shape of element in an imaginary domain to a more general and irregular shape of element in the physical domain. In addition, there are variety of choices of finite elements such as triangular or quadrilateral shapes in 2-D, or tetrahedral, triangular prism, or general six-sided solids in 3-D. As a result, the present study presents a new finite element formulation for the lattice Boltzmann equation using the general weighted residual technique. Among the weighted residual formulations, the collocation method, Galerkin method or method of moments are used to develop the finite element based LBM
Non-linear effects in the Boltzmann equation
International Nuclear Information System (INIS)
Barrachina, R.O.
1985-01-01
The Boltzmann equation is studied by defining an integral transformation of the energy distribution function for an isotropic and homogeneous gas. This transformation may be interpreted as a linear superposition of equilibrium states with variable temperatures. It is shown that the temporal evolution features of the distribution function are determined by the singularities of said transformation. This method is applied to Maxwell and Very Hard Particle interaction models. For the latter, the solution of the Boltzmann equation with the solution of its linearized version is compared, finding out many basic discrepancies and non-linear effects. This gives a hint to propose a new rational approximation method with a clear physical meaning. Applying this technique, the relaxation features of the BKW (Bobylev, Krook anf Wu) mode is analyzed, finding a conclusive counter-example for the Krook and Wu conjecture. The anisotropic Boltzmann equation for Maxwell models is solved as an expansion in terms of the eigenfunctions of the corresponding linearized collision operator, finding interesting transient overpopulation and underpopulation effects at thermal energies as well as a new preferential spreading effect. By analyzing the initial collision, a criterion is established to deduce the general features of the final approach to equilibrium. Finally, it is shown how to improve the convergence of the eigenfunction expansion for high energy underpopulated distribution functions. As an application of this theory, the linear cascade model for sputtering is analyzed, thus finding out that many differences experimentally observed are due to non-linear effects. (M.E.L.) [es
Low uncertainty Boltzmann constant determinations and the kelvin redefinition.
Fischer, J
2016-03-28
At its 25th meeting, the General Conference on Weights and Measures (CGPM) approved Resolution 1 'On the future revision of the International System of Units, the SI', which sets the path towards redefinition of four base units at the next CGPM in 2018. This constitutes a decisive advance towards the formal adoption of the new SI and its implementation. Kilogram, ampere, kelvin and mole will be defined in terms of fixed numerical values of the Planck constant, elementary charge, Boltzmann constant and Avogadro constant, respectively. The effect of the new definition of the kelvin referenced to the value of the Boltzmann constant k is that the kelvin is equal to the change of thermodynamic temperature T that results in a change of thermal energy kT by 1.380 65×10(-23) J. A value of the Boltzmann constant suitable for defining the kelvin is determined by fundamentally different primary thermometers such as acoustic gas thermometers, dielectric constant gas thermometers, noise thermometers and the Doppler broadening technique. Progress to date of the measurements and further perspectives are reported. Necessary conditions to be met before proceeding with changing the definition are given. The consequences of the new definition of the kelvin on temperature measurement are briefly outlined. © 2016 The Author(s).
Elementary principles in statistical mechanics
Gibbs, Josiah Willard
2014-01-01
Written by J. Willard Gibbs, the most distinguished American mathematical physicist of the nineteenth century, this book was the first to bring together and arrange in logical order the works of Clausius, Maxwell, Boltzmann, and Gibbs himself. The lucid, advanced-level text remains a valuable collection of fundamental equations and principles. Topics include the general problem and the fundamental equation of statistical mechanics, the canonical distribution of the average energy values in a canonical ensemble of systems, and formulas for evaluating important functions of the energies of a sys
Examples of the Application of Nonparametric Information Geometry to Statistical Physics
Directory of Open Access Journals (Sweden)
Giovanni Pistone
2013-09-01
Full Text Available We review a nonparametric version of Amari’s information geometry in which the set of positive probability densities on a given sample space is endowed with an atlas of charts to form a differentiable manifold modeled on Orlicz Banach spaces. This nonparametric setting is used to discuss the setting of typical problems in machine learning and statistical physics, such as black-box optimization, Kullback-Leibler divergence, Boltzmann-Gibbs entropy and the Boltzmann equation.
Nonextensive statistical mechanics and high energy physics
Directory of Open Access Journals (Sweden)
Tsallis Constantino
2014-04-01
Full Text Available The use of the celebrated Boltzmann-Gibbs entropy and statistical mechanics is justified for ergodic-like systems. In contrast, complex systems typically require more powerful theories. We will provide a brief introduction to nonadditive entropies (characterized by indices like q, which, in the q → 1 limit, recovers the standard Boltzmann-Gibbs entropy and associated nonextensive statistical mechanics. We then present somerecent applications to systems such as high-energy collisions, black holes and others. In addition to that, we clarify and illustrate the neat distinction that exists between Lévy distributions and q-exponential ones, a point which occasionally causes some confusion in the literature, very particularly in the LHC literature
Non-Boltzmann Ensembles and Monte Carlo Simulations
International Nuclear Information System (INIS)
Murthy, K. P. N.
2016-01-01
Boltzmann sampling based on Metropolis algorithm has been extensively used for simulating a canonical ensemble and for calculating macroscopic properties of a closed system at desired temperatures. An estimate of a mechanical property, like energy, of an equilibrium system, is made by averaging over a large number microstates generated by Boltzmann Monte Carlo methods. This is possible because we can assign a numerical value for energy to each microstate. However, a thermal property like entropy, is not easily accessible to these methods. The reason is simple. We can not assign a numerical value for entropy, to a microstate. Entropy is not a property associated with any single microstate. It is a collective property of all the microstates. Toward calculating entropy and other thermal properties, a non-Boltzmann Monte Carlo technique called Umbrella sampling was proposed some forty years ago. Umbrella sampling has since undergone several metamorphoses and we have now, multi-canonical Monte Carlo, entropic sampling, flat histogram methods, Wang-Landau algorithm etc . This class of methods generates non-Boltzmann ensembles which are un-physical. However, physical quantities can be calculated as follows. First un-weight a microstates of the entropic ensemble; then re-weight it to the desired physical ensemble. Carry out weighted average over the entropic ensemble to estimate physical quantities. In this talk I shall tell you of the most recent non- Boltzmann Monte Carlo method and show how to calculate free energy for a few systems. We first consider estimation of free energy as a function of energy at different temperatures to characterize phase transition in an hairpin DNA in the presence of an unzipping force. Next we consider free energy as a function of order parameter and to this end we estimate density of states g ( E , M ), as a function of both energy E , and order parameter M . This is carried out in two stages. We estimate g ( E ) in the first stage
Simulating density-dependent flows using the lattice Boltzmann method
Bardsley, K. J.; Sukop, M. C.
2008-12-01
Seawater intrusion is a classic density-dependent problem in hydrogeology. It must be fully understood in order to be able to predict and prevent groundwater deterioration in coastal areas. All of the current programs used to study this issue are either finite difference or finite element methods. Density-dependent flow problems are exceptionally challenging for conventional numerical methods due to inherent non-linearity; definitive solutions are often elusive and a completely different modeling approach may be advantageous. The lattice Boltzmann method (LBM) represents such a numerical tool because it is not based on discretization of a series of differential equations. Instead, its foundation lies in the kinetic theory of gasses as proposed by Boltzmann. A key advantage of lattice Boltzmann method is that it has the ability to solve the Navier-Stokes equations in larger conduits and pores. Recent advances in lattice Boltzmann modeling permit simulation of large-scale density-dependent ground water flow and heat/solute transport. These simulations can be accomplished while retaining the advantages of 'regular' lattice Boltzmann methods, such as solute/heat transport at high Reynolds numbers. Hence it allows for eddy diffusion brought on by inertial components of flow at higher Reynolds numbers, which may occur in some coastal aquifers. This may prove to be an advantage for freshwater/seawater interface simulations especially given the highly macroporous nature of the aquifers underlying south Florida. Simulation of these phenomena is not possible with traditional Darcy's law-based groundwater models. Some geologists and engineers have been able to successfully apply LBM to fluid flow and contaminant transport problems. There are only a handful of scientists attempting to apply LBM to density-dependent flows in general; even fewer have considered seawater intrusion. We show how this method can be applied to density-dependent flows. We present two sets of results
The Interaction of Boltzmann with Mach, Ostwald and Planck, and his influence on Nernst and Einstein
International Nuclear Information System (INIS)
Broda, E.
1981-01-01
Boltzmann esteemed both Mach and Ostwald personally and as experimentalists, but consistently fought them in epistemology. He represented atomism and realism against energism and positivism. In the early period Boltzmann also had to struggle against Planck as a phenomenologist, but he welcomed his quantum hypothesis. As a scientist Nernst was also under Boltzmann's influence. Einstein learned atomism from (Maxwell and) Boltzmann. After Einstein had overcome Mach's positivist influence, he unknowingly approached Boltzmann's philosophical views. Some sociopolitlcal aspects of the lives of the great physicists will be discussed. It will be shown how they all, and many of Boltzmann's most eminent students, in one way or other conflicted with evil tendencies and developments in existing society. (author)
Statistical mechanics of violent relaxation
International Nuclear Information System (INIS)
Shu, F.H.
1978-01-01
We reexamine the foundations of Lynden-Bell's statistical mechanical discussion of violent relaxation in collisionless stellar systems. We argue that Lynden-Bell's formulation in terms of a continuum description introduces unnecessary complications, and we consider a more conventional formulation in terms of particles. We then find the exclusion principle discovered by Lynden-Bell to be quantitatively important only at phase densities where two-body encounters are no longer negligible. Since the edynamical basis for the exclusion principle vanishes in such cases anyway, Lynden-Bell statistics always reduces in practice to Maxwell-Boltzmann statistics when applied to stellar systems. Lynden-Bell also found the equilibrium distribution function generally to be a sum of Maxwellians with velocity dispersions dependent on the phase density at star formation. We show that this difficulty vanishes in the particulate description for an encounterless stellar system as long as stars of different masses are initially well mixed in phase space. Our methods also demonstrate the equivalence between Gibbs's formalism which uses the microcanonical ensemble and Boltzmann's formalism which uses a coarse-grained continuum description. In addition, we clarify the concept of irreversible behavior on a macroscopic scale for an encounterless stellar system. Finally, we comment on the use of unusual macroscopic constraints to simulate the effects of incomplete relaxation
A Unified Theory of Non-Ideal Gas Lattice Boltzmann Models
Luo, Li-Shi
1998-01-01
A non-ideal gas lattice Boltzmann model is directly derived, in an a priori fashion, from the Enskog equation for dense gases. The model is rigorously obtained by a systematic procedure to discretize the Enskog equation (in the presence of an external force) in both phase space and time. The lattice Boltzmann model derived here is thermodynamically consistent and is free of the defects which exist in previous lattice Boltzmann models for non-ideal gases. The existing lattice Boltzmann models for non-ideal gases are analyzed and compared with the model derived here.
About the statistical description of gas-liquid flows
Energy Technology Data Exchange (ETDEWEB)
Sanz, D.; Guido-Lavalle, G.; Carrica, P. [Centro Atomico Bariloche and Instituto Balseiro (Argentina)] [and others
1995-09-01
Elements of the probabilistic geometry are used to derive the bubble coalescence term of the statistical description of gas liquid flows. It is shown that the Boltzmann`s hypothesis, that leads to the kinetic theory of dilute gases, is not appropriate for this kind of flows. The resulting integro-differential transport equation is numerically integrated to study the flow development in slender bubble columns. The solution remarkably predicts the transition from bubbly to slug flow pattern. Moreover, a bubbly bimodal size distribution is predicted, which has already been observed experimentally.
Jiang, Xue; Zhang, Han; Duan, Feng; Quan, Xiongwen
2017-10-11
Predicting disease-associated genes is helpful for understanding the molecular mechanisms during the disease progression. Since the pathological mechanisms of neurodegenerative diseases are very complex, traditional statistic-based methods are not suitable for identifying key genes related to the disease development. Recent studies have shown that the computational models with deep structure can learn automatically the features of biological data, which is useful for exploring the characteristics of gene expression during the disease progression. In this paper, we propose a deep learning approach based on the restricted Boltzmann machine to analyze the RNA-seq data of Huntington's disease, namely stacked restricted Boltzmann machine (SRBM). According to the SRBM, we also design a novel framework to screen the key genes during the Huntington's disease development. In this work, we assume that the effects of regulatory factors can be captured by the hierarchical structure and narrow hidden layers of the SRBM. First, we select disease-associated factors with different time period datasets according to the differentially activated neurons in hidden layers. Then, we select disease-associated genes according to the changes of the gene energy in SRBM at different time periods. The experimental results demonstrate that SRBM can detect the important information for differential analysis of time series gene expression datasets. The identification accuracy of the disease-associated genes is improved to some extent using the novel framework. Moreover, the prediction precision of disease-associated genes for top ranking genes using SRBM is effectively improved compared with that of the state of the art methods.
Lattice Boltzmann modeling and simulation of liquid jet breakup
Saito, Shimpei; Abe, Yutaka; Koyama, Kazuya
2017-07-01
A three-dimensional color-fluid lattice Boltzmann model for immiscible two-phase flows is developed in the framework of a three-dimensional 27-velocity (D3Q27) lattice. The collision operator comprises the D3Q27 versions of three suboperators: a multiple-relaxation-time (MRT) collision operator, a generalized Liu-Valocchi-Kang perturbation operator, and a Latva-Kokko-Rothman recoloring operator. A D3Q27 version of an enhanced equilibrium distribution function is also incorporated into this model to improve the Galilean invariance. Three types of numerical tests, namely, a static droplet, an oscillating droplet, and the Rayleigh-Taylor instability, show a good agreement with analytical solutions and numerical simulations. Following these numerical tests, this model is applied to liquid-jet-breakup simulations. The simulation conditions are matched to the conditions of the previous experiments. In this case, numerical stability is maintained throughout the simulation, although the kinematic viscosity for the continuous phase is set as low as 1.8 ×10-4 , in which case the corresponding Reynolds number is 3.4 ×103 ; the developed lattice Boltzmann model based on the D3Q27 lattice enables us to perform the simulation with parameters directly matched to the experiments. The jet's liquid column transitions from an asymmetrical to an axisymmetrical shape, and entrainment occurs from the side of the jet. The measured time history of the jet's leading-edge position shows a good agreement with the experiments. Finally, the reproducibility of the regime map for liquid-liquid systems is assessed. The present lattice Boltzmann simulations well reproduce the characteristics of predicted regimes, including varicose breakup, sinuous breakup, and atomization.
Lattice Boltzmann method for the fractional advection-diffusion equation
Zhou, J. G.; Haygarth, P. M.; Withers, P. J. A.; Macleod, C. J. A.; Falloon, P. D.; Beven, K. J.; Ockenden, M. C.; Forber, K. J.; Hollaway, M. J.; Evans, R.; Collins, A. L.; Hiscock, K. M.; Wearing, C.; Kahana, R.; Villamizar Velez, M. L.
2016-04-01
Mass transport, such as movement of phosphorus in soils and solutes in rivers, is a natural phenomenon and its study plays an important role in science and engineering. It is found that there are numerous practical diffusion phenomena that do not obey the classical advection-diffusion equation (ADE). Such diffusion is called abnormal or superdiffusion, and it is well described using a fractional advection-diffusion equation (FADE). The FADE finds a wide range of applications in various areas with great potential for studying complex mass transport in real hydrological systems. However, solution to the FADE is difficult, and the existing numerical methods are complicated and inefficient. In this study, a fresh lattice Boltzmann method is developed for solving the fractional advection-diffusion equation (LabFADE). The FADE is transformed into an equation similar to an advection-diffusion equation and solved using the lattice Boltzmann method. The LabFADE has all the advantages of the conventional lattice Boltzmann method and avoids a complex solution procedure, unlike other existing numerical methods. The method has been validated through simulations of several benchmark tests: a point-source diffusion, a boundary-value problem of steady diffusion, and an initial-boundary-value problem of unsteady diffusion with the coexistence of source and sink terms. In addition, by including the effects of the skewness β , the fractional order α , and the single relaxation time τ , the accuracy and convergence of the method have been assessed. The numerical predictions are compared with the analytical solutions, and they indicate that the method is second-order accurate. The method presented will allow the FADE to be more widely applied to complex mass transport problems in science and engineering.
Phase transitions in restricted Boltzmann machines with generic priors
Barra, Adriano; Genovese, Giuseppe; Sollich, Peter; Tantari, Daniele
2017-10-01
We study generalized restricted Boltzmann machines with generic priors for units and weights, interpolating between Boolean and Gaussian variables. We present a complete analysis of the replica symmetric phase diagram of these systems, which can be regarded as generalized Hopfield models. We underline the role of the retrieval phase for both inference and learning processes and we show that retrieval is robust for a large class of weight and unit priors, beyond the standard Hopfield scenario. Furthermore, we show how the paramagnetic phase boundary is directly related to the optimal size of the training set necessary for good generalization in a teacher-student scenario of unsupervised learning.
Simulating Electric Double Layer Capacitance by Using Lattice Boltzmann Method
Sun, Ning; Gersappe, Dilip
2015-03-01
By using the Lattice Boltzmann Method (LBM) we studied diffuse-charge dynamics in electrochemical systems. We use the LBM to solve Poisson-Nernst-Planck equations (PNP) and Modified Poisson-Nernst-Planck equations (MPNP). The isotropic permittivity of electrolyte is modeled using the Booth model. The results show that both steric effect (MPNP) and isotropic permittivity (Booth model) can have large influence on diffuse-charge dynamics, especially when electrolyte concentration or applied potential is high. This model can be applied to simulate electric double layer capacitance of super capacitors with complex geometry and also incorporate other effects such as heat convection in a modular manner.
Lattice Boltzmann model for thermal binary-mixture gas flows.
Kang, Jinfen; Prasianakis, Nikolaos I; Mantzaras, John
2013-05-01
A lattice Boltzmann model for thermal gas mixtures is derived. The kinetic model is designed in a way that combines properties of two previous literature models, namely, (a) a single-component thermal model and (b) a multicomponent isothermal model. A comprehensive platform for the study of various practical systems involving multicomponent mixture flows with large temperature differences is constructed. The governing thermohydrodynamic equations include the mass, momentum, energy conservation equations, and the multicomponent diffusion equation. The present model is able to simulate mixtures with adjustable Prandtl and Schmidt numbers. Validation in several flow configurations with temperature and species concentration ratios up to nine is presented.
Boltzmann equation and hydrodynamics beyond Navier-Stokes.
Bobylev, A V
2018-04-28
We consider in this paper the problem of derivation and regularization of higher (in Knudsen number) equations of hydrodynamics. The author's approach based on successive changes of hydrodynamic variables is presented in more detail for the Burnett level. The complete theory is briefly discussed for the linearized Boltzmann equation. It is shown that the best results in this case can be obtained by using the 'diagonal' equations of hydrodynamics. Rigorous estimates of accuracy of the Navier-Stokes and Burnett approximations are also presented.This article is part of the theme issue 'Hilbert's sixth problem'. © 2018 The Author(s).
A large eddy lattice Boltzmann simulation of magnetohydrodynamic turbulence
Flint, Christopher; Vahala, George
2018-02-01
Large eddy simulations (LES) of a lattice Boltzmann magnetohydrodynamic (LB-MHD) model are performed for the unstable magnetized Kelvin-Helmholtz jet instability. This algorithm is an extension of Ansumali et al. [1] to MHD in which one performs first an expansion in the filter width on the kinetic equations followed by the usual low Knudsen number expansion. These two perturbation operations do not commute. Closure is achieved by invoking the physical constraint that subgrid effects occur at transport time scales. The simulations are in very good agreement with direct numerical simulations.
Jet propagation within a Linearized Boltzmann Transport model
Energy Technology Data Exchange (ETDEWEB)
Luo, Tan; He, Yayun [Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079 (China); Wang, Xin-Nian [Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079 (China); Nuclear Science Division, Mailstop 70R0319, Lawrence Berkeley National Laboratory, Berkeley, CA 94740 (United States); Zhu, Yan [Departamento de Física de Partículas and IGFAE, Universidade de Santiago de Compostela, E-15706 Santiago de Compostela, Galicia (Spain)
2014-12-15
A Linearized Boltzmann Transport (LBT) model has been developed for the study of parton propagation inside quark–gluon plasma. Both leading and thermal recoiled partons are tracked in order to include the effect of jet-induced medium excitation. In this talk, we present a study within the LBT model in which we implement the complete set of elastic parton scattering processes. We investigate elastic parton energy loss and their energy and length dependence. We further investigate energy loss and transverse shape of reconstructed jets. Contributions from the recoiled thermal partons and jet-induced medium excitations are found to have significant influences on the jet energy loss and transverse profile.
Energy Technology Data Exchange (ETDEWEB)
Zabadal, Jorge; Borges, Volnei; Van der Laan, Flavio T., E-mail: jorge.zabadal@ufrgs.br, E-mail: borges@ufrgs.br, E-mail: ftvdl@ufrgs.br [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Departamento de Engenharia Mecanica. Grupo de Pesquisas Radiologicas; Ribeiro, Vinicius G., E-mail: vinicius_ribeiro@uniritter.edu.br [Centro Universitario Ritter dos Reis (UNIRITTER), Porto Alegre, RS (Brazil); Santos, Marcio G., E-mail: phd.marcio@gmail.com [Universidade Federal do Rio Grande do Sul (UFRGS), Tramandai, RS (Brazil). Departamento Interdisciplinar do Campus Litoral Norte
2015-07-01
This work presents a new analytical method for solving the Boltzmann equation. In this formulation, a linear differential operator is applied over the Boltzmann model, in order to produce a partial differential equation in which the scattering term is absent. This auxiliary equation is solved via reduction of order. The exact solution obtained is employed to define a precursor for the buildup factor. (author)
DEFF Research Database (Denmark)
Pingen, Georg; Evgrafov, Anton; Maute, Kurt
2009-01-01
We present an adjoint parameter sensitivity analysis formulation and solution strategy for the lattice Boltzmann method (LBM). The focus is on design optimization applications, in particular topology optimization. The lattice Boltzmann method is briefly described with an in-depth discussion of so...
L2-stability of the Vlasov-Maxwell-Boltzmann system near global Maxwellians
International Nuclear Information System (INIS)
Ha, Seung-Yeal; Xiao, Qinghua; Xiong, Linjie; Zhao, Huijiang
2013-01-01
We present a L 2 -stability theory of the Vlasov-Maxwell-Boltzmann system for the two-species collisional plasma. We show that in a perturbative regime of a global Maxwellian, the L 2 -distance between two strong solutions can be controlled by that between initial data in a Lipschitz manner. Our stability result extends earlier results [Ha, S.-Y. and Xiao, Q.-H., “A revisiting to the L 2 -stability theory of the Boltzmann equation near global Maxwellians,” (submitted) and Ha, S.-Y., Yang, X.-F., and Yun, S.-B., “L 2 stability theory of the Boltzmann equation near a global Maxwellian,” Arch. Ration. Mech. Anal. 197, 657–688 (2010)] on the L 2 -stability of the Boltzmann equation to the Boltzmann equation coupled with self-consistent external forces. As a direct application of our stability result, we show that classical solutions in Duan et al. [“Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space,” Commun. Pure Appl. Math. 24, 1497–1546 (2011)] and Guo [“The Vlasov-Maxwell-Boltzmann system near Maxwellians,” Invent. Math. 153(3), 593–630 (2003)] satisfy a uniform L 2 -stability estimate. This is the first result on the L 2 -stability of the Boltzmann equation coupled with self-consistent field equations in three dimensions
Lattice Boltzmann model capable of mesoscopic vorticity computation.
Peng, Cheng; Guo, Zhaoli; Wang, Lian-Ping
2017-11-01
It is well known that standard lattice Boltzmann (LB) models allow the strain-rate components to be computed mesoscopically (i.e., through the local particle distributions) and as such possess a second-order accuracy in strain rate. This is one of the appealing features of the lattice Boltzmann method (LBM) which is of only second-order accuracy in hydrodynamic velocity itself. However, no known LB model can provide the same quality for vorticity and pressure gradients. In this paper, we design a multiple-relaxation time LB model on a three-dimensional 27-discrete-velocity (D3Q27) lattice. A detailed Chapman-Enskog analysis is presented to illustrate all the necessary constraints in reproducing the isothermal Navier-Stokes equations. The remaining degrees of freedom are carefully analyzed to derive a model that accommodates mesoscopic computation of all the velocity and pressure gradients from the nonequilibrium moments. This way of vorticity calculation naturally ensures a second-order accuracy, which is also proven through an asymptotic analysis. We thus show, with enough degrees of freedom and appropriate modifications, the mesoscopic vorticity computation can be achieved in LBM. The resulting model is then validated in simulations of a three-dimensional decaying Taylor-Green flow, a lid-driven cavity flow, and a uniform flow passing a fixed sphere. Furthermore, it is shown that the mesoscopic vorticity computation can be realized even with single relaxation parameter.
A modified Poisson-Boltzmann equation applied to protein adsorption.
Gama, Marlon de Souza; Santos, Mirella Simões; Lima, Eduardo Rocha de Almeida; Tavares, Frederico Wanderley; Barreto, Amaro Gomes Barreto
2018-01-05
Ion-exchange chromatography has been widely used as a standard process in purification and analysis of protein, based on the electrostatic interaction between the protein and the stationary phase. Through the years, several approaches are used to improve the thermodynamic description of colloidal particle-surface interaction systems, however there are still a lot of gaps specifically when describing the behavior of protein adsorption. Here, we present an improved methodology for predicting the adsorption equilibrium constant by solving the modified Poisson-Boltzmann (PB) equation in bispherical coordinates. By including dispersion interactions between ions and protein, and between ions and surface, the modified PB equation used can describe the Hofmeister effects. We solve the modified Poisson-Boltzmann equation to calculate the protein-surface potential of mean force, treated as spherical colloid-plate system, as a function of process variables. From the potential of mean force, the Henry constants of adsorption, for different proteins and surfaces, are calculated as a function of pH, salt concentration, salt type, and temperature. The obtained Henry constants are compared with experimental data for several isotherms showing excellent agreement. We have also performed a sensitivity analysis to verify the behavior of different kind of salts and the Hofmeister effects. Copyright © 2017 Elsevier B.V. All rights reserved.
High order spectral difference lattice Boltzmann method for incompressible hydrodynamics
Li, Weidong
2017-09-01
This work presents a lattice Boltzmann equation (LBE) based high order spectral difference method for incompressible flows. In the present method, the spectral difference (SD) method is adopted to discretize the convection and collision term of the LBE to obtain high order (≥3) accuracy. Because the SD scheme represents the solution as cell local polynomials and the solution polynomials have good tensor-product property, the present spectral difference lattice Boltzmann method (SD-LBM) can be implemented on arbitrary unstructured quadrilateral meshes for effective and efficient treatment of complex geometries. Thanks to only first oder PDEs involved in the LBE, no special techniques, such as hybridizable discontinuous Galerkin method (HDG), local discontinuous Galerkin method (LDG) and so on, are needed to discrete diffusion term, and thus, it simplifies the algorithm and implementation of the high order spectral difference method for simulating viscous flows. The proposed SD-LBM is validated with four incompressible flow benchmarks in two-dimensions: (a) the Poiseuille flow driven by a constant body force; (b) the lid-driven cavity flow without singularity at the two top corners-Burggraf flow; and (c) the unsteady Taylor-Green vortex flow; (d) the Blasius boundary-layer flow past a flat plate. Computational results are compared with analytical solutions of these cases and convergence studies of these cases are also given. The designed accuracy of the proposed SD-LBM is clearly verified.
Polyelectrolyte Microcapsules: Ion Distributions from a Poisson-Boltzmann Model
Tang, Qiyun; Denton, Alan R.; Rozairo, Damith; Croll, Andrew B.
2014-03-01
Recent experiments have shown that polystyrene-polyacrylic-acid-polystyrene (PS-PAA-PS) triblock copolymers in a solvent mixture of water and toluene can self-assemble into spherical microcapsules. Suspended in water, the microcapsules have a toluene core surrounded by an elastomer triblock shell. The longer, hydrophilic PAA blocks remain near the outer surface of the shell, becoming charged through dissociation of OH functional groups in water, while the shorter, hydrophobic PS blocks form a networked (glass or gel) structure. Within a mean-field Poisson-Boltzmann theory, we model these polyelectrolyte microcapsules as spherical charged shells, assuming different dielectric constants inside and outside the capsule. By numerically solving the nonlinear Poisson-Boltzmann equation, we calculate the radial distribution of anions and cations and the osmotic pressure within the shell as a function of salt concentration. Our predictions, which can be tested by comparison with experiments, may guide the design of microcapsules for practical applications, such as drug delivery. This work was supported by the National Science Foundation under Grant No. DMR-1106331.
The Fluid Dynamical Limits of the Linearized Boltzmann Equation.
Campini, Marco
The old question concerning the mathematical formulation of the fluid dynamic limits of kinetic theory is examined by studying the solution of the Cauchy problem for two differently scaled linearized Boltzmann equations on periodic domain as the mean free path of the particles becomes small. Under minimal assumptions on the initial data, by using an a priori estimate, it is possible, in a Hilbert space functional frame, to prove the weak convergence of solutions toward a function that has the form of an infinitesimal maxwellian in the velocity variable. The velocity moments of this function are then proved to satisfy either the linearized Euler or the Stokes system of equations (depending on the chosen scaling), by passing to the limit in the conservation relations derived from the Boltzmann equation. A theorem injecting continuously the intersection of certain weak spaces into a normed one is proved. Together with properties of the Euler semigroup, this allows to show strong convergence of the first three moments of the distribution function toward the macroscopic quantities density, bulk velocity and temperature, solutions of the linearized Euler system. The Stokes case is treated somewhat differently, through the introduction of a result, proved by using the adjoint formulation for linear kinetic equations, that extends the averaging theory of Golse-Lions-Perthame-Sentis. The desired convergence for the divergence-free component of the second moment toward the macroscopic velocity is then shown.
A Boltzmann Transport Simulation Using Open Source Physics
Hasbun, Javier
2004-03-01
The speed of a charged particle, under an applied electric field, in a conducting media, is, usually, simply modelled by writing Newton's 2nd law in the form mfrac ddtv=qE-mfrac vτ ; (1), where v is the speed, E is the applied electric field, q is the charge, m is the mass, and τ is the scattering time between collisions. Here, we simulate a numerical solution of the Boltzmann transport equation,frac partial partial tf+ vot nabla _rf+Fot nabla _pf=frac partial partial tf|_coll (2), where in general the Boltzmann distribution function f=f(r,p,t) depends on position, momentum, and time. Our numerical solution is made possible by neglecting the 2nd term on the LHS, and by modelling the RHS collision term as fracpartial partial tf|_coll=-frac 1τ . With these approximations, in addition to considering only one dimension, we find, our numerical solution of (2). The average velocity numerically obtained through the resulting distribution is compared to that obtained by the analytic solution of (1). An efficient method of carrying out the numerical solution of (2) due to P. Drallos and M. Wadehra [Journal of Applied Physics 63, 5601(1988)] is incorporated here. A final version of an applet that performs the full Java simulation will be located at http://www.westga.edu/ jhasbun/osp/osp.htm.
Saltwater Intrusion Simulation in Heterogeneous Aquifer Using Lattice Boltzmann Method
Servan-Camas, B.; Tsai, F. T.
2006-12-01
This study develops a saltwater intrusion simulation model using a lattice Boltzmann method (LBM) in a two- dimensional coastal confined aquifer. The saltwater intrusion phenomenon is described by density-varied groundwater flow and mass transport equations, where a freshwater-saltwater mixing zone is considered. Although primarily developed using the mesoscopic approach to solve macroscopic fluid dynamic problems (e.g. Navier-Stoke equation), LBM is able to be adopted to solve physical-based diffusion-type governing equations as for the groundwater flow and mass transport equations. The challenge of using LBM in saltwater intrusion modeling is to recover hydraulic conductivity heterogeneity. In this study, the Darcy equation and the advection-dispersion equation (ADE) are recovered in the lattice Boltzmann modeling. Specifically, the hydraulic conductivity heterogeneity is represented by the speed of sound in LBM. Under the consideration on the steady-state groundwater flow due to low storativity, in each time step the flow problem is modified to be a Poisson equation and solved by LBM. Nevertheless, the groundwater flow is still a time-marching problem with spatial-temporal variation in salinity concentration as well as density. The Henry problem is used to compare the LBM results against the Henry analytic solution and SUTRA result. Also, we show that LBM is capable of handling the Dirichlet, Neumann, and Cauchy concentration boundary conditions at the sea side. Finally, we compare the saltwater intrusion results using LBM in the Henry problem when heterogeneous hydraulic conductivity is considered.
Well-Posedness of the Iterative Boltzmann Inversion
Hanke, Martin
2018-02-01
The iterative Boltzmann inversion is a fixed point iteration to determine an effective pair potential for an ensemble of identical particles in thermal equilibrium from the corresponding radial distribution function. Although the method is reported to work reasonably well in practice, it still lacks a rigorous convergence analysis. In this paper we provide some first steps towards such an analysis, and we show under quite general assumptions that the associated fixed point operator is Lipschitz continuous (in fact, differentiable) in a suitable neighborhood of the true pair potential, assuming that such a potential exists. In other words, the iterative Boltzmann inversion is well-defined in the sense that if the kth iterate of the scheme is sufficiently close to the true pair potential then the k+1st iterate is an admissible pair potential, which again belongs to the domain of the fixed point operator. On our way we establish important properties of the cavity distribution function and provide a proof of a statement formulated by Groeneveld concerning the rate of decay at infinity of the Ursell function associated with a Lennard-Jones type potential.
An interpolation boundary treatment for the Lattice Boltzmann method
Deladisma, Marnico D.; Smith, Marc K.
2003-11-01
A new boundary condition for the Lattice Boltzmann method based on bounce-back and spatial interpolations is presented. The boundary condition allows for the placement of a boundary at any position between nodes and tracks the exact position of that boundary. Multi-dimensional interpolation of streaming and bounce-back particle distribution functions from surrounding boundary nodes is used to solve for new distribution values. This allows more information from surrounding nodes to be incorporated into the boundary treatment calculation. Calculations of flow within a 2D rotating annulus (with and without an obstacle placed in the flow) using the present boundary condition are compared with calculations done with the commercial CFD solver Fluent. Results show that the boundary condition is accurate and robust for these cases. The boundary condition also allows for moving boundaries and is easily extended to 3D, which facilitates the simulation of moving 3D particles. The new boundary condition will allow a Lattice Boltzmann simulation of a rotating wall vessel bioreactor with freely suspended tissue constructs whose length scale is about 1 cm.
Element Free Lattice Boltzmann Method for Fluid-Flow Problems
Energy Technology Data Exchange (ETDEWEB)
Jo, Jong Chull; Roh, Kyung Wan; Yune, Young Gill; Kim, Hho Jhung [Korea Institute of Nuclear Safety, Daejeon (Korea, Republic of); Kwon, Young Kwon [US Naval Postgraduate School, New York (United States)
2007-10-15
The Lattice Boltzmann Method (LBM) has been developed for application to thermal-fluid problems. Most of the those studies considered a regular shape of lattice or mesh like square and cubic grids. In order to apply the LBM to more practical cases, it is necessary to be able to solve complex or irregular shapes of problem domains. Some techniques were based on the finite element method. Generally, the finite element method is very powerful for solving two or three-dimensional complex or irregular shapes of domains using the iso-parametric element formulation which is based on a mathematical mapping from a regular shape of element in an imaginary domain to a more general and irregular shape of element in the physical domain. In addition, the element free technique is also quite useful to analyze a complex shape of domain because there is no need to divide a domain by a compatible finite element mesh. This paper presents a new finite element and element free formulations for the lattice Boltzmann equation using the general weighted residual technique. Then, a series of validation examples are presented.
Lattice Boltzmann model capable of mesoscopic vorticity computation
Peng, Cheng; Guo, Zhaoli; Wang, Lian-Ping
2017-11-01
It is well known that standard lattice Boltzmann (LB) models allow the strain-rate components to be computed mesoscopically (i.e., through the local particle distributions) and as such possess a second-order accuracy in strain rate. This is one of the appealing features of the lattice Boltzmann method (LBM) which is of only second-order accuracy in hydrodynamic velocity itself. However, no known LB model can provide the same quality for vorticity and pressure gradients. In this paper, we design a multiple-relaxation time LB model on a three-dimensional 27-discrete-velocity (D3Q27) lattice. A detailed Chapman-Enskog analysis is presented to illustrate all the necessary constraints in reproducing the isothermal Navier-Stokes equations. The remaining degrees of freedom are carefully analyzed to derive a model that accommodates mesoscopic computation of all the velocity and pressure gradients from the nonequilibrium moments. This way of vorticity calculation naturally ensures a second-order accuracy, which is also proven through an asymptotic analysis. We thus show, with enough degrees of freedom and appropriate modifications, the mesoscopic vorticity computation can be achieved in LBM. The resulting model is then validated in simulations of a three-dimensional decaying Taylor-Green flow, a lid-driven cavity flow, and a uniform flow passing a fixed sphere. Furthermore, it is shown that the mesoscopic vorticity computation can be realized even with single relaxation parameter.
Detection of Hypertension Retinopathy Using Deep Learning and Boltzmann Machines
Triwijoyo, B. K.; Pradipto, Y. D.
2017-01-01
hypertensive retinopathy (HR) in the retina of the eye is disturbance caused by high blood pressure disease, where there is a systemic change of arterial in the blood vessels of the retina. Most heart attacks occur in patients caused by high blood pressure symptoms of undiagnosed. Hypertensive retinopathy Symptoms such as arteriolar narrowing, retinal haemorrhage and cotton wool spots. Based on this reasons, the early diagnosis of the symptoms of hypertensive retinopathy is very urgent to aim the prevention and treatment more accurate. This research aims to develop a system for early detection of hypertension retinopathy stage. The proposed method is to determine the combined features artery and vein diameter ratio (AVR) as well as changes position with Optic Disk (OD) in retinal images to review the classification of hypertensive retinopathy using Deep Neural Networks (DNN) and Boltzmann Machines approach. We choose this approach of because based on previous research DNN models were more accurate in the image pattern recognition, whereas Boltzmann machines selected because It requires speedy iteration in the process of learning neural network. The expected results from this research are designed a prototype system early detection of hypertensive retinopathy stage and analysed the effectiveness and accuracy of the proposed methods.
Lattice Boltzmann simulation of antiplane shear loading of a stationary crack
Schlüter, Alexander; Kuhn, Charlotte; Müller, Ralf
2018-01-01
In this work, the lattice Boltzmann method is applied to study the dynamic behaviour of linear elastic solids under antiplane shear deformation. In this case, the governing set of partial differential equations reduces to a scalar wave equation for the out of plane displacement in a two dimensional domain. The lattice Boltzmann approach developed by Guangwu (J Comput Phys 161(1):61-69, 2000) in 2006 is used to solve the problem numerically. Some aspects of the scheme are highlighted, including the treatment of the boundary conditions. Subsequently, the performance of the lattice Boltzmann scheme is tested for a stationary crack problem for which an analytic solution exists. The treatment of cracks is new compared to the examples that are discussed in Guangwu's work. Furthermore, the lattice Boltzmann simulations are compared to finite element computations. Finally, the influence of the lattice Boltzmann relaxation parameter on the stability of the scheme is illustrated.
Cosmic Statistics of Statistics
Szapudi, I.; Colombi, S.; Bernardeau, F.
1999-01-01
The errors on statistics measured in finite galaxy catalogs are exhaustively investigated. The theory of errors on factorial moments by Szapudi & Colombi (1996) is applied to cumulants via a series expansion method. All results are subsequently extended to the weakly non-linear regime. Together with previous investigations this yields an analytic theory of the errors for moments and connected moments of counts in cells from highly nonlinear to weakly nonlinear scales. The final analytic formu...
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An introduction to statistical mechanics and thermodynamics
Swendsen, Robert H
2012-01-01
This text presents the two complementary aspects of thermal physics as an integrated theory of the properties of matter. Conceptual understanding is promoted by thorough development of basic concepts. In contrast to many texts, statistical mechanics, including discussion of the required probability theory, is presented first. This provides a statistical foundation for the concept of entropy, which is central to thermal physics. A unique feature of the book is the development ofentropy based on Boltzmann's 1877 definition; this avoids contradictions or ad hoc corrections found in other texts. D
A viscosity adaption method for Lattice Boltzmann simulations
Conrad, Daniel; Schneider, Andreas; Böhle, Martin
2014-11-01
In this work, we consider the limited fitness for practical use of the Lattice Boltzmann Method for non-Newtonian fluid flows. Several authors have shown that the LBM is capable of correctly simulating those fluids. However, due to stability reasons the modeled viscosity range has to be truncated. The resulting viscosity boundaries are chosen arbitrarily, because the correct simulation Mach number for the physical problem is unknown a priori. This easily leads to corrupt simulation results. A viscosity adaption method (VAM) is derived which drastically improves the applicability of LBM for non-Newtonian fluid flows by adaption of the modeled viscosity range to the actual physical problem. This is done through tuning of the global Mach number to the solution-dependent shear rate. We demonstrate that the VAM can be used to accelerate LBM simulations and improve their accuracy, for both steady state and transient cases.
Lattice Boltzmann modeling an introduction for geoscientists and engineers
Sukop, Michael C
2005-01-01
Lattice Boltzmann models have a remarkable ability to simulate single- and multi-phase fluids and transport processes within them. A rich variety of behaviors, including higher Reynolds numbers flows, phase separation, evaporation, condensation, cavitation, buoyancy, and interactions with surfaces can readily be simulated. This book provides a basic introduction that emphasizes intuition and simplistic conceptualization of processes. It avoids the more difficult mathematics that underlies LB models. The model is viewed from a particle perspective where collisions, streaming, and particle-particle/particle-surface interactions constitute the entire conceptual framework. Beginners and those with more interest in model application than detailed mathematical foundations will find this a powerful "quick start" guide. Example simulations, exercises, and computer codes are included. Working code is provided on the Internet.
Simulating condensation on microstructured surfaces using Lattice Boltzmann Method
Alexeev, Alexander; Vasyliv, Yaroslav
2017-11-01
We simulate a single component fluid condensing on 2D structured surfaces with different wettability. To simulate the two phase fluid, we use the athermal Lattice Boltzmann Method (LBM) driven by a pseudopotential force. The pseudopotential force results in a non-ideal equation of state (EOS) which permits liquid-vapor phase change. To account for thermal effects, the athermal LBM is coupled to a finite volume discretization of the temperature evolution equation obtained using a thermal energy rate balance for the specific internal energy. We use the developed model to probe the effect of surface structure and surface wettability on the condensation rate in order to identify microstructure topographies promoting condensation. Financial support is acknowledged from Kimberly-Clark.
Distribution Learning in Evolutionary Strategies and Restricted Boltzmann Machines
DEFF Research Database (Denmark)
Krause, Oswin
The thesis is concerned with learning distributions in the two settings of Evolutionary Strategies (ESs) and Restricted Boltzmann Machines (RBMs). In both cases, the distributions are learned from samples, albeit with different goals. Evolutionary Strategies are concerned with finding an optimum...... of an objective function for which the gradient is not available. The algorithm samples function values from a search distribution and adapts the parameters of the distribution during the optimization process. In the thesis, new update schemes for the covariance matrix used by the CMA-ES are investigated....... An update rule using a triangular Cholesky factor is introduced and the additive covariance matrix update is replaced by a multiplicative rule. Experiments show that the proposed methods improve performance of the CMA-ES either computationally or by allowing simpler handling of constraints. The second part...
Exact results for the Boltzmann equation and Smoluchowski's coagulation equation
International Nuclear Information System (INIS)
Hendriks, E.M.
1983-01-01
Almost no analytical solutions have been found for realistic intermolecular forces, largely due to the complicated structure of the collision term which calls for the construction of simplified models, in which as many physical properties are maintained as possible. In the first three chapters of this thesis such model Boltzmann equations are studied. Only spatially homogeneous gases with isotropic distribution functions are considered. Chapter I considers transition kernels, chapter II persistent scattering models and chapter III very hard particles. The second part of this dissertation deals with Smoluchowski's coagulation equation for the size distribution function in a coagulating system, with chapters devoted to the following topics: kinetics of gelation and universality, coagulation equations with gelation and exactly soluble models of nucleation. (Auth./C.F.)
Spreading Dynamics of Nanodrops: a Lattice Boltzmann Study
Gross, Markus; Varnik, Fathollah
2014-01-01
Spreading of nano-droplets is an interesting and technologically relevant phenomenon, where thermal fluctuations lead to unexpected deviations from well-known deterministic laws. Here, we apply the newly developed fluctuating nonideal lattice Boltzmann (LB) method [M. Gross, M. E. Cates, F. Varnik and R. Adhikari, J. Stat. Mech.2011, P03030 (2011)] for the study of this issue. Confirming the predictions of Davidovich and coworkers [Phys. Rev. Lett.95, 244905 (2005)], we provide the first independent evidence for the existence of an asymptotic, self-similar noise-driven spreading regime in both two- (2D) and three-dimensional (3D) geometry. The cross over from the deterministic Tanner's law, where the drop's base radius b grows (in 3D) with time as b t1/10 and the noise dominated regime, where b t1/6 is also observed by tuning the strength of thermal noise.
Lattice Boltzmann model for melting with natural convection
International Nuclear Information System (INIS)
Huber, Christian; Parmigiani, Andrea; Chopard, Bastien; Manga, Michael; Bachmann, Olivier
2008-01-01
We develop a lattice Boltzmann method to couple thermal convection and pure-substance melting. The transition from conduction-dominated heat transfer to fully-developed convection is analyzed and scaling laws and previous numerical results are reproduced by our numerical method. We also investigate the limit in which thermal inertia (high Stefan number) cannot be neglected. We use our results to extend the scaling relations obtained at low Stefan number and establish the correlation between the melting front propagation and the Stefan number for fully-developed convection. We conclude by showing that the model presented here is particularly well-suited to study convection melting in geometrically complex media with many applications in geosciences
Supersymmetric electroweak baryogenesis, nonequilibrium field theory and quantum Boltzmann equations
Riotto, Antonio
1998-01-01
The closed time-path (CPT) formalism is a powerful Green's function formulation to describe nonequilibrium phenomena in field theory and it leads to a complete nonequilibrium quantum kinetic theory. In this paper we make use of the CPT formalism to write down a set of quantum Boltzmann equations describing the local number density asymmetries of the particles involved in supersymmetric electroweak baryogenesis. These diffusion equations automatically and self-consistently incorporate the CP-violating sources which fuel baryogenesis when transport properties allow the CP-violating charges to diffuse in front of the bubble wall separating the broken from the unbroken phase at the electroweak phase transition. This is a significant improvement with respect to recent approaches where the CP-violating sources are inserted by hand into the diffusion equations. Furthermore, the CP-violating sources and the particle number changing interactions manifest ``memory'' effects which are typical of the quantum transp ort t...
Lattice Boltzmann Simulation of Multiple Bubbles Motion under Gravity
Directory of Open Access Journals (Sweden)
Deming Nie
2015-01-01
Full Text Available The motion of multiple bubbles under gravity in two dimensions is numerically studied through the lattice Boltzmann method for the Eotvos number ranging from 1 to 12. Two kinds of initial arrangement are taken into account: vertical and horizontal arrangement. In both cases the effects of Eotvos number on the bubble coalescence and rising velocity are investigated. For the vertical arrangement, it has been found that the coalescence pattern is similar. The first coalescence always takes place between the two uppermost bubbles. And the last coalescence always takes place between the coalesced bubble and the bottommost bubble. For four bubbles in a horizontal arrangement, the outermost bubbles travel into the wake of the middle bubbles in all cases, which allows the bubbles to coalesce. The coalescence pattern is more complex for the case of eight bubbles, which strongly depends on the Eotvos number.
Boundary Slip and Surface Interaction: A Lattice Boltzmann Simulation
International Nuclear Information System (INIS)
Yan-Yan, Chen; Hua-Bing, Li; Hou-Hui, Yi
2008-01-01
The factors affecting slip length in Couette geometry flows are analysed by means of a two-phase mesoscopic lattice Boltzmann model including non-ideal fluid-fluid and fluid-wall interactions. The main factors influencing the boundary slip are the strength of interactions between fluid-fluid and fluid-wall particles. Other factors, such as fluid viscosity, bulk pressure may also change the slip length. We find that boundary slip only occurs under a certain density (bulk pressure). If the density is large enough, the slip length will tend to zero. In our simulations, a low density layer near the wall does not need to be postulated a priori but emerges naturally from the underlying non-ideal mesoscopic dynamics. It is the low density layer that induces the boundary slip. The results may be helpful to understand recent experimental observations on the slippage of micro flows
Multimesh anisotropic adaptivity for the Boltzmann transport equation
International Nuclear Information System (INIS)
Baker, C.M.J.; Buchan, A.G.; Pain, C.C.; Farrell, P.E.; Eaton, M.D.; Warner, P.
2013-01-01
Highlights: ► We solve the Boltzmann transport equation using anisotropically adaptive finite element meshes. ► The finite element mesh is resolved with minimal user input. ► Anisotropic adaptivity uses less elements than adaptive mesh refinement for the same finite element error. ► This paper also demonstrates the use of separate meshes for each energy group within the multigroup discretisation. ► The methods are applied to a range of fixed source and eigenvalue problems. - Abstract: This article presents a new adaptive finite element based method for the solution of the spatial dimensions of the Boltzmann transport equation. The method applies a curvature based error metric to locate the under and over resolved regions of a solution and this, in turn, is used to guide the refinement and coarsening of the spatial mesh. The error metrics and re-meshing procedures are designed such that they enable anisotropic resolution to form in the mesh should it be appropriate to do so. The adaptive mesh enables the appropriate resolution to be applied throughout the whole domain of a problem and so increase the efficiency of the solution procedure. Another new approach is also described that allows independent adaptive meshes to form for each of the energy group fluxes. The use of independent meshes can significantly improve computational efficiency when solving problems where the different group fluxes require high resolution over different regions. The mesh to mesh interpolation is made possible through the use of a ‘supermeshing’ procedure that ensures the conservation of particles when calculating the group to group scattering sources. Finally it is shown how these methods can be incorporated within a solver to resolve both fixed source and eigenvalue problems. A selection of both fixed source and eigenvalue problems are solved in order to demonstrate the capabilities of these methods
Immersed Boundary-Lattice Boltzmann Method Using Two Relaxation Times
Directory of Open Access Journals (Sweden)
Kosuke Hayashi
2012-06-01
Full Text Available An immersed boundary-lattice Boltzmann method (IB-LBM using a two-relaxation time model (TRT is proposed. The collision operator in the lattice Boltzmann equation is modeled using two relaxation times. One of them is used to set the fluid viscosity and the other is for numerical stability and accuracy. A direct-forcing method is utilized for treatment of immersed boundary. A multi-direct forcing method is also implemented to precisely satisfy the boundary conditions at the immersed boundary. Circular Couette flows between a stationary cylinder and a rotating cylinder are simulated for validation of the proposed method. The method is also validated through simulations of circular and spherical falling particles. Effects of the functional forms of the direct-forcing term and the smoothed-delta function, which interpolates the fluid velocity to the immersed boundary and distributes the forcing term to fixed Eulerian grid points, are also examined. As a result, the following conclusions are obtained: (1 the proposed method does not cause non-physical velocity distribution in circular Couette flows even at high relaxation times, whereas the single-relaxation time (SRT model causes a large non-physical velocity distortion at a high relaxation time, (2 the multi-direct forcing reduces the errors in the velocity profile of a circular Couette flow at a high relaxation time, (3 the two-point delta function is better than the four-point delta function at low relaxation times, but worse at high relaxation times, (4 the functional form of the direct-forcing term does not affect predictions, and (5 circular and spherical particles falling in liquids are well predicted by using the proposed method both for two-dimensional and three-dimensional cases.
Conditional High-Order Boltzmann Machines for Supervised Relation Learning.
Huang, Yan; Wang, Wei; Wang, Liang; Tan, Tieniu
2017-09-01
Relation learning is a fundamental problem in many vision tasks. Recently, high-order Boltzmann machine and its variants have shown their great potentials in learning various types of data relation in a range of tasks. But most of these models are learned in an unsupervised way, i.e., without using relation class labels, which are not very discriminative for some challenging tasks, e.g., face verification. In this paper, with the goal to perform supervised relation learning, we introduce relation class labels into conventional high-order multiplicative interactions with pairwise input samples, and propose a conditional high-order Boltzmann Machine (CHBM), which can learn to classify the data relation in a binary classification way. To be able to deal with more complex data relation, we develop two improved variants of CHBM: 1) latent CHBM, which jointly performs relation feature learning and classification, by using a set of latent variables to block the pathway from pairwise input samples to output relation labels and 2) gated CHBM, which untangles factors of variation in data relation, by exploiting a set of latent variables to multiplicatively gate the classification of CHBM. To reduce the large number of model parameters generated by the multiplicative interactions, we approximately factorize high-order parameter tensors into multiple matrices. Then, we develop efficient supervised learning algorithms, by first pretraining the models using joint likelihood to provide good parameter initialization, and then finetuning them using conditional likelihood to enhance the discriminant ability. We apply the proposed models to a series of tasks including invariant recognition, face verification, and action similarity labeling. Experimental results demonstrate that by exploiting supervised relation labels, our models can greatly improve the performance.
Lattice Boltzmann formulation for conjugate heat transfer in heterogeneous media.
Karani, Hamid; Huber, Christian
2015-02-01
In this paper, we propose an approach for studying conjugate heat transfer using the lattice Boltzmann method (LBM). The approach is based on reformulating the lattice Boltzmann equation for solving the conservative form of the energy equation. This leads to the appearance of a source term, which introduces the jump conditions at the interface between two phases or components with different thermal properties. The proposed source term formulation conserves conductive and advective heat flux simultaneously, which makes it suitable for modeling conjugate heat transfer in general multiphase or multicomponent systems. The simple implementation of the source term approach avoids any correction of distribution functions neighboring the interface and provides an algorithm that is independent from the topology of the interface. Moreover, our approach is independent of the choice of lattice discretization and can be easily applied to different advection-diffusion LBM solvers. The model is tested against several benchmark problems including steady-state convection-diffusion within two fluid layers with parallel and normal interfaces with respect to the flow direction, unsteady conduction in a three-layer stratified domain, and steady conduction in a two-layer annulus. The LBM results are in excellent agreement with analytical solution. Error analysis shows that our model is first-order accurate in space, but an extension to a second-order scheme is straightforward. We apply our LBM model to heat transfer in a two-component heterogeneous medium with a random microstructure. This example highlights that the method we propose is independent of the topology of interfaces between the different phases and, as such, is ideally suited for complex natural heterogeneous media. We further validate the present LBM formulation with a study of natural convection in a porous enclosure. The results confirm the reliability of the model in simulating complex coupled fluid and thermal dynamics
Direct numerical simulation of turbulent pipe flow using the lattice Boltzmann method
Peng, Cheng; Geneva, Nicholas; Guo, Zhaoli; Wang, Lian-Ping
2018-03-01
In this paper, we present a first direct numerical simulation (DNS) of a turbulent pipe flow using the mesoscopic lattice Boltzmann method (LBM) on both a D3Q19 lattice grid and a D3Q27 lattice grid. DNS of turbulent pipe flows using LBM has never been reported previously, perhaps due to inaccuracy and numerical stability associated with the previous implementations of LBM in the presence of a curved solid surface. In fact, it was even speculated that the D3Q19 lattice might be inappropriate as a DNS tool for turbulent pipe flows. In this paper, we show, through careful implementation, accurate turbulent statistics can be obtained using both D3Q19 and D3Q27 lattice grids. In the simulation with D3Q19 lattice, a few problems related to the numerical stability of the simulation are exposed. Discussions and solutions for those problems are provided. The simulation with D3Q27 lattice, on the other hand, is found to be more stable than its D3Q19 counterpart. The resulting turbulent flow statistics at a friction Reynolds number of Reτ = 180 are compared systematically with both published experimental and other DNS results based on solving the Navier-Stokes equations. The comparisons cover the mean-flow profile, the r.m.s. velocity and vorticity profiles, the mean and r.m.s. pressure profiles, the velocity skewness and flatness, and spatial correlations and energy spectra of velocity and vorticity. Overall, we conclude that both D3Q19 and D3Q27 simulations yield accurate turbulent flow statistics. The use of the D3Q27 lattice is shown to suppress the weak secondary flow pattern in the mean flow due to numerical artifacts.
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Directory of Open Access Journals (Sweden)
Jiang Lei
2015-01-01
Full Text Available Direct numerical simulation (DNS of a round jet in crossflow based on lattice Boltzmann method (LBM is carried out on multi-GPU cluster. Data parallel SIMT (single instruction multiple thread characteristic of GPU matches the parallelism of LBM well, which leads to the high efficiency of GPU on the LBM solver. With present GPU settings (6 Nvidia Tesla K20M, the present DNS simulation can be completed in several hours. A grid system of 1.5 × 108 is adopted and largest jet Reynolds number reaches 3000. The jet-to-free-stream velocity ratio is set as 3.3. The jet is orthogonal to the mainstream flow direction. The validated code shows good agreement with experiments. Vortical structures of CRVP, shear-layer vortices and horseshoe vortices, are presented and analyzed based on velocity fields and vorticity distributions. Turbulent statistical quantities of Reynolds stress are also displayed. Coherent structures are revealed in a very fine resolution based on the second invariant of the velocity gradients.
An introduction to the Boltzmann equation and transport processes in gases
Kremer, Gilberto M; Colton, David
2010-01-01
This book covers classical kinetic theory of gases, presenting basic principles in a self-contained framework and from a more rigorous approach based on the Boltzmann equation. Uses methods in kinetic theory for determining the transport coefficients of gases.
An improved FMM Algorithm of the 3d-linearized Poisson-Boltzmann Equation
Directory of Open Access Journals (Sweden)
Mehrez issa
2015-06-01
Full Text Available This paper presents a new FMM algorithm for the linearized Poisson-Boltzmann equation in three dimensions. The performance of the proposed algorithm is assessed on a example in three dimensions and compared with the direct method. The numerical results show the power of the new method, that allow to achieve the best schemes to reduce the time of the particle interactions, which are based on diagonal form of translation operators for linearized Poisson-Boltzmann equation.
On kinetic Boltzmann equations and related hydrodynamic flows with dry viscosity
Directory of Open Access Journals (Sweden)
Nikolai N. Bogoliubov (Jr.
2007-01-01
Full Text Available A two-component particle model of Boltzmann-Vlasov type kinetic equations in the form of special nonlinear integro-differential hydrodynamic systems on an infinite-dimensional functional manifold is discussed. We show that such systems are naturally connected with the nonlinear kinetic Boltzmann-Vlasov equations for some one-dimensional particle flows with pointwise interaction potential between particles. A new type of hydrodynamic two-component Benney equations is constructed and their Hamiltonian structure is analyzed.
Boţan, Vitalie; Ustach, Vincent D; Leonhard, Kai; Faller, Roland
2017-11-16
The polymer poly(N-isopropylacrylamide) (PNIPAM) is studied using a novel combination of multiscale modeling methodologies. We develop an iterative Boltzmann inversion potential of concentrated PNIPAM solutions and combine it with lattice Boltzmann as a Navier-Stokes equation solver for the solvent. We study in detail the influence of the methodology on statics and dynamics of the system. The combination is successful and significantly simpler and faster than other mapping techniques for polymer solution while keeping the correct hydrodynamics. The model can semiquantitatively describe the correct phase behavior and polymer dynamics.
International Nuclear Information System (INIS)
Gamba, Irene M.; Haack, Jeffrey R.
2014-01-01
We present the formulation of a conservative spectral method for the Boltzmann collision operator with anisotropic scattering cross-sections. The method is an extension of the conservative spectral method of Gamba and Tharkabhushanam [17,18], which uses the weak form of the collision operator to represent the collisional term as a weighted convolution in Fourier space. The method is tested by computing the collision operator with a suitably cut-off angular cross section and comparing the results with the solution of the Landau equation. We analytically study the convergence rate of the Fourier transformed Boltzmann collision operator in the grazing collisions limit to the Fourier transformed Landau collision operator under the assumption of some regularity and decay conditions of the solution to the Boltzmann equation. Our results show that the angular singularity which corresponds to the Rutherford scattering cross section is the critical singularity for which a grazing collision limit exists for the Boltzmann operator. Additionally, we numerically study the differences between homogeneous solutions of the Boltzmann equation with the Rutherford scattering cross section and an artificial cross section, which give convergence to solutions of the Landau equation at different asymptotic rates. We numerically show the rate of the approximation as well as the consequences for the rate of entropy decay for homogeneous solutions of the Boltzmann equation and Landau equation
Statistical mechanics out of equilibrium the irreversibility
International Nuclear Information System (INIS)
Alvarez Estrada, R. F.
2001-01-01
A Round Table about the issue of Irreversibility and related matters has taken place during the last (20th) Statistical Mechanics Conference, held in Paris (July 1998). This article tries to provide a view (necessarily limited, and hence, uncompleted) of some approaches to the subject: the one based upon deterministic chaos (which is currently giving rise to a very active research) and the classical interpretation due to Boltzmann. An attempt has been made to write this article in a self-contained way, and to avoid a technical presentation wherever possible. (Author) 29 refs
Lattice Boltzmann heat transfer model for permeable voxels
Pereira, Gerald G.; Wu, Bisheng; Ahmed, Shakil
2017-12-01
We develop a gray-scale lattice Boltzmann (LB) model to study fluid flow combined with heat transfer for flow through porous media where voxels may be partially solid (or void). Heat transfer in rocks may lead to deformation, which in turn can modulate the fluid flow and so has significant contribution to rock permeability. The LB temperature field is compared to a finite difference solution of the continuum partial differential equations for fluid flow in a channel. Excellent quantitative agreement is found for both Poiseuille channel flow and Brinkman flow. The LB model is then applied to sample porous media such as packed beds and also more realistic sandstone rock sample, and both the convective and diffusive regimes are recovered when varying the thermal diffusivity. It is found that while the rock permeability can be comparatively small (order milli-Darcy), the temperature field can show significant variation depending on the thermal convection of the fluid. This LB method has significant advantages over other numerical methods such as finite and boundary element methods in dealing with coupled fluid flow and heat transfer in rocks which have irregular and nonsmooth pore spaces.
Entropic Lattice Boltzmann: an implicit Large-Eddy Simulation?
Tauzin, Guillaume; Biferale, Luca; Sbragaglia, Mauro; Gupta, Abhineet; Toschi, Federico; Ehrhardt, Matthias; Bartel, Andreas
2017-11-01
We study the modeling of turbulence implied by the unconditionally stable Entropic Lattice Boltzmann Method (ELBM). We first focus on 2D homogeneous turbulence, for which we conduct numerical simulations for a wide range of relaxation times τ. For these simulations, we analyze the effective viscosity obtained by numerically differentiating the kinetic energy and enstrophy balance equations averaged over sub-domains of the computational grid. We aim at understanding the behavior of the implied sub-grid scale model and verify a formulation previously derived using Chapman-Enskog expansion. These ELBM benchmark simulations are thus useful to understand the range of validity of ELBM as a turbulence model. Finally, we will discuss an extension of the previously obtained results to the 3D case. Supported by the European Unions Framework Programme for Research and Innovation Horizon 2020 (2014-2020) under the Marie Sklodowska-Curie Grant Agreement No. 642069 and by the European Research Council under the ERC Grant Agreement No. 339032.
From Lattice Boltzmann to hydrodynamics in dissipative relativistic fluids
Gabbana, Alessandro; Mendoza, Miller; Succi, Sauro; Tripiccione, Raffaele
2017-11-01
Relativistic fluid dynamics is currently applied to several fields of modern physics, covering many physical scales, from astrophysics, to atomic scales (e.g. in the study of effective 2D systems such as graphene) and further down to subnuclear scales (e.g. quark-gluon plasmas). This talk focuses on recent progress in the largely debated connection between kinetic transport coefficients and macroscopic hydrodynamic parameters in dissipative relativistic fluid dynamics. We use a new relativistic Lattice Boltzmann method (RLBM), able to handle from ultra-relativistic to almost non-relativistic flows, and obtain strong evidence that the Chapman-Enskog expansion provides the correct pathway from kinetic theory to hydrodynamics. This analysis confirms recently obtained theoretical results, which can be used to obtain accurate calibrations for RLBM methods applied to realistic physics systems in the relativistic regime. Using this calibration methodology, RLBM methods are able to deliver improved physical accuracy in the simulation of the physical systems described above. European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No. 642069.
A dynamically adaptive lattice Boltzmann method for thermal convection problems
Directory of Open Access Journals (Sweden)
Feldhusen Kai
2016-12-01
Full Text Available Utilizing the Boussinesq approximation, a double-population incompressible thermal lattice Boltzmann method (LBM for forced and natural convection in two and three space dimensions is developed and validated. A block-structured dynamic adaptive mesh refinement (AMR procedure tailored for the LBM is applied to enable computationally efficient simulations of moderate to high Rayleigh number flows which are characterized by a large scale disparity in boundary layers and free stream flow. As test cases, the analytically accessible problem of a two-dimensional (2D forced convection flow through two porous plates and the non-Cartesian configuration of a heated rotating cylinder are considered. The objective of the latter is to advance the boundary conditions for an accurate treatment of curved boundaries and to demonstrate the effect on the solution. The effectiveness of the overall approach is demonstrated for the natural convection benchmark of a 2D cavity with differentially heated walls at Rayleigh numbers from 103 up to 108. To demonstrate the benefit of the employed AMR procedure for three-dimensional (3D problems, results from the natural convection in a cubic cavity at Rayleigh numbers from 103 up to 105 are compared with benchmark results.
Corner-transport-upwind lattice Boltzmann model for bubble cavitation
Sofonea, V.; Biciuşcǎ, T.; Busuioc, S.; Ambruş, Victor E.; Gonnella, G.; Lamura, A.
2018-02-01
Aiming to study the bubble cavitation problem in quiescent and sheared liquids, a third-order isothermal lattice Boltzmann model that describes a two-dimensional (2D) fluid obeying the van der Waals equation of state, is introduced. The evolution equations for the distribution functions in this off-lattice model with 16 velocities are solved using the corner-transport-upwind (CTU) numerical scheme on large square lattices (up to 6144 ×6144 nodes). The numerical viscosity and the regularization of the model are discussed for first- and second-order CTU schemes finding that the latter choice allows to obtain a very accurate phase diagram of a nonideal fluid. In a quiescent liquid, the present model allows us to recover the solution of the 2D Rayleigh-Plesset equation for a growing vapor bubble. In a sheared liquid, we investigated the evolution of the total bubble area, the bubble deformation, and the bubble tilt angle, for various values of the shear rate. A linear relation between the dimensionless deformation coefficient D and the capillary number Ca is found at small Ca but with a different factor than in equilibrium liquids. A nonlinear regime is observed for Ca≳0.2 .
Axisymmetric Lattice Boltzmann Model of Droplet Impact on Solid Surfaces
Dalgamoni, Hussein; Yong, Xin
2017-11-01
Droplet impact is a ubiquitous fluid phenomena encountered in scientific and engineering applications such as ink-jet printing, coating, electronics manufacturing, and many others. It is of great technological importance to understand the detailed dynamics of drop impact on various surfaces. The lattice Boltzmann method (LBM) emerges as an efficient method for modeling complex fluid systems involving rapidly evolving fluid-fluid and fluid-solid interfaces with complex geometries. In this work, we model droplet impact on flat solid substrates with well-defined wetting behavior using a two-phase axisymmetric LBM with high density and viscosity contrasts. We extend the two-dimensional Lee and Liu model to capture axisymmetric effect in the normal impact. First we compare the 2D axisymmetric results with the 2D and 3D results reported by Lee and Liu to probe the effect of axisymmetric terms. Then, we explore the effects of Weber number, Ohnesorge number, and droplet-surface equilibrium contact angle on the impact. The dynamic contact angle and spreading factor of the droplet during impact are investigated to qualitatively characterize the impact dynamics.
Equivalence of restricted Boltzmann machines and tensor network states
Chen, Jing; Cheng, Song; Xie, Haidong; Wang, Lei; Xiang, Tao
2018-02-01
The restricted Boltzmann machine (RBM) is one of the fundamental building blocks of deep learning. RBM finds wide applications in dimensional reduction, feature extraction, and recommender systems via modeling the probability distributions of a variety of input data including natural images, speech signals, and customer ratings, etc. We build a bridge between RBM and tensor network states (TNS) widely used in quantum many-body physics research. We devise efficient algorithms to translate an RBM into the commonly used TNS. Conversely, we give sufficient and necessary conditions to determine whether a TNS can be transformed into an RBM of given architectures. Revealing these general and constructive connections can cross fertilize both deep learning and quantum many-body physics. Notably, by exploiting the entanglement entropy bound of TNS, we can rigorously quantify the expressive power of RBM on complex data sets. Insights into TNS and its entanglement capacity can guide the design of more powerful deep learning architectures. On the other hand, RBM can represent quantum many-body states with fewer parameters compared to TNS, which may allow more efficient classical simulations.
Stable lattice Boltzmann model for Maxwell equations in media
Hauser, A.; Verhey, J. L.
2017-12-01
The present work shows a method for stable simulations via the lattice Boltzmann (LB) model for electromagnetic waves (EM) transiting homogeneous media. LB models for such media were already presented in the literature, but they suffer from numerical instability when the media transitions are sharp. We use one of these models in the limit of pure vacuum derived from Liu and Yan [Appl. Math. Model. 38, 1710 (2014), 10.1016/j.apm.2013.09.009] and apply an extension that treats the effects of polarization and magnetization separately. We show simulations of simple examples in which EM waves travel into media to quantify error scaling, stability, accuracy, and time scaling. For conductive media, we use the Strang splitting and check the simulations accuracy at the example of the skin effect. Like pure EM propagation, the error for the static limits, which are constructed with a current density added in a first-order scheme, can be less than 1 % . The presented method is an easily implemented alternative for the stabilization of simulation for EM waves propagating in spatially complex structured media properties and arbitrary transitions.
Lattice Boltzmann study of droplet motion inside a grooved channel
Huang, Jun Jie; Shu, Chang; Chew, Yong Tian
2009-02-01
A droplet moving inside a grooved channel is studied by using a new lattice Boltzmann model for multiphase flows with large density ratio. A constant body force is applied to drive the droplet. Flows under different surface tensions, driving forces, density ratios, wall wettabilities, and groove geometries are investigated. It is found that the drag on the droplet and the flow pattern are strongly affected by the wall wettability and topography when the system scale is small. The effects of the driving force on the droplet are investigated through comparison of two different ways of applying it. Besides, the density ratio is varied over a wide range to assess its effects in the present setup. Special attention is paid to grooved hydrophilic walls which tend to enhance the droplet-wall contact. For such walls, two distinctive types of shape of the interface inside the groove are found and series of numerical investigations are carried out to find the critical wall contact angle, groove width and depth that determine which kind of shape the droplet assumes. Some typical cases are chosen for detailed analyses and compared to some other work. This study is expected to improve our understanding on the lotus effect and the physics of small scale flows near rough walls.
Multiple-relaxation-time lattice Boltzmann model for compressible fluids
International Nuclear Information System (INIS)
Chen Feng; Xu Aiguo; Zhang Guangcai; Li Yingjun
2011-01-01
We present an energy-conserving multiple-relaxation-time finite difference lattice Boltzmann model for compressible flows. The collision step is first calculated in the moment space and then mapped back to the velocity space. The moment space and corresponding transformation matrix are constructed according to the group representation theory. Equilibria of the nonconserved moments are chosen according to the need of recovering compressible Navier-Stokes equations through the Chapman-Enskog expansion. Numerical experiments showed that compressible flows with strong shocks can be well simulated by the present model. The new model works for both low and high speeds compressible flows. It contains more physical information and has better numerical stability and accuracy than its single-relaxation-time version. - Highlights: → We present an energy-conserving MRT finite-difference LB model. → The moment space is constructed according to the group representation theory. → The new model works for both low and high speeds compressible flows. → It has better numerical stability and wider applicable range than its SRT version.
Lattice-Boltzmann Modeling of Interfacial Dynamics in Porous Media
Porter, M. L.; Coon, E. T.; Kang, Q.; Carey, J. W.
2012-12-01
Traditional continuum scale multiphase flow models rely heavily on average properties and constitutive relationships that do not always accurately represent the underlying physics affecting flow and transport at the pore scale. These models are typically based on heuristic extensions of Darcy's law, rather than formally upscaling conservation principles that account for the microscale physics. As a result, constitutive relationships, such as capillary pressure and relative permeability, are highly simplified. It has been recognized that continuum scale multiphase flow models must include gradients of saturation and specific fluid-fluid interfacial area, in addition to the Darcy pressure gradient, as driving forces for the flow of multiple fluids in porous media. In this work, we investigate interfacial dynamics in porous media using a multicomponent lattice-Boltzmann simulator. We present simulations of drainage and imbibition in 2D and 3D heterogeneous porous media. We validate the simulations by comparing specific interfacial area estimates with those obtained from experiments. In addition, we present estimates of continuum scale interfacial velocity and the production/destruction of specific interfacial area.
Lattice Boltzmann Simulation Optimization on Leading Multicore Platforms
Energy Technology Data Exchange (ETDEWEB)
Williams, Samuel; Carter, Jonathan; Oliker, Leonid; Shalf, John; Yelick, Katherine
2008-02-01
We present an auto-tuning approach to optimize application performance on emerging multicore architectures. The methodology extends the idea of search-based performance optimizations, popular in linear algebra and FFT libraries, to application-specific computational kernels. Our work applies this strategy to a lattice Boltzmann application (LBMHD) that historically has made poor use of scalar microprocessors due to its complex data structures and memory access patterns. We explore one of the broadest sets of multicore architectures in the HPC literature, including the Intel Clovertown, AMD Opteron X2, Sun Niagara2, STI Cell, as well as the single core Intel Itanium2. Rather than hand-tuning LBMHD for each system, we develop a code generator that allows us identify a highly optimized version for each platform, while amortizing the human programming effort. Results show that our auto-tuned LBMHD application achieves up to a 14x improvement compared with the original code. Additionally, we present detailed analysis of each optimization, which reveal surprising hardware bottlenecks and software challenges for future multicore systems and applications.
Lattice Boltzmann simulation optimization on leading multicore platforms
Energy Technology Data Exchange (ETDEWEB)
Williams, S. [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Univ. of California, Berkeley, CA (United States); Carter, J. [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Oliker, L. [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Shalf, J. [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Yelick, K. [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Univ. of California, Berkeley, CA (United States)
2008-01-01
We present an auto-tuning approach to optimize application performance on emerging multicore architectures. The methodology extends the idea of searchbased performance optimizations, popular in linear algebra and FFT libraries, to application-specific computational kernels. Our work applies this strategy to a lattice Boltzmann application (LBMHD) that historically has made poor use of scalar microprocessors due to its complex data structures and memory access patterns. We explore one of the broadest sets of multicore architectures in the HPC literature, including the Intel Clovertown, AMD Opteron X2, Sun Niagara2, STI Cell, as well as the single core Intel Itanium2. Rather than hand-tuning LBMHD for each system, we develop a code generator that allows us identify a highly optimized version for each platform, while amortizing the human programming effort. Results show that our autotuned LBMHD application achieves up to a 14 improvement compared with the original code. Additionally, we present detailed analysis of each optimization, which reveal surprising hardware bottlenecks and software challenges for future multicore systems and applications.
Lattice-Boltzmann simulations in reconstructed parametrized porous media
Ahrenholz, Benjamin; Tölke, Jonas; Krafczyk, Manfred
2006-07-01
Computations of flows in explicitly resolved porous media reported in the literature so far are based on binarized porous media data mapped to uniform Cartesian grids. The voxel set is directly being used as the computational grid and thus the geometrical representation is usually only first-order accurate due to stair-case patterns. In this work, we pursue a more elaborate approach: starting from a highly resolved tomographic grey value data set we utilize a Marching Cube algorithm to reconstruct the surface of the porous medium as a set of planar triangles. The numerical resolution of the Cartesian grid for the simulation can then be chosen independently from the voxel set. As we take into account the subgrid distances between the nodes of the Cartesian grid and the planar triangle surfaces, one can utilize a second-order accurate lattice Boltzmann flow solver to efficiently compute, e.g. permeabilities. As these interpolation-based no-slip boundary conditions are not mass preserving, we also present a local modification of the no-slip boundary condition restoring mass conservation. Our numerical results demonstrate that for saturated flow simulations this coupled approach allows a substantial acceleration of saturated flow computations in porous media.
Massively parallel simulations of multiphase flows using Lattice Boltzmann methods
Ahrenholz, Benjamin
2010-03-01
In the last two decades the lattice Boltzmann method (LBM) has matured as an alternative and efficient numerical scheme for the simulation of fluid flows and transport problems. Unlike conventional numerical schemes based on discretizations of macroscopic continuum equations, the LBM is based on microscopic models and mesoscopic kinetic equations. The fundamental idea of the LBM is to construct simplified kinetic models that incorporate the essential physics of microscopic or mesoscopic processes so that the macroscopic averaged properties obey the desired macroscopic equations. Especially applications involving interfacial dynamics, complex and/or changing boundaries and complicated constitutive relationships which can be derived from a microscopic picture are suitable for the LBM. In this talk a modified and optimized version of a Gunstensen color model is presented to describe the dynamics of the fluid/fluid interface where the flow field is based on a multi-relaxation-time model. Based on that modeling approach validation studies of contact line motion are shown. Due to the fact that the LB method generally needs only nearest neighbor information, the algorithm is an ideal candidate for parallelization. Hence, it is possible to perform efficient simulations in complex geometries at a large scale by massively parallel computations. Here, the results of drainage and imbibition (Degree of Freedom > 2E11) in natural porous media gained from microtomography methods are presented. Those fully resolved pore scale simulations are essential for a better understanding of the physical processes in porous media and therefore important for the determination of constitutive relationships.
Statistical mechanics and the description of the early universe I
DEFF Research Database (Denmark)
Pessah, Martin Elias; F. Torres, Diego; Vucetich, H.
2001-01-01
We analyze how the thermal history of the universe is influenced by the statistical description, assuming a deviation from the usual Bose-Einstein, Fermi-Dirac and Boltzmann-Gibbs distribution functions. These deviations represent the possible appearance of non-extensive effects related with the ......We analyze how the thermal history of the universe is influenced by the statistical description, assuming a deviation from the usual Bose-Einstein, Fermi-Dirac and Boltzmann-Gibbs distribution functions. These deviations represent the possible appearance of non-extensive effects related...... with the existence of long range forces, memory effects, or evolution in fractal or multi-fractal space. In the early universe, it is usually assumed that the distribution functions are the standard ones. Then, considering the evolution in a larger theoretical framework will allow to test this assumption...
Pestman, Wiebe R
2009-01-01
This textbook provides a broad and solid introduction to mathematical statistics, including the classical subjects hypothesis testing, normal regression analysis, and normal analysis of variance. In addition, non-parametric statistics and vectorial statistics are considered, as well as applications of stochastic analysis in modern statistics, e.g., Kolmogorov-Smirnov testing, smoothing techniques, robustness and density estimation. For students with some elementary mathematical background. With many exercises. Prerequisites from measure theory and linear algebra are presented.
Statistical mechanics in the context of special relativity. II.
Kaniadakis, G
2005-09-01
The special relativity laws emerge as one-parameter (light speed) generalizations of the corresponding laws of classical physics. These generalizations, imposed by the Lorentz transformations, affect both the definition of the various physical observables (e.g., momentum, energy, etc.), as well as the mathematical apparatus of the theory. Here, following the general lines of [Phys. Rev. E 66, 056125 (2002)], we show that the Lorentz transformations impose also a proper one-parameter generalization of the classical Boltzmann-Gibbs-Shannon entropy. The obtained relativistic entropy permits us to construct a coherent and self-consistent relativistic statistical theory, preserving the main features of the ordinary statistical theory, which is recovered in the classical limit. The predicted distribution function is a one-parameter continuous deformation of the classical Maxwell-Boltzmann distribution and has a simple analytic form, showing power law tails in accordance with the experimental evidence. Furthermore, this statistical mechanics can be obtained as the stationary case of a generalized kinetic theory governed by an evolution equation obeying the H theorem and reproducing the Boltzmann equation of the ordinary kinetics in the classical limit.
Dynamic permeability of porous media by the lattice Boltzmann method
Adler, P.; Pazdniakou, A.
2012-04-01
The main objective of our work is to determine the dynamic permeability of three dimensional porous media by means of the Lattice Boltzmann method (LBM). The Navier-Stokes equation can be numerically solved by LBM which is widely used to address various fluid dynamics problems. Space is discretized by a three-dimensional cubic lattice and time is discretized as well. The generally accepted notation for lattice Boltzmann models is DdQq where D stands for space dimension and Q for the number of discrete velocities. The present model is denoted by D3Q19. Moreover, the Two Relaxation Times variant of the Multi Relaxation Times model is implemented. Bounce back boundary conditions are used on the solid-fluid interfaces. The porous medium is spatially periodic. Reconstructed media were used; they are obtained by imposing a porosity and a correlation function characterized by a correlation length. Real samples can be obtained by MicroCT. In contrast with other previous contributions, the dynamic permeability K(omega) which is a complex number, is derived by imposing an oscillating body force of pulsation omega on the unit cell and by deriving the amplitude and the phase shift of the resulting time dependent seepage velocity. The influence of two limiting parameters, namely the Knudsen number Kn and the discretization for high frequencies, on K(omega) is carefully studied for the first time. Kn is proportional to nu/(cs H) where nu is the kinematic viscosity, cs the speed of sound in the fluid and H a characteristic length scale of the porous medium. Several porous media such as the classical plane Poiseuille flow and the reconstructed media are used to show that it is only for small enough values of Kn that reliable results are obtained. Otherwise, the data depend on Kn and may even be totally unphysical. However, it should be noticed that the limiting value of Kn could not be derived in general since it depends very much on the structure of the medium. Problems occur at
Multicomponent gas mixture air bearing modeling via lattice Boltzmann method
Tae Kim, Woo; Kim, Dehee; Hari Vemuri, Sesha; Kang, Soo-Choon; Seung Chung, Pil; Jhon, Myung S.
2011-04-01
As the demand for ultrahigh recording density increases, development of an integrated head disk interface (HDI) modeling tool, which considers the air bearing and lubricant film morphology simultaneously is of paramount importance. To overcome the shortcomings of the existing models based on the modified Reynolds equation (MRE), the lattice Boltzmann method (LBM) is a natural choice in modeling high Knudsen number (Kn) flows owing to its advantages over conventional methods. The transient and parallel nature makes this LBM an attractive tool for the next generation air bearing design. Although LBM has been successfully applied to single component systems, a multicomponent system analysis has been thwarted because of the complexity in coupling the terms for each component. Previous studies have shown good results in modeling immiscible component mixtures by use of an interparticle potential. In this paper, we extend our LBM model to predict the flow rate of high Kn pressure-driven flows in multicomponent gas mixture air bearings, such as the air-helium system. For accurate modeling of slip conditions near the wall, we adopt our LBM scheme with spatially dependent relaxation times for air bearings in HDIs. To verify the accuracy of our code, we tested our scheme via simple two-dimensional benchmark flows. In the pressure-driven flow of an air-helium mixture, we found that the simple linear combination of pure helium and pure air flow rates, based on helium and air mole fraction, gives considerable error when compared to our LBM calculation. Hybridization with the existing MRE database can be adopted with the procedure reported here to develop the state-of-the-art slider design software.
Peristaltic particle transport using the Lattice Boltzmann method
Energy Technology Data Exchange (ETDEWEB)
Connington, Kevin William [Los Alamos National Laboratory; Kang, Qinjun [Los Alamos National Laboratory; Viswanathan, Hari S [Los Alamos National Laboratory; Abdel-fattah, Amr [Los Alamos National Laboratory; Chen, Shiyi [JOHNS HOPKINS UNIV.
2009-01-01
Peristaltic transport refers to a class of internal fluid flows where the periodic deformation of flexible containing walls elicits a non-negligible fluid motion. It is a mechanism used to transport fluid and immersed solid particles in a tube or channel when it is ineffective or impossible to impose a favorable pressure gradient or desirous to avoid contact between the transported mixture and mechanical moving parts. Peristaltic transport occurs in many physiological situations and has myriad industrial applications. We focus our study on the peristaltic transport of a macroscopic particle in a two-dimensional channel using the lattice Boltzmann method. We systematically investigate the effect of variation of the relevant dimensionless parameters of the system on the particle transport. We find, among other results, a case where an increase in Reynolds number can actually lead to a slight increase in particle transport, and a case where, as the wall deformation increases, the motion of the particle becomes non-negative only. We examine the particle behavior when the system exhibits the peculiar phenomenon of fluid trapping. Under these circumstances, the particle may itself become trapped where it is subsequently transported at the wave speed, which is the maximum possible transport in the absence of a favorable pressure gradient. Finally, we analyze how the particle presence affects stress, pressure, and dissipation in the fluid in hopes of determining preferred working conditions for peristaltic transport of shear-sensitive particles. We find that the levels of shear stress are most hazardous near the throat of the channel. We advise that shear-sensitive particles should be transported under conditions where trapping occurs as the particle is typically situated in a region of innocuous shear stress levels.
Polar-coordinate lattice Boltzmann modeling of compressible flows
Lin, Chuandong; Xu, Aiguo; Zhang, Guangcai; Li, Yingjun; Succi, Sauro
2014-01-01
We present a polar coordinate lattice Boltzmann kinetic model for compressible flows. A method to recover the continuum distribution function from the discrete distribution function is indicated. Within the model, a hybrid scheme being similar to, but different from, the operator splitting is proposed. The temporal evolution is calculated analytically, and the convection term is solved via a modified Warming-Beam (MWB) scheme. Within the MWB scheme a suitable switch function is introduced. The current model works not only for subsonic flows but also for supersonic flows. It is validated and verified via the following well-known benchmark tests: (i) the rotational flow, (ii) the stable shock tube problem, (iii) the Richtmyer-Meshkov (RM) instability, and (iv) the Kelvin-Helmholtz instability. As an original application, we studied the nonequilibrium characteristics of the system around three kinds of interfaces, the shock wave, the rarefaction wave, and the material interface, for two specific cases. In one of the two cases, the material interface is initially perturbed, and consequently the RM instability occurs. It is found that the macroscopic effects due to deviating from thermodynamic equilibrium around the material interface differ significantly from those around the mechanical interfaces. The initial perturbation at the material interface enhances the coupling of molecular motions in different degrees of freedom. The amplitude of deviation from thermodynamic equilibrium around the shock wave is much higher than those around the rarefaction wave and material interface. By comparing each component of the high-order moments and its value in equilibrium, we can draw qualitatively the main behavior of the actual distribution function. These results deepen our understanding of the mechanical and material interfaces from a more fundamental level, which is indicative for constructing macroscopic models and other kinds of kinetic models.
Implementing the lattice Boltzmann model on commodity graphics hardware
International Nuclear Information System (INIS)
Kaufman, Arie; Fan, Zhe; Petkov, Kaloian
2009-01-01
Modern graphics processing units (GPUs) can perform general-purpose computations in addition to the native specialized graphics operations. Due to the highly parallel nature of graphics processing, the GPU has evolved into a many-core coprocessor that supports high data parallelism. Its performance has been growing at a rate of squared Moore's law, and its peak floating point performance exceeds that of the CPU by an order of magnitude. Therefore, it is a viable platform for time-sensitive and computationally intensive applications. The lattice Boltzmann model (LBM) computations are carried out via linear operations at discrete lattice sites, which can be implemented efficiently using a GPU-based architecture. Our simulations produce results comparable to the CPU version while improving performance by an order of magnitude. We have demonstrated that the GPU is well suited for interactive simulations in many applications, including simulating fire, smoke, lightweight objects in wind, jellyfish swimming in water, and heat shimmering and mirage (using the hybrid thermal LBM). We further advocate the use of a GPU cluster for large scale LBM simulations and for high performance computing. The Stony Brook Visual Computing Cluster has been the platform for several applications, including simulations of real-time plume dispersion in complex urban environments and thermal fluid dynamics in a pressurized water reactor. Major GPU vendors have been targeting the high performance computing market with GPU hardware implementations. Software toolkits such as NVIDIA CUDA provide a convenient development platform that abstracts the GPU and allows access to its underlying stream computing architecture. However, software programming for a GPU cluster remains a challenging task. We have therefore developed the Zippy framework to simplify GPU cluster programming. Zippy is based on global arrays combined with the stream programming model and it hides the low-level details of the
Eliazar, Iddo
2017-05-01
The exponential, the normal, and the Poisson statistical laws are of major importance due to their universality. Harmonic statistics are as universal as the three aforementioned laws, but yet they fall short in their 'public relations' for the following reason: the full scope of harmonic statistics cannot be described in terms of a statistical law. In this paper we describe harmonic statistics, in their full scope, via an object termed harmonic Poisson process: a Poisson process, over the positive half-line, with a harmonic intensity. The paper reviews the harmonic Poisson process, investigates its properties, and presents the connections of this object to an assortment of topics: uniform statistics, scale invariance, random multiplicative perturbations, Pareto and inverse-Pareto statistics, exponential growth and exponential decay, power-law renormalization, convergence and domains of attraction, the Langevin equation, diffusions, Benford's law, and 1/f noise.
A lattice Boltzmann model for solute transport in open channel flow
Wang, Hongda; Cater, John; Liu, Haifei; Ding, Xiangyi; Huang, Wei
2018-01-01
A lattice Boltzmann model of advection-dispersion problems in one-dimensional (1D) open channel flows is developed for simulation of solute transport and pollutant concentration. The hydrodynamics are calculated based on a previous lattice Boltzmann approach to solving the 1D Saint-Venant equations (LABSVE). The advection-dispersion model is coupled with the LABSVE using the lattice Boltzmann method. Our research recovers the advection-dispersion equations through the Chapman-Enskog expansion of the lattice Boltzmann equation. The model differs from the existing schemes in two points: (1) the lattice Boltzmann numerical method is adopted to solve the advection-dispersion problem by meso-scopic particle distribution; (2) and the model describes the relation between discharge, cross section area and solute concentration, which increases the applicability of the water quality model in practical engineering. The model is verified using three benchmark tests: (1) instantaneous solute transport within a short distance; (2) 1D point source pollution with constant velocity; (3) 1D point source pollution in a dam break flow. The model is then applied to a 50-year flood point source pollution accident on the Yongding River, which showed good agreement with a MIKE 11 solution and gauging data.
Sadovskii, Michael V
2012-01-01
This volume provides a compact presentation of modern statistical physics at an advanced level. Beginning with questions on the foundations of statistical mechanics all important aspects of statistical physics are included, such as applications to ideal gases, the theory of quantum liquids and superconductivity and the modern theory of critical phenomena. Beyond that attention is given to new approaches, such as quantum field theory methods and non-equilibrium problems.
Energy Technology Data Exchange (ETDEWEB)
Eliazar, Iddo, E-mail: eliazar@post.tau.ac.il
2017-05-15
The exponential, the normal, and the Poisson statistical laws are of major importance due to their universality. Harmonic statistics are as universal as the three aforementioned laws, but yet they fall short in their ‘public relations’ for the following reason: the full scope of harmonic statistics cannot be described in terms of a statistical law. In this paper we describe harmonic statistics, in their full scope, via an object termed harmonic Poisson process: a Poisson process, over the positive half-line, with a harmonic intensity. The paper reviews the harmonic Poisson process, investigates its properties, and presents the connections of this object to an assortment of topics: uniform statistics, scale invariance, random multiplicative perturbations, Pareto and inverse-Pareto statistics, exponential growth and exponential decay, power-law renormalization, convergence and domains of attraction, the Langevin equation, diffusions, Benford’s law, and 1/f noise. - Highlights: • Harmonic statistics are described and reviewed in detail. • Connections to various statistical laws are established. • Connections to perturbation, renormalization and dynamics are established.
Goodman, Joseph W
2015-01-01
This book discusses statistical methods that are useful for treating problems in modern optics, and the application of these methods to solving a variety of such problems This book covers a variety of statistical problems in optics, including both theory and applications. The text covers the necessary background in statistics, statistical properties of light waves of various types, the theory of partial coherence and its applications, imaging with partially coherent light, atmospheric degradations of images, and noise limitations in the detection of light. New topics have been introduced i
Szulc, Stefan
1965-01-01
Statistical Methods provides a discussion of the principles of the organization and technique of research, with emphasis on its application to the problems in social statistics. This book discusses branch statistics, which aims to develop practical ways of collecting and processing numerical data and to adapt general statistical methods to the objectives in a given field.Organized into five parts encompassing 22 chapters, this book begins with an overview of how to organize the collection of such information on individual units, primarily as accomplished by government agencies. This text then
Boltzmann-Fokker-Planck calculations using standard discrete-ordinates codes
International Nuclear Information System (INIS)
Morel, J.E.
1987-01-01
The Boltzmann-Fokker-Planck (BFP) equation can be used to describe both neutral and charged-particle transport. Over the past several years, the author and several collaborators have developed methods for representing Fokker-Planck operators with standard multigroup-Legendre cross-section data. When these data are input to a standard S/sub n/ code such as ONETRAN, the code actually solves the Boltzmann-Fokker-Planck equation rather than the Boltzmann equation. This is achieved wihout any modification to the S/sub n/ codes. Because BFP calculations can be more demanding from a numerical viewpoint than standard neutronics calculations, we have found it useful to implement new quadrature methods ad convergence acceleration methods in the standard discrete-ordinates code, ONETRAN. We discuss our BFP cross-section representation techniques, our improved quadrature and acceleration techniques, and present results from BFP coupled electron-photon transport calculations performed with ONETRAN. 19 refs., 7 figs
PB-AM: An open-source, fully analytical linear poisson-boltzmann solver
Energy Technology Data Exchange (ETDEWEB)
Felberg, Lisa E. [Department of Chemical and Biomolecular Engineering, University of California Berkeley, Berkeley California 94720; Brookes, David H. [Department of Chemistry, University of California Berkeley, Berkeley California 94720; Yap, Eng-Hui [Department of Systems and Computational Biology, Albert Einstein College of Medicine, Bronx New York 10461; Jurrus, Elizabeth [Division of Computational and Statistical Analytics, Pacific Northwest National Laboratory, Richland Washington 99352; Scientific Computing and Imaging Institute, University of Utah, Salt Lake City Utah 84112; Baker, Nathan A. [Advanced Computing, Mathematics, and Data Division, Pacific Northwest National Laboratory, Richland Washington 99352; Division of Applied Mathematics, Brown University, Providence Rhode Island 02912; Head-Gordon, Teresa [Department of Chemical and Biomolecular Engineering, University of California Berkeley, Berkeley California 94720; Department of Chemistry, University of California Berkeley, Berkeley California 94720; Department of Bioengineering, University of California Berkeley, Berkeley California 94720; Chemical Sciences Division, Lawrence Berkeley National Labs, Berkeley California 94720
2016-11-02
We present the open source distributed software package Poisson-Boltzmann Analytical Method (PB-AM), a fully analytical solution to the linearized Poisson Boltzmann equation. The PB-AM software package includes the generation of outputs files appropriate for visualization using VMD, a Brownian dynamics scheme that uses periodic boundary conditions to simulate dynamics, the ability to specify docking criteria, and offers two different kinetics schemes to evaluate biomolecular association rate constants. Given that PB-AM defines mutual polarization completely and accurately, it can be refactored as a many-body expansion to explore 2- and 3-body polarization. Additionally, the software has been integrated into the Adaptive Poisson-Boltzmann Solver (APBS) software package to make it more accessible to a larger group of scientists, educators and students that are more familiar with the APBS framework.
Viscous flow computations with the lattice-Boltzmann equation method
Yu, Dazhi
2002-09-01
The lattice Boltzmann equation (LBE) method is a kinetics-based approach for fluid flow computations, and it is amenable to parallel computing. Compared to the well-established Navier-Stokes (NS) approaches, critical issues remain with the LBE method, noticeably flexible spatial resolution, boundary treatments, and dispersion and relaxation time mode. Those issues are addressed in this dissertation with improved practice presented. At the formulation level, both the single-relaxation-time (SRT) and multiple-relaxation-time (MRT) models are analyzed. The SRT model involves no artificial parameters, with a constant relaxation time regulating the physical value of fluid viscosity. The MRT model allows different relaxation time scales for different variables. Computational assessment shows that the MRT model has advantages over the SRT model in maintaining stability, reducing the oscillation, and improving the convergence rate in the computation. A multi-block method is developed for both the SRT and MRT model to facilitate flexible spatial resolutions according to the flow structures. The formulae for information exchange at the interface between coarse and fine grids are derived to ensure the mass and momentum conservation while maintaining the second-order accuracy. A customized time matching between coarse and fine grids is also presented to ensure smooth exchange information. Results show that the multi-block method can greatly increase the computational efficiency of the LBE method without losing the accuracy. Two methods of force evaluation in LBE are examined: one based on stress integration on the solid boundary and the other momentum exchange between fluid and solid. The momentum exchange method is found to be simpler to implement while the integration of stress requires evaluation of the detailed surface geometry and extrapolation of stress-related variables to the same surface. The momentum exchange method performs better overall. Improved treatments for
An integrated Boltzmann+hydrodynamics approach to heavy ion collisions
Energy Technology Data Exchange (ETDEWEB)
Petersen, Hannah
2009-04-22
In this thesis the first fully integrated Boltzmann+hydrodynamics approach to relativistic heavy ion reactions has been developed. After a short introduction that motivates the study of heavy ion reactions as the tool to get insights about the QCD phase diagram, the most important theoretical approaches to describe the system are reviewed. The hadron-string transport approach that this work is based on is the Ultra-relativistic Quantum Molecular Dynamics (UrQMD) approach. Predictions for the charged particle multiplicities at LHC energies are made. The next step is the development of a new framework to calculate the baryon number density in a transport approach. Time evolutions of the net baryon number and the quark density have been calculated at AGS, SPS and RHIC energies. Studies of phase diagram trajectories using hydrodynamics are performed. The hybrid approach that has been developed as the main part of this thesis is based on the UrQMD transport approach with an intermediate hydrodynamical evolution for the hot and dense stage of the collision. The full (3+1) dimensional ideal relativistic one fluid dynamics evolution is solved using the SHASTA algorithm. Three different equations of state have been used, namely a hadron gas equation of state without a QGP phase transition, a chiral EoS and a bag model EoS including a strong first order phase transition. For the freeze-out transition from hydrodynamics to the cascade calculation two different set-ups are employed. The parameter dependences of the model are investigated and the time evolution of different quantities is explored. The hybrid model calculation is able to reproduce the experimentally measured integrated as well as transverse momentum dependent v{sub 2} values for charged particles. The multiplicity and mean transverse mass excitation function is calculated for pions, protons and kaons in the energy range from E{sub lab}=2-160 A GeV. The HBT correlation of the negatively charged pion source
Lyons, L.
2016-01-01
Accelerators and detectors are expensive, both in terms of money and human effort. It is thus important to invest effort in performing a good statistical anal- ysis of the data, in order to extract the best information from it. This series of five lectures deals with practical aspects of statistical issues that arise in typical High Energy Physics analyses.
DEFF Research Database (Denmark)
Tryggestad, Kjell
2004-01-01
The study aims is to describe how the inclusion and exclusion of materials and calculative devices construct the boundaries and distinctions between statistical facts and artifacts in economics. My methodological approach is inspired by John Graunt's (1667) Political arithmetic and more recent work...... within constructivism and the field of Science and Technology Studies (STS). The result of this approach is here termed reversible statistics, reconstructing the findings of a statistical study within economics in three different ways. It is argued that all three accounts are quite normal, albeit...... by accounting for the significance of the materials and the equipment that enters into the production of statistics. Key words: Reversible statistics, diverse materials, constructivism, economics, science, and technology....
A lattice Boltzmann coupled to finite volumes method for solving phase change problems
Directory of Open Access Journals (Sweden)
El Ganaoui Mohammed
2009-01-01
Full Text Available A numerical scheme coupling lattice Boltzmann and finite volumes approaches has been developed and qualified for test cases of phase change problems. In this work, the coupled partial differential equations of momentum conservation equations are solved with a non uniform lattice Boltzmann method. The energy equation is discretized by using a finite volume method. Simulations show the ability of this developed hybrid method to model the effects of convection, and to predict transfers. Benchmarking is operated both for conductive and convective situation dominating solid/liquid transition. Comparisons are achieved with respect to available analytical solutions and experimental results.
Energy Technology Data Exchange (ETDEWEB)
Uchaikin, V V; Sibatov, R T, E-mail: vuchaikin@gmail.com, E-mail: ren_sib@bk.ru [Ulyanovsk State University, 432000, 42 Leo Tolstoy str., Ulyanovsk (Russian Federation)
2011-04-08
The fractional Boltzmann equation for resonance radiation transport in plasma is proposed. We start with the standard Boltzmann equation; averaging over photon frequencies leads to the appearance of a fractional derivative. This fact is in accordance with the conception of latent variables leading to hereditary and non-local dynamics (in particular, fractional dynamics). The presence of a fractional material derivative in the equation is concordant with heavy tailed distribution of photon path lengths and with spatiotemporal coupling peculiar to the process. We discuss some methods of solving the obtained equation and demonstrate numerical results in some simple cases.
International Nuclear Information System (INIS)
Uchaikin, V V; Sibatov, R T
2011-01-01
The fractional Boltzmann equation for resonance radiation transport in plasma is proposed. We start with the standard Boltzmann equation; averaging over photon frequencies leads to the appearance of a fractional derivative. This fact is in accordance with the conception of latent variables leading to hereditary and non-local dynamics (in particular, fractional dynamics). The presence of a fractional material derivative in the equation is concordant with heavy tailed distribution of photon path lengths and with spatiotemporal coupling peculiar to the process. We discuss some methods of solving the obtained equation and demonstrate numerical results in some simple cases.
Two experiments to approach the Boltzmann factor: chemical reaction and viscous flow
International Nuclear Information System (INIS)
Fazio, Claudio; Battaglia, Onofrio R; Guastella, Ivan
2012-01-01
In this paper we discuss a pedagogical approach aimed at pointing out the role played by the Boltzmann factor in describing phenomena usually perceived as regulated by different mechanisms of functioning. Experimental results regarding some aspects of a chemical reaction and of the viscous flow of some liquids are analysed and described in terms of macroscopic variables whose temperature dependence is proportional to the Boltzmann factor. A description of a workshop implementing the approach in the framework of an undergraduate course for engineering education and some preliminary results about its pedagogical relevance are then reported. (paper)
Lattice Boltzmann equation calculation of internal, pressure-driven turbulent flow
International Nuclear Information System (INIS)
Hammond, L A; Halliday, I; Care, C M; Stevens, A
2002-01-01
We describe a mixing-length extension of the lattice Boltzmann approach to the simulation of an incompressible liquid in turbulent flow. The method uses a simple, adaptable, closure algorithm to bound the lattice Boltzmann fluid incorporating a law-of-the-wall. The test application, of an internal, pressure-driven and smooth duct flow, recovers correct velocity profiles for Reynolds number to 1.25 x 10 5 . In addition, the Reynolds number dependence of the friction factor in the smooth-wall branch of the Moody chart is correctly recovered. The method promises a straightforward extension to other curves of the Moody chart and to cylindrical pipe flow
International Nuclear Information System (INIS)
Schofield, S.L.
1988-01-01
Ackroyd's generalized least-squares method for solving the first-order Boltzmann equation is adapted to incorporate a potential treatment of voids. The adaptation comprises a direct least-squares minimization allied with a suitably-defined bilinear functional. The resulting formulation gives rise to a maximum principle whose functional does not contain terms of the type that have previously led to difficulties in treating void regions. The maximum principle is derived without requiring continuity of the flux at interfaces. The functional of the maximum principle is concluded to have an Euler-Lagrange equation given directly by the first-order Boltzmann equation. (author)
Contributions to the spectral theory of the linear Boltzmann operator for various geometries
International Nuclear Information System (INIS)
Protopopescu, V.
1975-01-01
The linear monoenergetic Boltzmann operator with isotropic scattering is studied for various geometries and boundary conditions as the infinitesimal generator of a positivity preserving contractive semigroup in an appropriate Hilbert space. General results about the existence and the uniqueness of the solutions of the corresponding evolution problems are reviewed. The spectrum of the Boltzmann operator is analyzed for semi-infinite, slab and parallelepipedic geometries with vacuum, periodic, perfectly reflecting, generalized and diffusely reflecting boundary condition respectively. The main features of these spectra, their importance for determining the asymptotic evolution and possible generalizations to more realistic models are put together in a final section. (author)
Punshon-Smith, Samuel; Smith, Scott
2018-02-01
This article studies the Cauchy problem for the Boltzmann equation with stochastic kinetic transport. Under a cut-off assumption on the collision kernel and a coloring hypothesis for the noise coefficients, we prove the global existence of renormalized (in the sense of DiPerna/Lions) martingale solutions to the Boltzmann equation for large initial data with finite mass, energy, and entropy. Our analysis includes a detailed study of weak martingale solutions to a class of linear stochastic kinetic equations. This study includes a criterion for renormalization, the weak closedness of the solution set, and tightness of velocity averages in {{L}1}.
Statistical Mechanics of Money, Income, and Wealth
Yakovenko, Victor
2006-03-01
In Ref. [1], we proposed an analogy between the exponential Boltzmann-Gibbs distribution of energy in physics and the equilibrium probability distribution of money in a closed economic system. Analogously to energy, money is locally conserved in interactions between economic agents, so the thermal Boltzmann-Gibbs distribution function is expected for money. Since then, many researchers followed and expanded this idea [2]. Much work was done on the analysis of empirical data, mostly on income, for which a lot of tax and census data is available. We demonstrated [3] that income distribution in the USA has a well-defined two-class structure. The majority of population (97-99%) belongs to the lower class characterized by the exponential Boltzmann-Gibbs (``thermal'') distribution. The upper class (1-3% of population) has a Pareto power-law (``superthermal'') distribution, whose parameters change in time with the rise and fall of stock market. We proposed a concept of equilibrium inequality in a society, based on the principle of maximal entropy, and quantitatively demonstrated that it applies to the majority of population. Income distribution in other countries shows similar patterns. For more references, see http://www2.physics.umd.edu/˜yakovenk/econophysics.html. References: [1] A. A. Dragulescu and V. M. Yakovenko, ``Statistical mechanics of money'', Eur. Phys. J. B 17, 723 (2000). [2] ``Econophysics of Wealth Distributions'', edited by A. Chatterjee, S. Yarlagadda, and B. K. Chakrabarti, Springer, 2005. [3] A. C. Silva and V. M. Yakovenko, ``Temporal evolution of the `thermal' and `superthermal' income classes in the USA during 1983-2001'', Europhys. Lett. 69, 304 (2005).
Application of Lattice Boltzmann Methods in Complex Mass Transfer Systems
Sun, Ning
Lattice Boltzmann Method (LBM) is a novel computational fluid dynamics method that can easily handle complex and dynamic boundaries, couple local or interfacial interactions/reactions, and be easily parallelized allowing for simulation of large systems. While most of the current studies in LBM mainly focus on fluid dynamics, however, the inherent power of this method makes it an ideal candidate for the study of mass transfer systems involving complex/dynamic microstructures and local reactions. In this thesis, LBM is introduced to be an alternative computational method for the study of electrochemical energy storage systems (Li-ion batteries (LIBs) and electric double layer capacitors (EDLCs)) and transdermal drug design on mesoscopic scale. Based on traditional LBM, the following in-depth studies have been carried out: (1) For EDLCs, the simulation of diffuse charge dynamics is carried out for both the charge and the discharge processes on 2D systems of complex random electrode geometries (pure random, random spheres and random fibers). Steric effect of concentrated solutions is considered by using modified Poisson-Nernst-Plank (MPNP) equations and compared with regular Poisson-Nernst-Plank (PNP) systems. The effects of electrode microstructures (electrode density, electrode filler morphology, filler size, etc.) on the net charge distribution and charge/discharge time are studied in detail. The influence of applied potential during discharging process is also discussed. (2) For the study of dendrite formation on the anode of LIBs, it is shown that the Lattice Boltzmann model can capture all the experimentally observed features of microstructure evolution at the anode, from smooth to mossy to dendritic. The mechanism of dendrite formation process in mesoscopic scale is discussed in detail and compared with the traditional Sand's time theories. It shows that dendrite formation is closely related to the inhomogeneous reactively at the electrode-electrolyte interface
Feiveson, Alan H.; Foy, Millennia; Ploutz-Snyder, Robert; Fiedler, James
2014-01-01
Do you have elevated p-values? Is the data analysis process getting you down? Do you experience anxiety when you need to respond to criticism of statistical methods in your manuscript? You may be suffering from Insufficient Statistical Support Syndrome (ISSS). For symptomatic relief of ISSS, come for a free consultation with JSC biostatisticians at our help desk during the poster sessions at the HRP Investigators Workshop. Get answers to common questions about sample size, missing data, multiple testing, when to trust the results of your analyses and more. Side effects may include sudden loss of statistics anxiety, improved interpretation of your data, and increased confidence in your results.
Wannier, Gregory Hugh
1966-01-01
Until recently, the field of statistical physics was traditionally taught as three separate subjects: thermodynamics, statistical mechanics, and kinetic theory. This text, a forerunner in its field and now a classic, was the first to recognize the outdated reasons for their separation and to combine the essentials of the three subjects into one unified presentation of thermal physics. It has been widely adopted in graduate and advanced undergraduate courses, and is recommended throughout the field as an indispensable aid to the independent study and research of statistical physics.Designed for
Blakemore, J S
1962-01-01
Semiconductor Statistics presents statistics aimed at complementing existing books on the relationships between carrier densities and transport effects. The book is divided into two parts. Part I provides introductory material on the electron theory of solids, and then discusses carrier statistics for semiconductors in thermal equilibrium. Of course a solid cannot be in true thermodynamic equilibrium if any electrical current is passed; but when currents are reasonably small the distribution function is but little perturbed, and the carrier distribution for such a """"quasi-equilibrium"""" co
An improved electronic determination of the Boltzmann constant by Johnson noise thermometry
Qu, Jifeng; Benz, Samuel P.; Coakley, Kevin; Rogalla, Horst; Tew, Weston L.; White, Rod; Zhou, Kunli; Zhou, Zhenyu
2017-08-01
Recent measurements using acoustic gas thermometry have determined the value of the Boltzmann constant, k, with a relative uncertainty less than 1 × 10-6. These results have been supported by a measurement with a relative uncertainty of 1.9 × 10-6 made with dielectric-constant gas thermometry. Together, the measurements meet the requirements of the International Committee for Weights and Measures and enable them to proceed with the redefinition of the kelvin in 2018. In further support, we provide a new determination of k using a purely electronic approach, Johnson noise thermometry, in which the thermal noise power generated by a sensing resistor immersed in a triple-point-of-water cell is compared to the noise power of a quantum-accurate pseudo-random noise waveform of nominally equal noise power. The experimental setup differs from that of the 2015 determination in several respects: a 100 Ω resistor is used as the thermal noise source, identical thin coaxial cables made of solid beryllium-copper conductors and foam dielectrics are used to connect the thermal and quantum-accurate noise sources to the correlator so as to minimize the temperature and frequency sensitivity of the impedances in the connecting leads, and no trimming capacitors or inductors are inserted into the connecting leads. The combination of reduced uncertainty due to spectral mismatches in the connecting leads and reduced statistical uncertainty due to a longer integration period of 100 d results in an improved determination of k = 1.380 649 7(37) × 10-23 J K-1 with a relative standard uncertainty of 2.7 × 10-6 and a relative offset of 0.89 × 10-6 from the CODATA 2014 recommended value. The most significant terms in the uncertainty budget, the statistical uncertainty and the spectral-mismatch uncertainty, are uncorrelated with the corresponding uncertainties in the 2015 measurements.
Serdobolskii, Vadim Ivanovich
2007-01-01
This monograph presents mathematical theory of statistical models described by the essentially large number of unknown parameters, comparable with sample size but can also be much larger. In this meaning, the proposed theory can be called "essentially multiparametric". It is developed on the basis of the Kolmogorov asymptotic approach in which sample size increases along with the number of unknown parameters.This theory opens a way for solution of central problems of multivariate statistics, which up until now have not been solved. Traditional statistical methods based on the idea of an infinite sampling often break down in the solution of real problems, and, dependent on data, can be inefficient, unstable and even not applicable. In this situation, practical statisticians are forced to use various heuristic methods in the hope the will find a satisfactory solution.Mathematical theory developed in this book presents a regular technique for implementing new, more efficient versions of statistical procedures. ...
Department of Homeland Security — Accident statistics available on the Coast Guard’s website by state, year, and one variable to obtain tables and/or graphs. Data from reports has been loaded for...
U.S. Department of Health & Human Services — The CMS Center for Strategic Planning produces an annual CMS Statistics reference booklet that provides a quick reference for summary information about health...
DEFF Research Database (Denmark)
Tryggestad, Kjell
2004-01-01
The study aims is to describe how the inclusion and exclusion of materials and calculative devices construct the boundaries and distinctions between statistical facts and artifacts in economics. My methodological approach is inspired by John Graunt's (1667) Political arithmetic and more recent work...... within constructivism and the field of Science and Technology Studies (STS). The result of this approach is here termed reversible statistics, reconstructing the findings of a statistical study within economics in three different ways. It is argued that all three accounts are quite normal, albeit...... in different ways. The presence and absence of diverse materials, both natural and political, is what distinguishes them from each other. Arguments are presented for a more symmetric relation between the scientific statistical text and the reader. I will argue that a more symmetric relation can be achieved...
Energy Technology Data Exchange (ETDEWEB)
Wendelberger, Laura Jean [Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
2017-08-08
In large datasets, it is time consuming or even impossible to pick out interesting images. Our proposed solution is to find statistics to quantify the information in each image and use those to identify and pick out images of interest.
MacKenzie, Dana
2004-01-01
The drawbacks of using 19th-century mathematics in physics and astronomy are illustrated. To continue with the expansion of the knowledge about the cosmos, the scientists will have to come in terms with modern statistics. Some researchers have deliberately started importing techniques that are used in medical research. However, the physicists need to identify the brand of statistics that will be suitable for them, and make a choice between the Bayesian and the frequentists approach. (Edited abstract).
Lyons, L.
2017-01-01
Accelerators and detectors are expensive, both in terms of money and human effort. It is thus important to invest effort in performing a good statistical anal- ysis of the data, in order to extract the best information from it. This series of five lectures deals with practical aspects of statistical issues that arise in typical High Energy Physics analyses. Accelerators and detectors are expensive, both in terms of money and human effort. It is thus important to invest effort in performing...
Statistical ensembles for money and debt
Viaggiu, Stefano; Lionetto, Andrea; Bargigli, Leonardo; Longo, Michele
2012-10-01
We build a statistical ensemble representation of two economic models describing respectively, in simplified terms, a payment system and a credit market. To this purpose we adopt the Boltzmann-Gibbs distribution where the role of the Hamiltonian is taken by the total money supply (i.e. including money created from debt) of a set of interacting economic agents. As a result, we can read the main thermodynamic quantities in terms of monetary ones. In particular, we define for the credit market model a work term which is related to the impact of monetary policy on credit creation. Furthermore, with our formalism we recover and extend some results concerning the temperature of an economic system, previously presented in the literature by considering only the monetary base as a conserved quantity. Finally, we study the statistical ensemble for the Pareto distribution.
A Lattice Boltzmann Approach to Multi-Phase Surface Reactions with Heat Effects
Kamali, M.R.
2013-01-01
The aim of the present research was to explore the promises and shift the limits of the numerical framework of lattice Boltzmann (LB) for studying the physics behind multi-component two-phase heterogeneous non-isothermal reactive flows under industrial conditions. An example of such an industrially
A Truly Second-Order and Unconditionally Stable Thermal Lattice Boltzmann Method
Directory of Open Access Journals (Sweden)
Zhen Chen
2017-03-01
Full Text Available An unconditionally stable thermal lattice Boltzmann method (USTLBM is proposed in this paper for simulating incompressible thermal flows. In USTLBM, solutions to the macroscopic governing equations that are recovered from lattice Boltzmann equation (LBE through Chapman–Enskog (C-E expansion analysis are resolved in a predictor–corrector scheme and reconstructed within lattice Boltzmann framework. The development of USTLBM is inspired by the recently proposed simplified thermal lattice Boltzmann method (STLBM. Comparing with STLBM which can only achieve the first-order of accuracy in time, the present USTLBM ensures the second-order of accuracy both in space and in time. Meanwhile, all merits of STLBM are maintained by USTLBM. Specifically, USTLBM directly updates macroscopic variables rather than distribution functions, which greatly saves virtual memories and facilitates implementation of physical boundary conditions. Through von Neumann stability analysis, it can be theoretically proven that USTLBM is unconditionally stable. It is also shown in numerical tests that, comparing to STLBM, lower numerical error can be expected in USTLBM at the same mesh resolution. Four typical numerical examples are presented to demonstrate the robustness of USTLBM and its flexibility on non-uniform and body-fitted meshes.
Patel, R.A.; Perko, J.; Jaques, D.; De Schutter, G.; Ye, G.; Van Breugel, K.
2013-01-01
A Lattice Boltzmann (LB) based reactive transport model intended to capture reactions and solid phase changes occurring at the pore scale is presented. The proposed approach uses LB method to compute multi component mass transport. The LB multi-component transport model is then coupled with the
DEFF Research Database (Denmark)
Johannessen, Kim
2014-01-01
The exact solution to the one-dimensional Poisson–Boltzmann equation with asymmetric boundary conditions can be expressed in terms of the Jacobi elliptic functions. The boundary conditions determine the modulus of the Jacobi elliptic functions. The boundary conditions can not be solved analytically...
Inelastic Quantum Transport in Superlattices: Success and Failure of the Boltzmann Equation
DEFF Research Database (Denmark)
Wacker, Andreas; Jauho, Antti-Pekka; Rott, Stephan
1999-01-01
the whole held range from linear response to negative differential conductivity. The quantum results are compared with the respective results obtained from a Monte Carlo solution of the Boltzmann equation. Our analysis thus sets the limits of validity for the semiclassical theory in a nonlinear transport...
Frausto-Solis, Juan; Liñán-García, Ernesto; Sánchez-Hernández, Juan Paulo; González-Barbosa, J Javier; González-Flores, Carlos; Castilla-Valdez, Guadalupe
2016-01-01
A new hybrid Multiphase Simulated Annealing Algorithm using Boltzmann and Bose-Einstein distributions (MPSABBE) is proposed. MPSABBE was designed for solving the Protein Folding Problem (PFP) instances. This new approach has four phases: (i) Multiquenching Phase (MQP), (ii) Boltzmann Annealing Phase (BAP), (iii) Bose-Einstein Annealing Phase (BEAP), and (iv) Dynamical Equilibrium Phase (DEP). BAP and BEAP are simulated annealing searching procedures based on Boltzmann and Bose-Einstein distributions, respectively. DEP is also a simulated annealing search procedure, which is applied at the final temperature of the fourth phase, which can be seen as a second Bose-Einstein phase. MQP is a search process that ranges from extremely high to high temperatures, applying a very fast cooling process, and is not very restrictive to accept new solutions. However, BAP and BEAP range from high to low and from low to very low temperatures, respectively. They are more restrictive for accepting new solutions. DEP uses a particular heuristic to detect the stochastic equilibrium by applying a least squares method during its execution. MPSABBE parameters are tuned with an analytical method, which considers the maximal and minimal deterioration of problem instances. MPSABBE was tested with several instances of PFP, showing that the use of both distributions is better than using only the Boltzmann distribution on the classical SA.
Second order bounce back boundary condition for the lattice Boltzmann fluid simulation
International Nuclear Information System (INIS)
Kim, In Chan
2000-01-01
A new bounce back boundary method of the second order in error is proposed for the lattice Boltzmann fluid simulation. This new method can be used for the arbitrarily irregular lattice geometry of a non-slip boundary. The traditional bounce back boundary condition for the lattice Boltzmann simulation is of the first order in error. Since the lattice Boltzmann method is the second order scheme by itself, a boundary technique of the second order has been desired to replace the first order bounce back method. This study shows that, contrary to the common belief that the bounce back boundary condition is unilaterally of the first order, the second order bounce back boundary condition can be realized. This study also shows that there exists a generalized bounce back technique that can be characterized by a single interpolation parameter. The second order bounce back method can be obtained by proper selection of this parameter in accordance with the detailed lattice geometry of the boundary. For an illustrative purpose, the transient Couette and the plane Poiseuille flows are solved by the lattice Boltzmann simulation with various boundary conditions. The results show that the generalized bounce back method yields the second order behavior in the error of the solution, provided that the interpolation parameter is properly selected. Coupled with its intuitive nature and the ease of implementation, the bounce back method can be as good as any second order boundary method
Models, Their Application, and Scientific Anticipation: Ludwig Boltzmann's Work as Tacit Knowing
Schmitt, Richard Henry
2011-01-01
Ludwig Boltzmann's work in theoretical physics exhibits an approach to the construction of theory that he transmitted to the succeeding generation by example. It involved the construction of clear models, allowed more than one, and was not based solely on the existing facts, with the intent of examining and criticizing the assumptions that made…
International Nuclear Information System (INIS)
Shan Ming-Lei; Zhu Chang-Ping; Yao Cheng; Yin Cheng; Jiang Xiao-Yan
2016-01-01
The dynamics of the cavitation bubble collapse is a fundamental issue for the bubble collapse application and prevention. In the present work, the modified forcing scheme for the pseudopotential multi-relaxation-time lattice Boltzmann model developed by Li Q et al. [Li Q, Luo K H and Li X J 2013 Phys. Rev. E 87 053301] is adopted to develop a cavitation bubble collapse model. In the respects of coexistence curves and Laplace law verification, the improved pseudopotential multi-relaxation-time lattice Boltzmann model is investigated. It is found that the thermodynamic consistency and surface tension are independent of kinematic viscosity. By homogeneous and heterogeneous cavitation simulation, the ability of the present model to describe the cavitation bubble development as well as the cavitation inception is verified. The bubble collapse between two parallel walls is simulated. The dynamic process of a collapsing bubble is consistent with the results from experiments and simulations by other numerical methods. It is demonstrated that the present pseudopotential multi-relaxation-time lattice Boltzmann model is applicable and efficient, and the lattice Boltzmann method is an alternative tool for collapsing bubble modeling. (paper)
Directory of Open Access Journals (Sweden)
Juan Frausto-Solis
2016-01-01
Full Text Available A new hybrid Multiphase Simulated Annealing Algorithm using Boltzmann and Bose-Einstein distributions (MPSABBE is proposed. MPSABBE was designed for solving the Protein Folding Problem (PFP instances. This new approach has four phases: (i Multiquenching Phase (MQP, (ii Boltzmann Annealing Phase (BAP, (iii Bose-Einstein Annealing Phase (BEAP, and (iv Dynamical Equilibrium Phase (DEP. BAP and BEAP are simulated annealing searching procedures based on Boltzmann and Bose-Einstein distributions, respectively. DEP is also a simulated annealing search procedure, which is applied at the final temperature of the fourth phase, which can be seen as a second Bose-Einstein phase. MQP is a search process that ranges from extremely high to high temperatures, applying a very fast cooling process, and is not very restrictive to accept new solutions. However, BAP and BEAP range from high to low and from low to very low temperatures, respectively. They are more restrictive for accepting new solutions. DEP uses a particular heuristic to detect the stochastic equilibrium by applying a least squares method during its execution. MPSABBE parameters are tuned with an analytical method, which considers the maximal and minimal deterioration of problem instances. MPSABBE was tested with several instances of PFP, showing that the use of both distributions is better than using only the Boltzmann distribution on the classical SA.
Two Experiments to Approach the Boltzmann Factor: Chemical Reaction and Viscous Flow
Fazio, Claudio; Battaglia, Onofrio R.; Guastella, Ivan
2012-01-01
In this paper we discuss a pedagogical approach aimed at pointing out the role played by the Boltzmann factor in describing phenomena usually perceived as regulated by different mechanisms of functioning. Experimental results regarding some aspects of a chemical reaction and of the viscous flow of some liquids are analysed and described in terms…
Lin, Luan; McKerrow, Wilson H; Richards, Bryce; Phonsom, Chukiat; Lawrence, Charles E
2018-03-05
The nearest neighbor model and associated dynamic programming algorithms allow for the efficient estimation of the RNA secondary structure Boltzmann ensemble. However because a given RNA secondary structure only contains a fraction of the possible helices that could form from a given sequence, the Boltzmann ensemble is multimodal. Several methods exist for clustering structures and finding those modes. However less focus is given to exploring the underlying reasons for this multimodality: the presence of conflicting basepairs. Information theory, or more specifically mutual information, provides a method to identify those basepairs that are key to the secondary structure. To this end we find most informative basepairs and visualize the effect of these basepairs on the secondary structure. Knowing whether a most informative basepair is present tells us not only the status of the particular pair but also provides a large amount of information about which other pairs are present or not present. We find that a few basepairs account for a large amount of the structural uncertainty. The identification of these pairs indicates small changes to sequence or stability that will have a large effect on structure. We provide a novel algorithm that uses mutual information to identify the key basepairs that lead to a multimodal Boltzmann distribution. We then visualize the effect of these pairs on the overall Boltzmann ensemble.
G. van Tulder (Gijs); M. de Bruijne (Marleen)
2016-01-01
textabstractThe choice of features greatly influences the performance of a tissue classification system. Despite this, many systems are built with standard, predefined filter banks that are not optimized for that particular application. Representation learning methods such as restricted Boltzmann
DEFF Research Database (Denmark)
Hygum, Morten Arnfeldt; Karlin, Iliya; Popok, Vladimir
2015-01-01
A model for vapor condensation on vertical hydrophilic surfaces is developed using the entropic lattice Boltzmann method extended with a free surface formulation of the evaporation–condensation problem. The model is validated with the steady liquid film formation on a flat vertical wall. It is sh...
Schwabl, Franz
2006-01-01
The completely revised new edition of the classical book on Statistical Mechanics covers the basic concepts of equilibrium and non-equilibrium statistical physics. In addition to a deductive approach to equilibrium statistics and thermodynamics based on a single hypothesis - the form of the microcanonical density matrix - this book treats the most important elements of non-equilibrium phenomena. Intermediate calculations are presented in complete detail. Problems at the end of each chapter help students to consolidate their understanding of the material. Beyond the fundamentals, this text demonstrates the breadth of the field and its great variety of applications. Modern areas such as renormalization group theory, percolation, stochastic equations of motion and their applications to critical dynamics, kinetic theories, as well as fundamental considerations of irreversibility, are discussed. The text will be useful for advanced students of physics and other natural sciences; a basic knowledge of quantum mechan...
Jana, Madhusudan
2015-01-01
Statistical mechanics is self sufficient, written in a lucid manner, keeping in mind the exam system of the universities. Need of study this subject and its relation to Thermodynamics is discussed in detail. Starting from Liouville theorem gradually, the Statistical Mechanics is developed thoroughly. All three types of Statistical distribution functions are derived separately with their periphery of applications and limitations. Non-interacting ideal Bose gas and Fermi gas are discussed thoroughly. Properties of Liquid He-II and the corresponding models have been depicted. White dwarfs and condensed matter physics, transport phenomenon - thermal and electrical conductivity, Hall effect, Magneto resistance, viscosity, diffusion, etc. are discussed. Basic understanding of Ising model is given to explain the phase transition. The book ends with a detailed coverage to the method of ensembles (namely Microcanonical, canonical and grand canonical) and their applications. Various numerical and conceptual problems ar...
An Implementation of Hydrostatic Boundary Conditions for Variable Density Lattice Boltzmann Methods
Bardsley, K. J.; Thorne, D. T.; Lee, J. S.; Sukop, M. C.
2006-12-01
Lattice Boltzmann Methods (LBMs) have been under development for the last two decades and have become another capable numerical method for simulating fluid flow. Recent advances in lattice Boltzmann applications involve simulation of density-dependent fluid flow in closed (Dixit and Babu, 2006; D'Orazio et al., 2004) or periodic (Guo and Zhao, 2005) domains. However, standard pressure boundary conditions (BCs) are incompatible with concentration-dependent density flow simulations that use a body force for gravity. An implementation of hydrostatic BCs for use under these conditions is proposed here. The basis of this new implementation is an additional term in the pressure BC. It is derived to account for the incorporation of gravity as a body force and the effect of varying concentration in the fluid. The hydrostatic BC expands the potential of density-dependent LBM to simulate domains with boundaries other than the closed or periodic boundaries that have appeared in previous literature on LBM simulations. With this new implementation, LBM will be able to simulate complex concentration-dependent density flows, such as salt water intrusion in the classic Henry and Henry-Hilleke problems. This is demonstrated using various examples, beginning with a closed box system, and ending with a system containing two solid walls, one velocity boundary and one pressure boundary, as in the Henry problem. References Dixit, H. N., V. Babu, (2006), Simulation of high Rayleigh number natural convection in a square cavity using the lattice Boltzmann method, Int. J. Heat Mass Transfer, 49, 727-739. D'Orazio, A., M. Corcione, G.P. Celata, (2004), Application to natural convection enclosed flows of a lattice Boltzmann BGK model coupled with a general purpose thermal boundary conditions, Int. J. Thermal Sci., 43, 575-586. Gou, Z., T.S. Zhao, (2005), Lattice Boltzmann simulation of natural convection with temperature-dependant viscosity in a porous cavity, Numerical Heat Transfer, Part B
Rohatgi, Vijay K
2003-01-01
Unified treatment of probability and statistics examines and analyzes the relationship between the two fields, exploring inferential issues. Numerous problems, examples, and diagrams--some with solutions--plus clear-cut, highlighted summaries of results. Advanced undergraduate to graduate level. Contents: 1. Introduction. 2. Probability Model. 3. Probability Distributions. 4. Introduction to Statistical Inference. 5. More on Mathematical Expectation. 6. Some Discrete Models. 7. Some Continuous Models. 8. Functions of Random Variables and Random Vectors. 9. Large-Sample Theory. 10. General Meth
Davidson, Norman
2003-01-01
Clear and readable, this fine text assists students in achieving a grasp of the techniques and limitations of statistical mechanics. The treatment follows a logical progression from elementary to advanced theories, with careful attention to detail and mathematical development, and is sufficiently rigorous for introductory or intermediate graduate courses.Beginning with a study of the statistical mechanics of ideal gases and other systems of non-interacting particles, the text develops the theory in detail and applies it to the study of chemical equilibrium and the calculation of the thermody
Levine-Wissing, Robin
2012-01-01
All Access for the AP® Statistics Exam Book + Web + Mobile Everything you need to prepare for the Advanced Placement® exam, in a study system built around you! There are many different ways to prepare for an Advanced Placement® exam. What's best for you depends on how much time you have to study and how comfortable you are with the subject matter. To score your highest, you need a system that can be customized to fit you: your schedule, your learning style, and your current level of knowledge. This book, and the online tools that come with it, will help you personalize your AP® Statistics prep
Mandl, Franz
1988-01-01
The Manchester Physics Series General Editors: D. J. Sandiford; F. Mandl; A. C. Phillips Department of Physics and Astronomy, University of Manchester Properties of Matter B. H. Flowers and E. Mendoza Optics Second Edition F. G. Smith and J. H. Thomson Statistical Physics Second Edition E. Mandl Electromagnetism Second Edition I. S. Grant and W. R. Phillips Statistics R. J. Barlow Solid State Physics Second Edition J. R. Hook and H. E. Hall Quantum Mechanics F. Mandl Particle Physics Second Edition B. R. Martin and G. Shaw The Physics of Stars Second Edition A. C. Phillips Computing for Scient
Indian Academy of Sciences (India)
Home; Journals; Resonance – Journal of Science Education; Volume 4; Issue 10. Statistical Computing - Understanding Randomness and Random Numbers. Sudhakar Kunte. Series Article Volume 4 Issue 10 October 1999 pp 16-21. Fulltext. Click here to view fulltext PDF. Permanent link:
Schrödinger, Erwin
1952-01-01
Nobel Laureate's brilliant attempt to develop a simple, unified standard method of dealing with all cases of statistical thermodynamics - classical, quantum, Bose-Einstein, Fermi-Dirac, and more.The work also includes discussions of Nernst theorem, Planck's oscillator, fluctuations, the n-particle problem, problem of radiation, much more.
International Nuclear Information System (INIS)
Anon.
1994-01-01
For the years 1992 and 1993, part of the figures shown in the tables of the Energy Review are preliminary or estimated. The annual statistics of the Energy Review appear in more detail from the publication Energiatilastot - Energy Statistics issued annually, which also includes historical time series over a longer period. The tables and figures shown in this publication are: Changes in the volume of GNP and energy consumption; Coal consumption; Natural gas consumption; Peat consumption; Domestic oil deliveries; Import prices of oil; Price development of principal oil products; Fuel prices for power production; Total energy consumption by source; Electricity supply; Energy imports by country of origin in 1993; Energy exports by recipient country in 1993; Consumer prices of liquid fuels; Consumer prices of hard coal and natural gas, prices of indigenous fuels; Average electricity price by type of consumer; Price of district heating by type of consumer and Excise taxes and turnover taxes included in consumer prices of some energy sources
Goodman, Joseph W.
2000-07-01
The Wiley Classics Library consists of selected books that have become recognized classics in their respective fields. With these new unabridged and inexpensive editions, Wiley hopes to extend the life of these important works by making them available to future generations of mathematicians and scientists. Currently available in the Series: T. W. Anderson The Statistical Analysis of Time Series T. S. Arthanari & Yadolah Dodge Mathematical Programming in Statistics Emil Artin Geometric Algebra Norman T. J. Bailey The Elements of Stochastic Processes with Applications to the Natural Sciences Robert G. Bartle The Elements of Integration and Lebesgue Measure George E. P. Box & Norman R. Draper Evolutionary Operation: A Statistical Method for Process Improvement George E. P. Box & George C. Tiao Bayesian Inference in Statistical Analysis R. W. Carter Finite Groups of Lie Type: Conjugacy Classes and Complex Characters R. W. Carter Simple Groups of Lie Type William G. Cochran & Gertrude M. Cox Experimental Designs, Second Edition Richard Courant Differential and Integral Calculus, Volume I RIchard Courant Differential and Integral Calculus, Volume II Richard Courant & D. Hilbert Methods of Mathematical Physics, Volume I Richard Courant & D. Hilbert Methods of Mathematical Physics, Volume II D. R. Cox Planning of Experiments Harold S. M. Coxeter Introduction to Geometry, Second Edition Charles W. Curtis & Irving Reiner Representation Theory of Finite Groups and Associative Algebras Charles W. Curtis & Irving Reiner Methods of Representation Theory with Applications to Finite Groups and Orders, Volume I Charles W. Curtis & Irving Reiner Methods of Representation Theory with Applications to Finite Groups and Orders, Volume II Cuthbert Daniel Fitting Equations to Data: Computer Analysis of Multifactor Data, Second Edition Bruno de Finetti Theory of Probability, Volume I Bruno de Finetti Theory of Probability, Volume 2 W. Edwards Deming Sample Design in Business Research
Piasecka-Belkhayat, Alicja; Korczak, Anna
2018-01-01
The interval coupled lattice Boltzmann equations for electrons and phonons are used to analyse the heating process of thin metal films. The interval lattice Boltzmann method (ILBM) with the uncertainly defined external source function associated with the laser irradiation is used to simulate the heat transfer. The solution of the interval Boltzmann transport equations has been obtained taking into account the rules of directed interval arithmetic. A similar analysis has been done using the sensitivity model where the Boltzmann transport equations and boundary-initial conditions have been differentiated with respect to the no-interval laser parameter. The knowledge of the sensitivity function distribution and the application of the Taylor formula allow one to find the border solutions of the problem analysed which correspond to the solution obtained assuming the uncertainly defined source function. In the final part of the paper the results of numerical computations obtained using both methods are presented.
Energy Technology Data Exchange (ETDEWEB)
EL Safadi, M
2007-03-15
We study the regularity of kinetic equations of Boltzmann type.We use essentially Littlewood-Paley method from harmonic analysis, consisting mainly in working with dyadics annulus. We shall mainly concern with the homogeneous case, where the solution f(t,x,v) depends only on the time t and on the velocities v, while working with realistic and singular cross-sections (non cutoff). In the first part, we study the particular case of Maxwellian molecules. Under this hypothesis, the structure of the Boltzmann operator and his Fourier transform write in a simple form. We show a global C{sup {infinity}} regularity. Then, we deal with the case of general cross-sections with 'hard potential'. We are interested in the Landau equation which is limit equation to the Boltzmann equation, taking in account grazing collisions. We prove that any weak solution belongs to Schwartz space S. We demonstrate also a similar regularity for the case of Boltzmann equation. Let us note that our method applies directly for all dimensions, and proofs are often simpler compared to other previous ones. Finally, we finish with Boltzmann-Dirac equation. In particular, we adapt the result of regularity obtained in Alexandre, Desvillettes, Wennberg and Villani work, using the dissipation rate connected with Boltzmann-Dirac equation. (author)
International Nuclear Information System (INIS)
Anon.
1989-01-01
World data from the United Nation's latest Energy Statistics Yearbook, first published in our last issue, are completed here. The 1984-86 data were revised and 1987 data added for world commercial energy production and consumption, world natural gas plant liquids production, world LP-gas production, imports, exports, and consumption, world residual fuel oil production, imports, exports, and consumption, world lignite production, imports, exports, and consumption, world peat production and consumption, world electricity production, imports, exports, and consumption (Table 80), and world nuclear electric power production
Sheffield, Scott
2009-01-01
In recent years, statistical mechanics has been increasingly recognized as a central domain of mathematics. Major developments include the Schramm-Loewner evolution, which describes two-dimensional phase transitions, random matrix theory, renormalization group theory and the fluctuations of random surfaces described by dimers. The lectures contained in this volume present an introduction to recent mathematical progress in these fields. They are designed for graduate students in mathematics with a strong background in analysis and probability. This book will be of particular interest to graduate students and researchers interested in modern aspects of probability, conformal field theory, percolation, random matrices and stochastic differential equations.
Reis, T.
2010-09-06
Existing lattice Boltzmann models that have been designed to recover a macroscopic description of immiscible liquids are only able to make predictions that are quantitatively correct when the interface that exists between the fluids is smeared over several nodal points. Attempts to minimise the thickness of this interface generally leads to a phenomenon known as lattice pinning, the precise cause of which is not well understood. This spurious behaviour is remarkably similar to that associated with the numerical simulation of hyperbolic partial differential equations coupled with a stiff source term. Inspired by the seminal work in this field, we derive a lattice Boltzmann implementation of a model equation used to investigate such peculiarities. This implementation is extended to different spacial discretisations in one and two dimensions. We shown that the inclusion of a quasi-random threshold dramatically delays the onset of pinning and facetting.
Lattice Boltzmann Simulations in the Slip and Transition Flow Regime with the Peano Framework
Neumann, Philipp
2012-01-01
We present simulation results of flows in the finite Knudsen range, which is in the slip and transition flow regime. Our implementations are based on the Lattice Boltzmann method and are accomplished within the Peano framework. We validate our code by solving two- and three-dimensional channel flow problems and compare our results with respective experiments from other research groups. We further apply our Lattice Boltzmann solver to the geometrical setup of a microreactor consisting of differently sized channels and a reactor chamber. Here, we apply static adaptive grids to fur-ther reduce computational costs. We further investigate the influence of using a simple BGK collision kernel in coarse grid regions which are further away from the slip boundaries. Our results are in good agreement with theory and non-adaptive simulations, demonstrating the validity and the capabilities of our adaptive simulation software for flow problems at finite Knudsen numbers.
Matin, Rastin; Misztal, Marek K.; Hernandez-Garcia, Anier; Mathiesen, Joachim
2015-11-01
Many hydrodynamic phenomena such as flows at micron scale in porous media, large Reynolds numbers flows, non-Newtonian and multiphase flows have been simulated numerically using the lattice Boltzmann method. By solving the Lattice Boltzmann Equation on three-dimensional unstructured meshes, we efficiently model single-phase fluid flow in real rock samples. We use the flow field to estimate the permeability and further investigate the anomalous dispersion of passive tracers in porous media. By extending our single-phase model with a free-energy based method, we are able to simulate binary systems with moderate density ratios in a thermodynamically consistent way. In this presentation we will present our recent results on both anomalous transport and multiphase segregation.
From Pore Scale to Turbulent Flow with the Unstructured Lattice Boltzmann Method
DEFF Research Database (Denmark)
Matin, Rastin
region depends on the method of time integration. The formulation based on the finite element method exhibits improved stability and is therefore used for two applications. Firstly, together with a free-energy model two-phase flow is simulated at large density and kinematic viscosity contrasts including......Abstract: The lattice Boltzmann method is a class of methods in computational fluid dynamics for simulating fluid flow. Implementations on unstructured grids are particularly relevant for various engineering applications, where geometric flexibility or high resolution near a body or a wall...... is required. The main topic of this thesis is to further develop unstructured lattice Boltzmann methods for simulations of Newtonian fluid flow in three dimensions, in particular porous flow. Two methods are considered in this thesis based on the finite volume method and finite element method, respectively...
From Newton's Law to the Linear Boltzmann Equation Without Cut-Off
Ayi, Nathalie
2017-03-01
We provide a rigorous derivation of the linear Boltzmann equation without cut-off starting from a system of particles interacting via a potential with infinite range as the number of particles N goes to infinity under the Boltzmann-Grad scaling. More particularly, we will describe the motion of a tagged particle in a gas close to global equilibrium. The main difficulty in our context is that, due to the infinite range of the potential, a non-integrable singularity appears in the angular collision kernel, making no longer valid the single-use of Lanford's strategy. Our proof relies then on a combination of Lanford's strategy, of tools developed recently by Bodineau, Gallagher and Saint-Raymond to study the collision process, and of new duality arguments to study the additional terms associated with the long-range interaction, leading to some explicit weak estimates.
Slip velocity and Knudsen layer in the lattice Boltzmann method for microscale flows.
Kim, Seung Hyun; Pitsch, Heinz; Boyd, Iain D
2008-02-01
We present mesoscopic fluid-wall interaction models for lattice Boltzmann (LB) model simulations of microscale flows. The exact solution of the slip velocity for the LB equation with the Bhatnagar-Gross-Krook collision operator is obtained for Poiseuille flow at finite Knudsen numbers. With a consistent definition of the Knudsen number, the slip coefficients of the LB equation with the standard D2Q9 scheme are found to be slightly larger than those of the Boltzmann equation with the same boundary condition, which makes the standard LB method remain quantitatively accurate only for small Knudsen numbers. By modifying the nonequilibrium energy flux or introducing the effective relaxation time, the LB method is analytically shown to reproduce the slip phenomena up to second order in the Knudsen number. For the standard LB method, the Knudsen layer is captured only with modification of the relaxation dynamics such as in the effective relaxation time model.
Steady-State Anderson Accelerated Coupling of Lattice Boltzmann and Navier–Stokes Solvers
Atanasov, Atanas
2016-10-17
We present an Anderson acceleration-based approach to spatially couple three-dimensional Lattice Boltzmann and Navier–Stokes (LBNS) flow simulations. This allows to locally exploit the computational features of both fluid flow solver approaches to the fullest extent and yields enhanced control to match the LB and NS degrees of freedom within the LBNS overlap layer. Designed for parallel Schwarz coupling, the Anderson acceleration allows for the simultaneous execution of both Lattice Boltzmann and Navier–Stokes solver. We detail our coupling methodology, validate it, and study convergence and accuracy of the Anderson accelerated coupling, considering three steady-state scenarios: plane channel flow, flow around a sphere and channel flow across a porous structure. We find that the Anderson accelerated coupling yields a speed-up (in terms of iteration steps) of up to 40% in the considered scenarios, compared to strictly sequential Schwarz coupling.
Prediction of sound absorption in rigid porous media with the lattice Boltzmann method
da Silva, Andrey Ricardo; Mareze, Paulo; Brandão, Eric
2016-02-01
In this work, sound absorption phenomena associated with the viscous shear stress within rigid porous media is investigated with a simple isothermal lattice Boltzmann BGK model. Simulations are conducted for different macroscopic material properties such as sample thickness and porosity and the results are compared with the exact analytical solution for materials with slit-like structure in terms of acoustic impedance and sound absorption coefficient. The numerical results agree very well with the exact solution, particularly for the sound absorption coefficient. The small deviations found in the low frequency limit for the real part of the acoustic impedance are attributed to the ratio between the thicknesses of the slit and the viscous boundary layer. The results suggest that the lattice Boltzmann method can be a very compelling numerical tool for simulating viscous sound absorption phenomena in the time domain, particularly due to its computational simplicity when compared to traditional continuum based techniques.
Prediction of sound absorption in rigid porous media with the lattice Boltzmann method
International Nuclear Information System (INIS)
Silva, Andrey Ricardo da; Mareze, Paulo; Brandão, Eric
2016-01-01
In this work, sound absorption phenomena associated with the viscous shear stress within rigid porous media is investigated with a simple isothermal lattice Boltzmann BGK model. Simulations are conducted for different macroscopic material properties such as sample thickness and porosity and the results are compared with the exact analytical solution for materials with slit-like structure in terms of acoustic impedance and sound absorption coefficient. The numerical results agree very well with the exact solution, particularly for the sound absorption coefficient. The small deviations found in the low frequency limit for the real part of the acoustic impedance are attributed to the ratio between the thicknesses of the slit and the viscous boundary layer. The results suggest that the lattice Boltzmann method can be a very compelling numerical tool for simulating viscous sound absorption phenomena in the time domain, particularly due to its computational simplicity when compared to traditional continuum based techniques. (paper)
Moment-based boundary conditions for lattice Boltzmann simulations of natural convection in cavities
Allen, Rebecca
2016-06-29
We study a multiple relaxation time lattice Boltzmann model for natural convection with moment-based boundary conditions. The unknown primary variables of the algorithm at a boundary are found by imposing conditions directly upon hydrodynamic moments, which are then translated into conditions for the discrete velocity distribution functions. The method is formulated so that it is consistent with the second order implementation of the discrete velocity Boltzmann equations for fluid flow and temperature. Natural convection in square cavities is studied for Rayleigh numbers ranging from 103 to 108. An excellent agreement with benchmark data is observed and the flow fields are shown to converge with second order accuracy. Copyright © 2016 Inderscience Enterprises Ltd.
Accelerated lattice Boltzmann model for colloidal suspensions rheology and interface morphology
Farhat, Hassan; Kondaraju, Sasidhar
2014-01-01
Colloids are ubiquitous in the food, medical, cosmetics, polymers, water purification, and pharmaceutical industries. The thermal, mechanical, and storage properties of colloids are highly dependent on their interface morphology and their rheological behavior. Numerical methods provide a convenient and reliable tool for the study of colloids. Accelerated Lattice Boltzmann Model for Colloidal Suspensions introduce the main building-blocks for an improved lattice Boltzmann–based numerical tool designed for the study of colloidal rheology and interface morphology. This book also covers the migrating multi-block used to simulate single component, multi-component, multiphase, and single component multiphase flows and their validation by experimental, numerical, and analytical solutions. Among other topics discussed are the hybrid lattice Boltzmann method (LBM) for surfactant-covered droplets; biological suspensions such as blood; used in conjunction with the suppression of coalescence for investigating the...
On the Stability of the Finite Difference based Lattice Boltzmann Method
El-Amin, Mohamed
2013-06-01
This paper is devoted to determining the stability conditions for the finite difference based lattice Boltzmann method (FDLBM). In the current scheme, the 9-bit two-dimensional (D2Q9) model is used and the collision term of the Bhatnagar- Gross-Krook (BGK) is treated implicitly. The implicitness of the numerical scheme is removed by introducing a new distribution function different from that being used. Therefore, a new explicit finite-difference lattice Boltzmann method is obtained. Stability analysis of the resulted explicit scheme is done using Fourier expansion. Then, stability conditions in terms of time and spatial steps, relaxation time and explicitly-implicitly parameter are determined by calculating the eigenvalues of the given difference system. The determined conditions give the ranges of the parameters that have stable solutions.
Numerical study of convection in phase change material based on Lattice-Boltzmann method
Zhang, Tianyu; Feng, Ying; Zhao, Zhening
2017-06-01
In this paper, the lattice Boltzmann method was studied for the phase change process with convective heat transfer in phase change energy storage materials. Firstly, the macroscopic heat transfer equations for the phase change process with convective heat transfer was given, by which we built the lattice Boltzmann equations for solving the problems. In the model, the speed model of D2Q9 was selected, and the boundary conditions including of non-equilibrium extrapolation and bounce back scheme were selected. Then, the effects of different Rayleigh number on the temperature field and velocity field were analyzed. Further research in a square cavity heat transfer processes with high temperature object and low temperature object were studied, in order to observe the effects of different temperature objects in the phase change process using the changes of phase field.
Directory of Open Access Journals (Sweden)
L.B.Bhuiyan
2005-01-01
Full Text Available The density functional and modified Poisson-Boltzmann descriptions of a spherical (electric double layer are compared and contrasted vis-a-vis existing Monte Carlo simulation data (for small ion diameter 4.25·10-10 m from the literature for a range of physical parameters such as macroion surface charge, macroion radius, valencies of the small ions, and electrolyte concentration. Overall, the theoretical predictions are seen to be remarkably consistent between themselves, being also in very good agreement with the simulations. Some modified Poisson-Boltzmann results for the zeta potential at small ion diameters of 3 and 2·10-10 m are also reported.
Magnetic nanoparticles in fluid environment: combining molecular dynamics and Lattice-Boltzmann
Melenev, Petr
2017-06-01
Hydrodynamic interactions between magnetic nanoparticles suspended in the Newtonian liquid are accounted for using a combination of the lattice Boltzmann method and molecular dynamics simulations. Nanoparticle is modelled by the system of molecular dynamics material points (which form structure resembles raspberry) coupled to the lattice Boltzmann fluid. The hydrodynamic coupling between the colloids is studied by simulations of the thermo-induced rotational diffusion of two raspberry objects. It was found that for the considered range of model parameters the approaching of the raspberries leads to slight retard of the relaxation process. The presence of the weak magnetic dipolar interaction between the objects leads to modest decrease of the relaxation time and the extent of the acceleration of the diffusion is intensified along with magnetic forces.
Introduction to probability with statistical applications
Schay, Géza
2016-01-01
Now in its second edition, this textbook serves as an introduction to probability and statistics for non-mathematics majors who do not need the exhaustive detail and mathematical depth provided in more comprehensive treatments of the subject. The presentation covers the mathematical laws of random phenomena, including discrete and continuous random variables, expectation and variance, and common probability distributions such as the binomial, Poisson, and normal distributions. More classical examples such as Montmort's problem, the ballot problem, and Bertrand’s paradox are now included, along with applications such as the Maxwell-Boltzmann and Bose-Einstein distributions in physics. Key features in new edition: * 35 new exercises * Expanded section on the algebra of sets * Expanded chapters on probabilities to include more classical examples * New section on regression * Online instructors' manual containing solutions to all exercises
Application of Boltzmann equation to electron transmission and seconary electron emission
International Nuclear Information System (INIS)
Lanteri, H.; Bindi, R.; Rostaing, P.
1979-01-01
A method is presented for numerical treatment of integro-differential equation, based upon finite difference techniques. This method allows to formulate in a satisfactory manner the Boltzmann's equation applied to backscattering, transmission and secondary emission of metallic targets, avoiding must of the restrictive hypothesis, used until now in these models. For aluminium, the calculated energy spectra, angular distribution, transmission and backscattering coefficients, and secondary emission yield, are found to be in good agreement with experiment [fr
On the asymptotic behavior of a boltzmann-type price formation model
Burger, Martin
2014-01-01
In this paper we study the asymptotic behavior of a Boltzmann-type price formation model, which describes the trading dynamics in a financial market. In many of these markets trading happens at high frequencies and low transaction costs. This observation motivates the study of the limit as the number of transactions k tends to infinity, the transaction cost a to zero and ka=const. Furthermore we illustrate the price dynamics with numerical simulations © 2014 International Press.
Directory of Open Access Journals (Sweden)
Peilin Zhang
2015-01-01
Full Text Available We present an algorithm of quantum restricted Boltzmann machine network based on quantum gates. The algorithm is used to initialize the procedure that adjusts the qubit and weights. After adjusting, the network forms an unsupervised generative model that gives better classification performance than other discriminative models. In addition, we show how the algorithm can be constructed with quantum circuit for quantum computer.
Dorschner, B.; Chikatamarla, S. S.; Karlin, I. V.
2017-06-01
Entropic lattice Boltzmann methods have been developed to alleviate intrinsic stability issues of lattice Boltzmann models for under-resolved simulations. Its reliability in combination with moving objects was established for various laminar benchmark flows in two dimensions in our previous work [B. Dorschner, S. Chikatamarla, F. Bösch, and I. Karlin, J. Comput. Phys. 295, 340 (2015), 10.1016/j.jcp.2015.04.017] as well as for three-dimensional one-way coupled simulations of engine-type geometries in B . Dorschner, F. Bösch, S. Chikatamarla, K. Boulouchos, and I. Karlin [J. Fluid Mech. 801, 623 (2016), 10.1017/jfm.2016.448] for flat moving walls. The present contribution aims to fully exploit the advantages of entropic lattice Boltzmann models in terms of stability and accuracy and extends the methodology to three-dimensional cases, including two-way coupling between fluid and structure and then turbulence and deforming geometries. To cover this wide range of applications, the classical benchmark of a sedimenting sphere is chosen first to validate the general two-way coupling algorithm. Increasing the complexity, we subsequently consider the simulation of a plunging SD7003 airfoil in the transitional regime at a Reynolds number of Re =40 000 and, finally, to access the model's performance for deforming geometries, we conduct a two-way coupled simulation of a self-propelled anguilliform swimmer. These simulations confirm the viability of the new fluid-structure interaction lattice Boltzmann algorithm to simulate flows of engineering relevance.
Dynamically adaptive Lattice Boltzmann simulation of shallow water flows with the Peano framework
Neumann, Philipp
2015-09-01
© 2014 Elsevier Inc. All rights reserved. We present a dynamically adaptive Lattice Boltzmann (LB) implementation for solving the shallow water equations (SWEs). Our implementation extends an existing LB component of the Peano framework. We revise the modular design with respect to the incorporation of new simulation aspects and LB models. The basic SWE-LB implementation is validated in different breaking dam scenarios. We further provide a numerical study on stability of the MRT collision operator used in our simulations.
Darquié Benoît; Mejri Sinda; Sow Papa Lat Tabara; Lemarchand Cyril; Triki Meriam; Tokunaga Sean K.; Bordé Christian J.; Chardonnet Christian; Daussy Christophe
2013-01-01
proceedings of the ICAP 2012 conference (23rd International Conference on Atomic Physics); International audience; Accurate molecular spectroscopy in the mid-infrared region allows precision measurements of fundamental constants. For instance, measuring the linewidth of an isolated Doppler-broadened absorption line of ammonia around 10 µm enables a determination of the Boltzmann constant k B. We report on our latest measurements. By fitting this lineshape to several models which include Dicke...
Some progress in the development of lattice Boltzmann methods for dissipative MHD
Czech Academy of Sciences Publication Activity Database
Macnab, A.; Vahala, G.; Vahala, L.; Pavlo, Pavol; Soe, M.
2002-01-01
Roč. 52, supplement D (2002), s. 59-64 ISSN 0011-4626. [Symposium on Plasma Physics and Technology/20th./. Prague, 10.06.2002-13.06.2002] R&D Projects: GA ČR GA202/00/1216 Institutional research plan: CEZ:AV0Z2043910 Keywords : Boltzmann models Subject RIV: BL - Plasma and Gas Discharge Physics Impact factor: 0.311, year: 2002
Stability of Nonlinear Wave Patterns to the Bipolar Vlasov-Poisson-Boltzmann System
Li, Hailiang; Wang, Yi; Yang, Tong; Zhong, Mingying
2018-04-01
The main purpose of the present paper is to investigate the nonlinear stability of viscous shock waves and rarefaction waves for the bipolar Vlasov-Poisson-Boltzmann (VPB) system. To this end, motivated by the micro-macro decomposition to the Boltzmann equation in Liu and Yu (Commun Math Phys 246:133-179, 2004) and Liu et al. (Physica D 188:178-192, 2004), we first set up a new micro-macro decomposition around the local Maxwellian related to the bipolar VPB system and give a unified framework to study the nonlinear stability of the basic wave patterns to the system. Then, as applications of this new decomposition, the time-asymptotic stability of the two typical nonlinear wave patterns, viscous shock waves and rarefaction waves are proved for the 1D bipolar VPB system. More precisely, it is first proved that the linear superposition of two Boltzmann shock profiles in the first and third characteristic fields is nonlinearly stable to the 1D bipolar VPB system up to some suitable shifts without the zero macroscopic mass conditions on the initial perturbations. Then the time-asymptotic stability of the rarefaction wave fan to compressible Euler equations is proved for the 1D bipolar VPB system. These two results are concerned with the nonlinear stability of wave patterns for Boltzmann equation coupled with additional (electric) forces, which together with spectral analysis made in Li et al. (Indiana Univ Math J 65(2):665-725, 2016) sheds light on understanding the complicated dynamic behaviors around the wave patterns in the transportation of charged particles under the binary collisions, mutual interactions, and the effect of the electrostatic potential forces.
International Nuclear Information System (INIS)
Battaglia, Onofrio Rosario; Di Paola, Benedetto
2015-01-01
This paper describes a quantitative method to analyse an openended questionnaire. Student responses to a specially designed written questionnaire are quantitatively analysed by not hierarchical clustering called k-means method. Through this we can characterise behaviour students with respect their expertise to formulate explanations for phenomena or processes and/or use a given model in the different context. The physics topic is about the Boltzmann Factor, which allows the students to have a unifying view of different phenomena in different contexts.
Numerical Treatment of the Boltzmann Equation for Self-Propelled Particle Systems
Directory of Open Access Journals (Sweden)
Florian Thüroff
2014-11-01
Full Text Available Kinetic theories constitute one of the most promising tools to decipher the characteristic spatiotemporal dynamics in systems of actively propelled particles. In this context, the Boltzmann equation plays a pivotal role, since it provides a natural translation between a particle-level description of the system’s dynamics and the corresponding hydrodynamic fields. Yet, the intricate mathematical structure of the Boltzmann equation substantially limits the progress toward a full understanding of this equation by solely analytical means. Here, we propose a general framework to numerically solve the Boltzmann equation for self-propelled particle systems in two spatial dimensions and with arbitrary boundary conditions. We discuss potential applications of this numerical framework to active matter systems and use the algorithm to give a detailed analysis to a model system of self-propelled particles with polar interactions. In accordance with previous studies, we find that spatially homogeneous isotropic and broken-symmetry states populate two distinct regions in parameter space, which are separated by a narrow region of spatially inhomogeneous, density-segregated moving patterns. We find clear evidence that these three regions in parameter space are connected by first-order phase transitions and that the transition between the spatially homogeneous isotropic and polar ordered phases bears striking similarities to liquid-gas phase transitions in equilibrium systems. Within the density-segregated parameter regime, we find a novel stable limit-cycle solution of the Boltzmann equation, which consists of parallel lanes of polar clusters moving in opposite directions, so as to render the overall symmetry of the system’s ordered state nematic, despite purely polar interactions on the level of single particles.
Lattice Boltzmann method used to simulate particle motion in a conduit
Czech Academy of Sciences Publication Activity Database
Dolanský, Jindřich; Chára, Zdeněk; Vlasák, Pavel; Kysela, Bohuš
2017-01-01
Roč. 65, č. 2 (2017), s. 105-113 ISSN 0042-790X R&D Projects: GA ČR GA15-18870S Institutional support: RVO:67985874 Keywords : Lattice Boltzmann method * particle motion * particle–fluid interaction * PIV * particle tracking Subject RIV: BK - Fluid Dynamics OBOR OECD: Fluids and plasma physics (including surface physics) Impact factor: 1.654, year: 2016
An Improved Immersed-Boundary Algorithm for Fluid-Solid Interaction un Lattice-Boltzmann Simulations
Boroni, Gustavo Adolfo; Dottori, Javier Alejandro; Dalponte, Diego David; Rinaldi, Pablo Rafael; Clausse, Alejandro
2015-01-01
An improved algorithm combining the features of the lattice Boltzmann and the immersed-boundary methods is presented. Following previous formulations, the method represents a fluid con-strained by flexible boundaries by means of a force term acting on the cells adjacent to the boundary, which in turn is moved by the fluid. The present al-gorithm introduces a more efficient iteration proce-dure to calculate the fluid-boundary interaction, which facilitates the implementation. The simulations w...
Lattice Boltzmann simulation of fluid flow induced by thermal effect in heterogeneity porous media
Directory of Open Access Journals (Sweden)
Hou Peng
2017-01-01
Full Text Available In this paper, a coupled lattice Boltzmann model is used to visually study fluid flow induced by thermal effect in heterogeneity porous media reconstructed by the quartet structure generation set. The fluid flow behavior inside porous media is presented and analyzed under different conditions. The simulation results indicate that the pore morphological properties of porous media and the Rayleigh number have noticeable impact on the velocity distribution and flow rate of fluid.
Effects of nanoparticles on melting process with phase-change using the lattice Boltzmann method
Directory of Open Access Journals (Sweden)
Ahmed M. Ibrahem
Full Text Available In this work, the problem of nanoparticles dispersion effects on coupled heat transfer and solid-liquid phase change has been studied. The lattice Boltzmann method (LBM enthalpy-based is employed. The collision model of lattice Bhatnagar-Gross-Krook (LBGK is used to solve the problem of 1D melting by conduction. On the other hand, we use the model of multi-distribution functions (MDF to calculate the density, the velocity and the temperature for the problem of 2D melting by free convection, associated with different boundary conditions. In these simulations, the volume fractions of copper nanoparticles (0â2% added to water-base fluid and Rayleigh numbers of 103â105. We use the Chapman-Enskog expansion to derive the governing macroscopic quantities from the mesoscopic lattice Boltzmann equation. The results obtained by these models have been compared to an analytical solution or other numerical methods. The effects of nanoparticles on conduction and natural convection during the melting process have been investigated. Moreover, the influences of nanoparticles on moving of the phase change front, the thermal conductivity and the latent heat of fusion are also studied. Keywords: Lattice Boltzmann method, Nanofluids, Conduction melting, Convection melting, BGK collision model
Munafò, A; Panesi, M; Magin, T E
2014-02-01
A Boltzmann rovibrational collisional coarse-grained model is proposed to reduce a detailed kinetic mechanism database developed at NASA Ames Research Center for internal energy transfer and dissociation in N(2)-N interactions. The coarse-grained model is constructed by lumping the rovibrational energy levels of the N(2) molecule into energy bins. The population of the levels within each bin is assumed to follow a Boltzmann distribution at the local translational temperature. Excitation and dissociation rate coefficients for the energy bins are obtained by averaging the elementary rate coefficients. The energy bins are treated as separate species, thus allowing for non-Boltzmann distributions of their populations. The proposed coarse-grained model is applied to the study of nonequilibrium flows behind normal shock waves and within converging-diverging nozzles. In both cases, the flow is assumed inviscid and steady. Computational results are compared with those obtained by direct solution of the master equation for the rovibrational collisional model and a more conventional multitemperature model. It is found that the proposed coarse-grained model is able to accurately resolve the nonequilibrium dynamics of internal energy excitation and dissociation-recombination processes with only 20 energy bins. Furthermore, the proposed coarse-grained model provides a superior description of the nonequilibrium phenomena occurring in shock heated and nozzle flows when compared with the conventional multitemperature models.
Feng, Yue
Plasma is currently a hot topic and it has many significant applications due to its composition of both positively and negatively charged particles. The energy distribution function is important in plasma science since it characterizes the ability of the plasma to affect chemical reactions, affect physical outcomes, and drive various applications. The Boltzmann Transport Equation is an important kinetic equation that provides an accurate basis for characterizing the distribution function---both in energy and space. This dissertation research proposes a multi-term approximation to solve the Boltzmann Transport Equation by treating the relaxation process using an expansion of the electron distribution function in Legendre polynomials. The elastic and 29 inelastic cross sections for electron collisions with nitrogen molecules (N2) and singly ionized nitrogen molecules ( N+2 ) have been used in this application of the Boltzmann Transport Equation. Different numerical methods have been considered to compare the results. The numerical methods discussed in this thesis are the implicit time-independent method, the time-dependent Euler method, the time-dependent Runge-Kutta method, and finally the implicit time-dependent relaxation method by generating the 4-way grid with a matrix solver. The results show that the implicit time-dependent relaxation method is the most accurate and stable method for obtaining reliable results. The results were observed to match with the published experimental data rather well.
Magnetic nanoparticles in fluid environment: combining molecular dynamics and Lattice-Boltzmann
Energy Technology Data Exchange (ETDEWEB)
Melenev, Petr, E-mail: melenev@icmm.ru [Ural Federal University, 4, Turgeneva str., 620000 Ekaterinburg (Russian Federation); Institute of Continuous Media Mechanics, 1, Koroleva str., 614013 Perm (Russian Federation)
2017-06-01
Hydrodynamic interactions between magnetic nanoparticles suspended in the Newtonian liquid are accounted for using a combination of the lattice Boltzmann method and molecular dynamics simulations. Nanoparticle is modelled by the system of molecular dynamics material points (which form structure resembles raspberry) coupled to the lattice Boltzmann fluid. The hydrodynamic coupling between the colloids is studied by simulations of the thermo-induced rotational diffusion of two raspberry objects. It was found that for the considered range of model parameters the approaching of the raspberries leads to slight retard of the relaxation process. The presence of the weak magnetic dipolar interaction between the objects leads to modest decrease of the relaxation time and the extent of the acceleration of the diffusion is intensified along with magnetic forces. - Highlights: • The combination of molecular dynamics and lattice Boltzmann method is utilized for the reveal of the role of hydrodynamic interaction in rotational dynamics of colloid particles. • The verification of the model parameters is done based on the comparison with the results of Langevin dynamics. • For the task of free rotational diffusion of the pair of colloid particles the influence of the hydrodynamic interactions on the relaxation time is examined in the case of nonmagnetic particles and at the presence of weak dipolar interaction.
A generalized Poisson and Poisson-Boltzmann solver for electrostatic environments
Energy Technology Data Exchange (ETDEWEB)
Fisicaro, G., E-mail: giuseppe.fisicaro@unibas.ch; Goedecker, S. [Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel (Switzerland); Genovese, L. [University of Grenoble Alpes, CEA, INAC-SP2M, L-Sim, F-38000 Grenoble (France); Andreussi, O. [Institute of Computational Science, Università della Svizzera Italiana, Via Giuseppe Buffi 13, CH-6904 Lugano (Switzerland); Theory and Simulations of Materials (THEOS) and National Centre for Computational Design and Discovery of Novel Materials (MARVEL), École Polytechnique Fédérale de Lausanne, Station 12, CH-1015 Lausanne (Switzerland); Marzari, N. [Theory and Simulations of Materials (THEOS) and National Centre for Computational Design and Discovery of Novel Materials (MARVEL), École Polytechnique Fédérale de Lausanne, Station 12, CH-1015 Lausanne (Switzerland)
2016-01-07
The computational study of chemical reactions in complex, wet environments is critical for applications in many fields. It is often essential to study chemical reactions in the presence of applied electrochemical potentials, taking into account the non-trivial electrostatic screening coming from the solvent and the electrolytes. As a consequence, the electrostatic potential has to be found by solving the generalized Poisson and the Poisson-Boltzmann equations for neutral and ionic solutions, respectively. In the present work, solvers for both problems have been developed. A preconditioned conjugate gradient method has been implemented for the solution of the generalized Poisson equation and the linear regime of the Poisson-Boltzmann, allowing to solve iteratively the minimization problem with some ten iterations of the ordinary Poisson equation solver. In addition, a self-consistent procedure enables us to solve the non-linear Poisson-Boltzmann problem. Both solvers exhibit very high accuracy and parallel efficiency and allow for the treatment of periodic, free, and slab boundary conditions. The solver has been integrated into the BigDFT and Quantum-ESPRESSO electronic-structure packages and will be released as an independent program, suitable for integration in other codes.
A generalized Poisson and Poisson-Boltzmann solver for electrostatic environments
International Nuclear Information System (INIS)
Fisicaro, G.; Goedecker, S.; Genovese, L.; Andreussi, O.; Marzari, N.
2016-01-01
The computational study of chemical reactions in complex, wet environments is critical for applications in many fields. It is often essential to study chemical reactions in the presence of applied electrochemical potentials, taking into account the non-trivial electrostatic screening coming from the solvent and the electrolytes. As a consequence, the electrostatic potential has to be found by solving the generalized Poisson and the Poisson-Boltzmann equations for neutral and ionic solutions, respectively. In the present work, solvers for both problems have been developed. A preconditioned conjugate gradient method has been implemented for the solution of the generalized Poisson equation and the linear regime of the Poisson-Boltzmann, allowing to solve iteratively the minimization problem with some ten iterations of the ordinary Poisson equation solver. In addition, a self-consistent procedure enables us to solve the non-linear Poisson-Boltzmann problem. Both solvers exhibit very high accuracy and parallel efficiency and allow for the treatment of periodic, free, and slab boundary conditions. The solver has been integrated into the BigDFT and Quantum-ESPRESSO electronic-structure packages and will be released as an independent program, suitable for integration in other codes
Analysis of a bubble coalescence in the multiphase lattice Boltzmann method
International Nuclear Information System (INIS)
Ryu, Seung Yeob; Park, Cheon Tae; Lee, Chung Chan; Kim, Keung Koo
2008-01-01
Recently, the lattice Boltzmann method (LBM) has gained much attention for its ability to simulate fluid flows, and for its potential advantages over a conventional CFD method. The key advantages of LBM are, (1) suitability for parallel computations, (2) absence of the need to solve the time-consuming Poisson equation for a pressure, and (3) an ease with multiphase flows, complex geometries and interfacial dynamics may be treated. To study the effect of the mobility coefficient Γ and the width of the interface layer, two stationary bubbles without a collision are considered. The gap of the two bubbles is taken as 4, while the width of the interface (w) and the mobility coefficient Γ are varied. In the present work, the lattice Boltzmann model for multiphase flows proposed by Zheng et al. is used for simulating two stationary bubbles without a collision. By adopting a finite difference gradient operator of a sufficient isotropy, the spurious currents can be made smaller. The main objective of the present work is to establish the lattice Boltzmann method as a viable tool for the simulation of multiphase or multi-component flows
Urano, C.; Yamazawa, K.; Kaneko, N.-H.
2017-12-01
We report on our measurement of the Boltzmann constant by Johnson noise thermometry (JNT) using an integrated quantum voltage noise source (IQVNS) that is fully implemented with superconducting integrated circuit technology. The IQVNS generates calculable pseudo white noise voltages to calibrate the JNT system. The thermal noise of a sensing resistor placed at the temperature of the triple point of water was measured precisely by the IQVNS-based JNT. We accumulated data of more than 429 200 s in total (over 6 d) and used the Akaike information criterion to estimate the fitting frequency range for the quadratic model to calculate the Boltzmann constant. Upon detailed evaluation of the uncertainty components, the experimentally obtained Boltzmann constant was k=1.380 6436× {{10}-23} J K-1 with a relative combined uncertainty of 10.22× {{10}-6} . The value of k is relatively -3.56× {{10}-6} lower than the CODATA 2014 value (Mohr et al 2016 Rev. Mod. Phys. 88 035009).
A Lattice-Boltzmann model for simulating bedform-induced hyporheic exchange
Dapelo, D.; Bridgeman, J.; Krause, S.
2016-12-01
Bedform-induced hyporheic exchange plays a fundamental role in the ecohydrological and biogeochemical functioning of aquifer-river interfaces. The understanding of the complex interchange of hyporheic exchange fluxes, solute and energy transport between surface and groundwater is fundamental to design effective management, restoration and pollution mitigation strategies. For the first time, the Lattice-Boltzmann method was used to simulate 2D hyporheic exchange flow across a succession of dunes. The velocity field in both surface and groundwater was simulated directly; then, residence times were computed through post-processing. As a novelty to most previous applications of similar computational fluid dynamics models, a grid-independence test was performed for to analyse independence of the results from the mesh choice. The Lattice-Boltzmann simulation results are compared to previous fluid dynamic models of similar bedforms, and the impact of the bedform on hyporheic exchange flow dynamics is discussed. As an advantage, both the free-flow and the hyporheic exchange flow are simulated within the same model, thus removing the need of developing two distinct models as well as the coupling between them: the model dynamically reproduces turbulent Navier-Stokes (surface water) or generalized Darcian (groundwater) flow, depending only on the local value of the porosity field. Through this model, the critical advantages of the Lattice-Boltzmann method, consisting of unparalleled computational parsimony, meshing simplicity and attitude towards diffuse computing, are made available for a wide range of similar applications.
Lattice Boltzmann simulation for temperature-sensitive magnetic fluids in a porous square cavity
International Nuclear Information System (INIS)
Jin Licong; Zhang Xinrong; Niu Xiaodong
2012-01-01
A lattice Boltzmann method is developed to simulate temperature-sensitive magnetic fluids in a porous cavity. In the simulation, the magnetic force, efficient gravity, viscous loss term and geometric loss term in porous medium are imported to the momentum equation. To test the reliability of the method, a validation with water in porous cavity is carried out. Good agreements with the previous results verify that the present lattice Boltzmann method is promising for simulation of magnetic fluids in porous medium. In this study, we investigate the change of magnetization with external magnetic field, and we present numerical results for the streamlines, isotherms, and magnetization at vertical or horizontal mid-profiles for different values of Ram. In addition, Nusselt numbers changing with magnetic Rayleigh numbers are also investigated. - Highlights: → Developed a lattice Boltzmann method for magnetic nano-fluids in porous cavity. → Clarified flow and heat transfer for different values of (magnetic) Rayleigh numbers. → Heat transfer enhancement for magnetic fluid in porous cavity.
International Nuclear Information System (INIS)
Rodriguez, Barbara D. do Amaral; Vilhena, Marco Tullio
2009-01-01
Questions regarding accuracy and efficiency of deterministic transport methods are still on our mind today, even with modern supercomputers. The most versatile and widely used deterministic methods are the P N approximation, the S N method (discrete ordinates method) and their variants. In the discrete ordinates (S N ) formulations of the transport equation, it is assumed that the linearized Boltzmann equation only holds for a set of distinct numerical values of the direction-of-motion variables. In this work, looking forward to confirm the capabilities of deterministic methods in obtaining accurate results, we present a general overview of deterministic methods to solve the Boltzmann transport equation for neutral and charged particles. First, we describe a review in the Laplace transform technique applied to S N two dimensional transport equation in a rectangular domain considering Compton scattering. Next, we solved the Fokker-Planck (FP) equation, an alternative approach for the Boltzmann transport equation, assuming a monoenergetic electron beam in a rectangular domain. The main idea relies on applying the P N approximation, a recent advance in the class of deterministic methods, in the angular variable, to the two dimensional Fokker-Planck equation and then applying the Laplace Transform in the spatial x-variable. Numerical results are given to illustrate the accuracy of deterministic methods presented. (author)
Magnetic nanoparticles in fluid environment: combining molecular dynamics and Lattice-Boltzmann
International Nuclear Information System (INIS)
Melenev, Petr
2017-01-01
Hydrodynamic interactions between magnetic nanoparticles suspended in the Newtonian liquid are accounted for using a combination of the lattice Boltzmann method and molecular dynamics simulations. Nanoparticle is modelled by the system of molecular dynamics material points (which form structure resembles raspberry) coupled to the lattice Boltzmann fluid. The hydrodynamic coupling between the colloids is studied by simulations of the thermo-induced rotational diffusion of two raspberry objects. It was found that for the considered range of model parameters the approaching of the raspberries leads to slight retard of the relaxation process. The presence of the weak magnetic dipolar interaction between the objects leads to modest decrease of the relaxation time and the extent of the acceleration of the diffusion is intensified along with magnetic forces. - Highlights: • The combination of molecular dynamics and lattice Boltzmann method is utilized for the reveal of the role of hydrodynamic interaction in rotational dynamics of colloid particles. • The verification of the model parameters is done based on the comparison with the results of Langevin dynamics. • For the task of free rotational diffusion of the pair of colloid particles the influence of the hydrodynamic interactions on the relaxation time is examined in the case of nonmagnetic particles and at the presence of weak dipolar interaction.
Student understanding of Taylor series expansions in statistical mechanics
Directory of Open Access Journals (Sweden)
Trevor I. Smith
2013-08-01
Full Text Available One goal of physics instruction is to have students learn to make physical meaning of specific mathematical expressions, concepts, and procedures in different physical settings. As part of research investigating student learning in statistical physics, we are developing curriculum materials that guide students through a derivation of the Boltzmann factor using a Taylor series expansion of entropy. Using results from written surveys, classroom observations, and both individual think-aloud and teaching interviews, we present evidence that many students can recognize and interpret series expansions, but they often lack fluency in creating and using a Taylor series appropriately, despite previous exposures in both calculus and physics courses.
Student understanding of Taylor series expansions in statistical mechanics
Smith, Trevor I.; Thompson, John R.; Mountcastle, Donald B.
2013-12-01
One goal of physics instruction is to have students learn to make physical meaning of specific mathematical expressions, concepts, and procedures in different physical settings. As part of research investigating student learning in statistical physics, we are developing curriculum materials that guide students through a derivation of the Boltzmann factor using a Taylor series expansion of entropy. Using results from written surveys, classroom observations, and both individual think-aloud and teaching interviews, we present evidence that many students can recognize and interpret series expansions, but they often lack fluency in creating and using a Taylor series appropriately, despite previous exposures in both calculus and physics courses.
A new lattice Boltzmann equation to simulate density-driven convection of carbon dioxide
Allen, Rebecca
2013-01-01
The storage of CO2 in fluid-filled geological formations has been carried out for more than a decade in locations around the world. After CO2 has been injected into the aquifer and has moved laterally under the aquifer\\'s cap-rock, density-driven convection becomes an important transport process to model. However, the challenge lies in simulating this transport process accurately with high spatial resolution and low CPU cost. This issue can be addressed by using the lattice Boltzmann equation (LBE) to formulate a model for a similar scenario when a solute diffuses into a fluid and density differences lead to convective mixing. The LBE is a promising alternative to the traditional methods of computational fluid dynamics. Rather than discretizing the system of partial differential equations of classical continuum mechanics directly, the LBE is derived from a velocity-space truncation of the Boltzmann equation of classical kinetic theory. We propose an extension to the LBE, which can accurately predict the transport of dissolved CO2 in water, as a step towards fluid-filled porous media simulations. This is achieved by coupling two LBEs, one for the fluid flow and one for the convection and diffusion of CO2. Unlike existing lattice Boltzmann equations for porous media flow, our model is derived from a system of moment equations and a Crank-Nicolson discretization of the velocity-truncated Boltzmann equation. The forcing terms are updated locally without the need for additional central difference approximation. Therefore our model preserves all the computational advantages of the single-phase lattice Boltzmann equation and is formally second-order accurate in both space and time. Our new model also features a novel implementation of boundary conditions, which is simple to implement and does not suffer from the grid-dependent error that is present in the standard "bounce-back" condition. The significance of using the LBE in this work lies in the ability to efficiently
A mass-conserving multiphase lattice Boltzmann model for simulation of multiphase flows
Niu, Xiao-Dong; Li, You; Ma, Yi-Ren; Chen, Mu-Feng; Li, Xiang; Li, Qiao-Zhong
2018-01-01
In this study, a mass-conserving multiphase lattice Boltzmann (LB) model is proposed for simulating the multiphase flows. The proposed model developed in the present study is to improve the model of Shao et al. ["Free-energy-based lattice Boltzmann model for simulation of multiphase flows with density contrast," Phys. Rev. E 89, 033309 (2014)] by introducing a mass correction term in the lattice Boltzmann model for the interface. The model of Shao et al. [(the improved Zheng-Shu-Chew (Z-S-C model)] correctly considers the effect of the local density variation in momentum equation and has an obvious improvement over the Zheng-Shu-Chew (Z-S-C) model ["A lattice Boltzmann model for multiphase flows with large density ratio," J. Comput. Phys. 218(1), 353-371 (2006)] in terms of solution accuracy. However, due to the physical diffusion and numerical dissipation, the total mass of each fluid phase cannot be conserved correctly. To solve this problem, a mass correction term, which is similar to the one proposed by Wang et al. ["A mass-conserved diffuse interface method and its application for incompressible multiphase flows with large density ratio," J. Comput. Phys. 290, 336-351 (2015)], is introduced into the lattice Boltzmann equation for the interface to compensate the mass losses or offset the mass increase. Meanwhile, to implement the wetting boundary condition and the contact angle, a geometric formulation and a local force are incorporated into the present mass-conserving LB model. The proposed model is validated by verifying the Laplace law, simulating both one and two aligned droplets splashing onto a liquid film, droplets standing on an ideal wall, droplets with different wettability splashing onto smooth wax, and bubbles rising under buoyancy. Numerical results show that the proposed model can correctly simulate multiphase flows. It was found that the mass is well-conserved in all cases considered by the model developed in the present study. The developed
Statistical characterization of the standard map
Ruiz, Guiomar; Tirnakli, Ugur; Borges, Ernesto P.; Tsallis, Constantino
2017-06-01
The standard map, paradigmatic conservative system in the (x, p) phase space, has been recently shown (Tirnakli and Borges (2016 Sci. Rep. 6 23644)) to exhibit interesting statistical behaviors directly related to the value of the standard map external parameter K. A comprehensive statistical numerical description is achieved in the present paper. More precisely, for large values of K (e.g. K = 10) where the Lyapunov exponents are neatly positive over virtually the entire phase space consistently with Boltzmann-Gibbs (BG) statistics, we verify that the q-generalized indices related to the entropy production q{ent} , the sensitivity to initial conditions q{sen} , the distribution of a time-averaged (over successive iterations) phase-space coordinate q{stat} , and the relaxation to the equilibrium final state q{rel} , collapse onto a fixed point, i.e. q{ent}=q{sen}=q{stat}=q{rel}=1 . In remarkable contrast, for small values of K (e.g. K = 0.2) where the Lyapunov exponents are virtually zero over the entire phase space, we verify q{ent}=q{sen}=0 , q{stat} ≃ 1.935 , and q{rel} ≃1.4 . The situation corresponding to intermediate values of K, where both stable orbits and a chaotic sea are present, is discussed as well. The present results transparently illustrate when BG behavior and/or q-statistical behavior are observed.
Tellinghuisen, Joel
2008-01-01
The method of least squares is probably the most powerful data analysis tool available to scientists. Toward a fuller appreciation of that power, this work begins with an elementary review of statistics fundamentals, and then progressively increases in sophistication as the coverage is extended to the theory and practice of linear and nonlinear least squares. The results are illustrated in application to data analysis problems important in the life sciences. The review of fundamentals includes the role of sampling and its connection to probability distributions, the Central Limit Theorem, and the importance of finite variance. Linear least squares are presented using matrix notation, and the significance of the key probability distributions-Gaussian, chi-square, and t-is illustrated with Monte Carlo calculations. The meaning of correlation is discussed, including its role in the propagation of error. When the data themselves are correlated, special methods are needed for the fitting, as they are also when fitting with constraints. Nonlinear fitting gives rise to nonnormal parameter distributions, but the 10% Rule of Thumb suggests that such problems will be insignificant when the parameter is sufficiently well determined. Illustrations include calibration with linear and nonlinear response functions, the dangers inherent in fitting inverted data (e.g., Lineweaver-Burk equation), an analysis of the reliability of the van't Hoff analysis, the problem of correlated data in the Guggenheim method, and the optimization of isothermal titration calorimetry procedures using the variance-covariance matrix for experiment design. The work concludes with illustrations on assessing and presenting results.
Energy Technology Data Exchange (ETDEWEB)
Ortega J, R.; Valle G, E. del [IPN-ESFM, 07738 Mexico D.F. (Mexico)]. e-mail: roj@correo.azc.uam.mx
2003-07-01
There are carried out charge and energy calculations deposited due to the interaction of electrons with a plate of a certain material, solving numerically the electron transport equation for the Boltzmann-Fokker-Planck approach of first order in plate geometry with a computer program denominated TEOD-NodExp (Transport of Electrons in Discreet Ordinates, Nodal Exponentials), using the proposed method by the Dr. J. E. Morel to carry out the discretization of the variable energy and several spatial discretization schemes, denominated exponentials nodal. It is used the Fokker-Planck equation since it represents an approach of the Boltzmann transport equation that is been worth whenever it is predominant the dispersion of small angles, that is to say, resulting dispersion in small dispersion angles and small losses of energy in the transport of charged particles. Such electrons could be those that they face with a braking plate in a device of thermonuclear fusion. In the present work its are considered electrons of 1 MeV that impact isotropically on an aluminum plate. They were considered three different thickness of plate that its were designated as problems 1, 2 and 3. In the calculations it was used the discrete ordinate method S{sub 4} with expansions of the dispersion cross sections until P{sub 3} order. They were considered 25 energy groups of uniform size between the minimum energy of 0.1 MeV and the maximum of 1.0 MeV; the one spatial intervals number it was considered variable and it was assigned the values of 10, 20 and 30. (Author)
On the laws of thermodynamics from the escort average and on the uniqueness of statistical factors
International Nuclear Information System (INIS)
Yamano, Takuya
2003-01-01
We consider the relation between the statistical weight and the laws of thermodynamics. Our path bases on the infinitesimal perturbation from outside of the thermodynamical system. It is fair to say that the form of the first laws of thermodynamics and the Clausius' definition of thermodynamic entropy are commensurately altered once we employ the escort average of statistical weight but preserves the forms in the limit of the usual weight. We also see an example for the unique determination of the statistical factor (so-called the Gibbs theorem) in addition to the Boltzmann and the Tsallis ones
Energy Technology Data Exchange (ETDEWEB)
Ayissi, Raoul Domingo, E-mail: raoulayissi@yahoo.fr; Noutchegueme, Norbert, E-mail: nnoutch@yahoo.fr [Department of Mathematics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde (Cameroon)
2015-01-15
Global solutions regular for the Einstein-Boltzmann equation on a magnetized Bianchi type-I cosmological model with the cosmological constant are investigated. We suppose that the metric is locally rotationally symmetric. The Einstein-Boltzmann equation has been already considered by some authors. But, in general Bancel and Choquet-Bruhat [Ann. Henri Poincaré XVIII(3), 263 (1973); Commun. Math. Phys. 33, 83 (1973)], they proved only the local existence, and in the case of the nonrelativistic Boltzmann equation. Mucha [Global existence of solutions of the Einstein-Boltzmann equation in the spatially homogeneous case. Evolution equation, existence, regularity and singularities (Banach Center Publications, Institute of Mathematics, Polish Academy of Science, 2000), Vol. 52] obtained a global existence result, for the relativistic Boltzmann equation coupled with the Einstein equations and using the Yosida operator, but confusing unfortunately with the nonrelativistic case. Noutchegueme and Dongho [Classical Quantum Gravity 23, 2979 (2006)] and Noutchegueme, Dongho, and Takou [Gen. Relativ. Gravitation 37, 2047 (2005)], have obtained a global solution in time, but still using the Yosida operator and considering only the uncharged case. Noutchegueme and Ayissi [Adv. Stud. Theor. Phys. 4, 855 (2010)] also proved a global existence of solutions to the Maxwell-Boltzmann system using the characteristic method. In this paper, we obtain using a method totally different from those used in the works of Noutchegueme and Dongho [Classical Quantum Gravity 23, 2979 (2006)], Noutchegueme, Dongho, and Takou [Gen. Relativ. Gravitation 37, 2047 (2005)], Noutchegueme and Ayissi [Adv. Stud. Theor. Phys. 4, 855 (2010)], and Mucha [Global existence of solutions of the Einstein-Boltzmann equation in the spatially homogeneous case. Evolution equation, existence, regularity and singularities (Banach Center Publications, Institute of Mathematics, Polish Academy of Science, 2000), Vol. 52] the
Statistical mechanics in the context of special relativity.
Kaniadakis, G
2002-11-01
In Ref. [Physica A 296, 405 (2001)], starting from the one parameter deformation of the exponential function exp(kappa)(x)=(sqrt[1+kappa(2)x(2)]+kappax)(1/kappa), a statistical mechanics has been constructed which reduces to the ordinary Boltzmann-Gibbs statistical mechanics as the deformation parameter kappa approaches to zero. The distribution f=exp(kappa)(-beta E+betamu) obtained within this statistical mechanics shows a power law tail and depends on the nonspecified parameter beta, containing all the information about the temperature of the system. On the other hand, the entropic form S(kappa)= integral d(3)p(c(kappa) f(1+kappa)+c(-kappa) f(1-kappa)), which after maximization produces the distribution f and reduces to the standard Boltzmann-Shannon entropy S0 as kappa-->0, contains the coefficient c(kappa) whose expression involves, beside the Boltzmann constant, another nonspecified parameter alpha. In the present effort we show that S(kappa) is the unique existing entropy obtained by a continuous deformation of S0 and preserving unaltered its fundamental properties of concavity, additivity, and extensivity. These properties of S(kappa) permit to determine unequivocally the values of the above mentioned parameters beta and alpha. Subsequently, we explain the origin of the deformation mechanism introduced by kappa and show that this deformation emerges naturally within the Einstein special relativity. Furthermore, we extend the theory in order to treat statistical systems in a time dependent and relativistic context. Then, we show that it is possible to determine in a self consistent scheme within the special relativity the values of the free parameter kappa which results to depend on the light speed c and reduces to zero as c--> infinity recovering in this way the ordinary statistical mechanics and thermodynamics. The statistical mechanics here presented, does not contain free parameters, preserves unaltered the mathematical and epistemological structure of
Realistic finite temperature simulations of magnetic systems using quantum statistics
Bergqvist, Lars; Bergman, Anders
2018-01-01
We have performed realistic atomistic simulations at finite temperatures using Monte Carlo and atomistic spin dynamics simulations incorporating quantum (Bose-Einstein) statistics. The description is much improved at low temperatures compared to classical (Boltzmann) statistics normally used in these kind of simulations, while at higher temperatures the classical statistics are recovered. This corrected low-temperature description is reflected in both magnetization and the magnetic specific heat, the latter allowing for improved modeling of the magnetic contribution to free energies. A central property in the method is the magnon density of states at finite temperatures, and we have compared several different implementations for obtaining it. The method has no restrictions regarding chemical and magnetic order of the considered materials. This is demonstrated by applying the method to elemental ferromagnetic systems, including Fe and Ni, as well as Fe-Co random alloys and the ferrimagnetic system GdFe3.
Statistical physics of balance theory.
Directory of Open Access Journals (Sweden)
Andres M Belaza
Full Text Available Triadic relationships are accepted to play a key role in the dynamics of social and political networks. Building on insights gleaned from balance theory in social network studies and from Boltzmann-Gibbs statistical physics, we propose a model to quantitatively capture the dynamics of the four types of triadic relationships in a network. Central to our model are the triads' incidence rates and the idea that those can be modeled by assigning a specific triadic energy to each type of triadic relation. We emphasize the role of the degeneracy of the different triads and how it impacts the degree of frustration in the political network. In order to account for a persistent form of disorder in the formation of the triadic relationships, we introduce the systemic variable temperature. In order to learn about the dynamics and motives, we propose a generic Hamiltonian with three terms to model the triadic energies. One term is connected with a three-body interaction that captures balance theory. The other terms take into account the impact of heterogeneity and of negative edges in the triads. The validity of our model is tested on four datasets including the time series of triadic relationships for the standings between two classes of alliances in a massively multiplayer online game (MMOG. We also analyze real-world data for the relationships between the "agents" involved in the Syrian civil war, and in the relations between countries during the Cold War era. We find emerging properties in the triadic relationships in a political network, for example reflecting itself in a persistent hierarchy between the four triadic energies, and in the consistency of the extracted parameters from comparing the model Hamiltonian to the data.
Statistical physics of balance theory.
Belaza, Andres M; Hoefman, Kevin; Ryckebusch, Jan; Bramson, Aaron; van den Heuvel, Milan; Schoors, Koen
2017-01-01
Triadic relationships are accepted to play a key role in the dynamics of social and political networks. Building on insights gleaned from balance theory in social network studies and from Boltzmann-Gibbs statistical physics, we propose a model to quantitatively capture the dynamics of the four types of triadic relationships in a network. Central to our model are the triads' incidence rates and the idea that those can be modeled by assigning a specific triadic energy to each type of triadic relation. We emphasize the role of the degeneracy of the different triads and how it impacts the degree of frustration in the political network. In order to account for a persistent form of disorder in the formation of the triadic relationships, we introduce the systemic variable temperature. In order to learn about the dynamics and motives, we propose a generic Hamiltonian with three terms to model the triadic energies. One term is connected with a three-body interaction that captures balance theory. The other terms take into account the impact of heterogeneity and of negative edges in the triads. The validity of our model is tested on four datasets including the time series of triadic relationships for the standings between two classes of alliances in a massively multiplayer online game (MMOG). We also analyze real-world data for the relationships between the "agents" involved in the Syrian civil war, and in the relations between countries during the Cold War era. We find emerging properties in the triadic relationships in a political network, for example reflecting itself in a persistent hierarchy between the four triadic energies, and in the consistency of the extracted parameters from comparing the model Hamiltonian to the data.
A multi-component lattice Boltzmann scheme: towards the mesoscale simulation of blood flow.
Dupin, M M; Halliday, I; Care, C M
2006-01-01
While blood at the macroscopic scale is frequently treated as a continuum by techniques such as computational fluid dynamics, its mesoscale behaviour is not so well investigated or understood. At this scale, the deformability of each cell within the plasma is important and cannot be ignored. However there is currently a lack of efficient computational techniques able to simulate a large number of deformable particles such as blood cells. This paper addresses this problem and demonstrates the applicability of the authors' recent multi-component lattice Boltzmann method for the simulation of a large number of mutually immiscible liquid species [Dupin MM, Halliday I, Care CM. Multi-component lattice boltzmann equation for mesoscale blood flow. J Phys A: Math Gen 2003;36:8517-34]. In here, biological cells are treated as immiscible, deformable, and relatively viscous drops (compared to the surrounding fluid). The validation of the model is based on the work of Goldsmith on the flow of solid particles, deformable particles and red blood cells [Goldsmith HL, Marlow JC. Flow behavior of erythrocytes. II. Particle motions in concentrated suspensions of ghost cells. J Colloid Interf Sci 1979;71:383-407]. We demonstrate, in particular, that the model recovers Goldsmith's observations on the flow properties of red blood cells and also the experimental observations of Frank on the flow of solid beads [Frank M, Anderson D, Weeks ER, Morris JF. Particle migration in pressure-driven flow of a brownian suspension. J Fluid Mech 2003;493:363-78]. The current article is the first validation of our new lattice Boltzmann model for a large number of deformable particles in this context and demonstrates that the method provides a new, and effective, approach for the modeling of mesoscale blood flow.
Bouchard, Hugo; Bielajew, Alex
2015-07-07
To establish a theoretical framework for generalizing Monte Carlo transport algorithms by adding external electromagnetic fields to the Boltzmann radiation transport equation in a rigorous and consistent fashion. Using first principles, the Boltzmann radiation transport equation is modified by adding a term describing the variation of the particle distribution due to the Lorentz force. The implications of this new equation are evaluated by investigating the validity of Fano's theorem. Additionally, Lewis' approach to multiple scattering theory in infinite homogeneous media is redefined to account for the presence of external electromagnetic fields. The equation is modified and yields a description consistent with the deterministic laws of motion as well as probabilistic methods of solution. The time-independent Boltzmann radiation transport equation is generalized to account for the electromagnetic forces in an additional operator similar to the interaction term. Fano's and Lewis' approaches are stated in this new equation. Fano's theorem is found not to apply in the presence of electromagnetic fields. Lewis' theory for electron multiple scattering and moments, accounting for the coupling between the Lorentz force and multiple elastic scattering, is found. However, further investigation is required to develop useful algorithms for Monte Carlo and deterministic transport methods. To test the accuracy of Monte Carlo transport algorithms in the presence of electromagnetic fields, the Fano cavity test, as currently defined, cannot be applied. Therefore, new tests must be designed for this specific application. A multiple scattering theory that accurately couples the Lorentz force with elastic scattering could improve Monte Carlo efficiency. The present study proposes a new theoretical framework to develop such algorithms.
Grid refinement for aeroacoustics in the lattice Boltzmann method: A directional splitting approach
Gendre, Félix; Ricot, Denis; Fritz, Guillaume; Sagaut, Pierre
2017-08-01
This study focuses on grid refinement techniques for the direct simulation of aeroacoustics, when using weakly compressible lattice Boltzmann models, such as the D3Q19 athermal velocity set. When it comes to direct noise computation, very small errors on the density or pressure field may have great negative consequences. Even strong acoustic density fluctuations have indeed a clearly lower amplitude than the hydrodynamic ones. This work deals with such very weak spurious fluctuations that emerge when a vortical structure crosses a refinement interface, which may contaminate the resulting aeroacoustic field. We show through an extensive literature review that, within the framework described above, this issue has never been addressed before. To tackle this problem, we develop an alternative algorithm and compare its behavior to a classical one, which fits our in-house vertex-centered data structure. Our main idea relies on a directional splitting of the continuous discrete velocity Boltzmann equation, followed by an integration over specific characteristics. This method can be seen as a specific coupling between finite difference and lattice Boltzmann, locally on the interface between the two grids. The method is assessed considering two cases: an acoustic pulse and a convected vortex. We show how very small errors on the density field arise and propagate throughout the domain when a vortical flow crosses the refinement interface. We also show that an increased free stream Mach number (but still within the weakly compressible regime) strongly deteriorates the situation, although the magnitude of the errors may remain negligible for purely aerodynamic studies. A drastically reduced level of error for the near-field spurious noise is obtained with our approach, especially for under-resolved simulations, a situation that is crucial for industrial applications. Thus, the vortex case is proved useful for aeroacoustic validations of any grid refinement algorithm.
Lattice Boltzmann Methods to Address Fundamental Boiling and Two-Phase Problems
Energy Technology Data Exchange (ETDEWEB)
Uddin, Rizwan
2012-01-01
This report presents the progress made during the fourth (no cost extension) year of this three-year grant aimed at the development of a consistent Lattice Boltzmann formulation for boiling and two-phase flows. During the first year, a consistent LBM formulation for the simulation of a two-phase water-steam system was developed. Results of initial model validation in a range of thermo-dynamic conditions typical for Boiling Water Reactors (BWRs) were shown. Progress was made on several fronts during the second year. Most important of these included the simulation of the coalescence of two bubbles including the surface tension effects. Work during the third year focused on the development of a new lattice Boltzmann model, called the artificial interface lattice Boltzmann model (AILB model) for the 3 simulation of two-phase dynamics. The model is based on the principle of free energy minimization and invokes the Gibbs-Duhem equation in the formulation of non-ideal forcing function. This was reported in detail in the last progress report. Part of the efforts during the last (no-cost extension) year were focused on developing a parallel capability for the 2D as well as for the 3D codes developed in this project. This will be reported in the final report. Here we report the work carried out on testing the AILB model for conditions including the thermal effects. A simplified thermal LB model, based on the thermal energy distribution approach, was developed. The simplifications are made after neglecting the viscous heat dissipation and the work done by pressure in the original thermal energy distribution model. Details of the model are presented here, followed by a discussion of the boundary conditions, and then results for some two-phase thermal problems.
Energy Technology Data Exchange (ETDEWEB)
Premnath, Kannan N [Department of Mechanical Engineering, University of Colorado Denver, 1200 Larimer Street, Denver, CO 80217 (United States); Pattison, Martin J [HyPerComp Inc., 2629 Townsgate Road, Suite 105, Westlake Village, CA 91361 (United States); Banerjee, Sanjoy, E-mail: kannan.premnath@ucdenver.edu, E-mail: kannan.np@gmail.com [Department of Chemical Engineering, City College of New York, City University of New York, New York, NY 10031 (United States)
2013-10-15
Lattice Boltzmann method (LBM) is a kinetic based numerical scheme for the simulation of fluid flow. While the approach has attracted considerable attention during the last two decades, there is a need for systematic investigation of its applicability for complex canonical turbulent flow problems of engineering interest, where the nature of the numerical properties of the underlying scheme plays an important role for their accurate solution. In this paper, we discuss and evaluate a LBM based on a multiblock approach for efficient large eddy simulation of three-dimensional external flow past a circular cylinder in the transitional regime characterized by the presence of multiple scales. For enhanced numerical stability at higher Reynolds numbers, a multiple relaxation time formulation is considered. The effect of subgrid scales is represented by means of a Smagorinsky eddy-viscosity model, where the model coefficient is computed locally by means of a dynamic procedure, providing better representation of flow physics with reduced empiricism. Simulations are performed for a Reynolds number of 3900 based on the free stream velocity and cylinder diameter for which prior data is available for comparison. The presence of laminar boundary layer which separates into a pair of shear layers that evolve into turbulent wakes impose particular challenge for numerical methods for this condition. The relatively low numerical dissipation introduced by the inherently parallel and second-order accurate LBM is an important computational asset in this regard. Computations using five different grid levels, where the various blocks are suitably aligned to resolve multiscale flow features show that the structure of the recirculation region is well reproduced and the statistics of the mean flow and turbulent fluctuations are in satisfactory agreement with prior data. (paper)
Lattice Boltzmann implementation of the three-dimensional Ben-Naim potential for water-like fluids.
Moradi, Nasrollah; Greiner, Andreas; Rao, Francesco; Succi, Sauro
2013-03-28
We develop a three-dimensional lattice Boltzmann (LB) model accounting for directional interactions between water-like molecules, based on the so-called Ben-Naim (BN) potential [A. Ben-Naim, Molecular Theory of Water and Aqueous Solutions: Part I: Understanding Water (World Scientific Publishing Company, 2010); "Statistical mechanics of 'waterlike' particles in two dimensions. I. Physical model and application of the Percus-Yevick equation," J. Chem. Phys. 54, 3682 (1971)]. The water-like molecules are represented by rigid tetrahedra, with two donors and two acceptors at the corners and interacting with neighboring tetrahedra, sitting on the nodes of a regular lattice. The tetrahedra are free to rotate about their centers under the drive of the torque arising from the interparticle potential. The orientations of the water molecules are evolved in time via an overdamped Langevin dynamics for the torque, which is solved by means of a quaternion technique. The resulting advection-diffusion-reaction equation for the quaternion components is solved by a LB method, acting as a dynamic minimizer for the global energy of the fluid. By adding thermal fluctuations to the torque equation, the model is shown to reproduce some microscopic features of real water, such as an average number of hydrogen bonds per molecules (HBs) between 3 and 4, in a qualitative agreement with microscopic water models. Albeit slower than a standard LB solver for ordinary fluids, the present scheme opens up potentially far-reaching scenarios for multiscale applications based on a coarse-grained representation of the water solvent.
International Nuclear Information System (INIS)
Premnath, Kannan N; Pattison, Martin J; Banerjee, Sanjoy
2013-01-01
Lattice Boltzmann method (LBM) is a kinetic based numerical scheme for the simulation of fluid flow. While the approach has attracted considerable attention during the last two decades, there is a need for systematic investigation of its applicability for complex canonical turbulent flow problems of engineering interest, where the nature of the numerical properties of the underlying scheme plays an important role for their accurate solution. In this paper, we discuss and evaluate a LBM based on a multiblock approach for efficient large eddy simulation of three-dimensional external flow past a circular cylinder in the transitional regime characterized by the presence of multiple scales. For enhanced numerical stability at higher Reynolds numbers, a multiple relaxation time formulation is considered. The effect of subgrid scales is represented by means of a Smagorinsky eddy-viscosity model, where the model coefficient is computed locally by means of a dynamic procedure, providing better representation of flow physics with reduced empiricism. Simulations are performed for a Reynolds number of 3900 based on the free stream velocity and cylinder diameter for which prior data is available for comparison. The presence of laminar boundary layer which separates into a pair of shear layers that evolve into turbulent wakes impose particular challenge for numerical methods for this condition. The relatively low numerical dissipation introduced by the inherently parallel and second-order accurate LBM is an important computational asset in this regard. Computations using five different grid levels, where the various blocks are suitably aligned to resolve multiscale flow features show that the structure of the recirculation region is well reproduced and the statistics of the mean flow and turbulent fluctuations are in satisfactory agreement with prior data. (paper)
Hilbert's sixth problem and the failure of the Boltzmann to Euler limit.
Slemrod, Marshall
2018-04-28
This paper addresses the main issue of Hilbert's sixth problem, namely the rigorous passage of solutions to the mesoscopic Boltzmann equation to macroscopic solutions of the Euler equations of compressible gas dynamics. The results of the paper are that (i) in general Hilbert's program will fail because of the appearance of van der Waals-Korteweg capillarity terms in a macroscopic description of motion of a gas, and (ii) the van der Waals-Korteweg theory itself might satisfy Hilbert's quest for a map from the 'atomistic view' to the laws of motion of continua.This article is part of the theme issue 'Hilbert's sixth problem'. © 2018 The Author(s).
Effect of Rolling Massage on the Vortex Flow in Blood Vessels with Lattice Boltzmann Simulation
Yi, Hou Hui
The rolling massage manipulation is a classic Chinese Medical Massage, which is a nature therapy in eliminating many diseases. Here, the effect of the rolling massage on the cavity flows in blood vessel under the rolling manipulation is studied by the lattice Boltzmann simulation. The simulation results show that the vortex flows are fully disturbed by the rolling massage. The flow behavior depends on the rolling velocity and the rolling depth. Rolling massage has a better effect on the flows in the cavity than that of the flows in a planar blood vessel. The result is helpful to understand the mechanism of the massage and develop the rolling techniques.
Demand Forecasting at Low Aggregation Levels using Factored Conditional Restricted Boltzmann Machine
DEFF Research Database (Denmark)
Mocanu, Elena; Nguyen, Phuong H.; Gibescu, Madeleine
2016-01-01
approaches have been proposed in the literature. As an evolution of neural network-based prediction methods, deep learning techniques are expected to increase the prediction accuracy by allowing stochastic formulations and bi-directional connections between neurons. In this paper, we investigate a newly...... developed deep learning model for time series prediction, namely Factored Conditional Restricted Boltzmann Machine (FCRBM), and extend it for electrical demand forecasting. The assessment is made on the EcoGrid dataset, originating from the Bornholm island experiment in Denmark, consisting of aggregated...
From Conformal Invariance towards Dynamical Symmetries of the Collisionless Boltzmann Equation
Directory of Open Access Journals (Sweden)
Stoimen Stoimenov
2015-09-01
Full Text Available Dynamical symmetries of the collisionless Boltzmann transport equation, or Vlasov equation, but under the influence of an external driving force, are derived from non-standard representations of the 2D conformal algebra. In the case without external forces, the symmetry of the conformally-invariant transport equation is first generalized by considering the particle momentum as an independent variable. This new conformal representation can be further extended to include an external force. The construction and possible physical applications are outlined.
Lattice Boltzmann simulations of leukocyte rolling and deformation in a three-dimensional shear flow
Luo, Ye; Qi, Dewei; He, Guowei
2013-11-01
Lattice Boltzmann simulation is used to simulate the motion of a leukocyte in fluid. The cell membrane is built by lattice spring model. The interaction between the fluid flow and the solid surface is treated by immersed boundary method. Stochastic Monte Carlo method is used to deal with receptor/ligand interaction. It is shown that the model can correctly predict the characteristic ``stop-and-g'' motion of rolling leukocytes. Effects of cell deformation, shear rates, bonding force, microvilli distribution on rolling are studied and compared with experiments.
New Fundamental Light Particle and Breakdown of Stefan-Boltzmann's Law
Directory of Open Access Journals (Sweden)
Samoilov V.
2011-04-01
Full Text Available Recently, we predicted the existence of fundamental particles in Nature, neutral Light Particles with spin 1 and rest mass m = 1.8 x 10^{-4} m_e, in addition to electrons, neutrons and protons. We call these particles Light Bosons because they create electromagnetic field which represents Planck's gas of massless photons together with a gas of Light Particles in the condensate. Such reasoning leads to a breakdown of Stefan-Boltzmann's law at low temperature. On the other hand, the existence of new fundamental neutral Light Particles leads to correction of such physical concepts as Bose-Einstein condensation of photons, polaritons and exciton polaritons.
Poisson-Boltzmann thermodynamics of counter-ions confined by curved hard walls
Samaj, Ladislav; Trizac, E.
2015-01-01
We consider a set of identical mobile point-like charges (counter-ions) confined to a domain with curved hard walls carrying a uniform fixed surface charge density, the system as a whole being electroneutral. Three domain geometries are considered: a pair of parallel plates, the cylinder and the sphere. The particle system in thermal equilibrium is assumed to be described by the nonlinear Poisson-Boltzmann theory. While the effectively 1D plates and the 2D cylinder have already been solved, t...
Thermodynamic aspect in using modified Boltzmann model as an acoustic probe for URu2Si2
Kwang-Hua, Chu Rainer
2018-05-01
The approximate system of equations describing ultrasonic attenuation propagating in many electrons of the heavy-fermion materials URu2Si2 under high magnetic fields were firstly derived and then calculated based on the modified Boltzmann model considering the microscopic contributions due to electronic fluids. A system of nonlinear partial differential coupled with integral equations were linearized firstly and approximately solved considering the perturbed thermodynamic equilibrium states. Our numerical data were compared with previous measurements using non-dimensional or normalized physical values. The rather good fit of our numerical calculations with experimental measurements confirms our present approach.
Lattice Boltzmann Simulation of Collision between 2D Circular Particles Suspension in Couette Flow
Directory of Open Access Journals (Sweden)
Li-Zhong Huang
2013-01-01
Full Text Available Collision between 2D circular particles suspension in Couette flow is simulated by using multiple-relaxation-time based lattice Boltzmann and direct forcing/fictitious domain method in this paper. The patterns of particle collisions are simulated and analyzed in detail by changing the velocity of top and bottom walls in the Couette flow. It can be seen from the simulation results that, while the velocity is large enough, the number of collisions between particles will change little as this velocity varies.
Fermion propagator in an out of equilibrium quantum-field system and the Boltzmann equation
International Nuclear Information System (INIS)
Niegawa, A.
2002-01-01
We aim to construct from first principles a perturbative framework for studying nonequilibrium quantum-field systems that include massless Dirac fermions. The system of our concern is a quasiuniform system near equilibrium or a nonequilibrium quasistationary system. We employ the closed-time-path formalism and use the so-called gradient approximation. Essentially no further approximation is introduced. We construct a fermion propagator, with which a well-defined perturbative framework is formulated. In the course of the construction of the framework, we obtain the generalized Boltzmann equation that describes the evolution of the number-density functions of (anti)fermionic quasiparticles
Cyclic Heating-Annealing and Boltzmann Distribution of Free Energies in a Spin-Glass System
International Nuclear Information System (INIS)
Zhou Haijun
2007-01-01
Ergodicity of a spin-glass is broken at low temperatures; the system is trapped in one of many ergodic configurational domains. Transitions between different ergodic domains are achievable through a heating-annealing procedure. If this experiment is repeated infinite times, all ergodic configurational domains will be visited with frequences that decreasing exponentially with their free energies. The mean free energy density of a spin-glass system on a random graph is calculated based on this free energy Boltzmann distribution in the present work, by means of the cavity approach.
The Influence of Ernst Mach and Ludwig Boltzmann on Albert Einstein
International Nuclear Information System (INIS)
Broda, E.
1979-01-01
This document, written by Engelbert Broda in 1979, analyses the influence of Ernst Mach and Ludwig Boltzmann on Albert Einstein. Broda describes how Einstein and his scientific thinking benefited from Mach’s criticism on classical mechanics and its basic concepts like absolute time and absolute space. This criticism encouraged Einstein in the time he worked on his special relativity. On the other side Broda writes about the influence of Ludwig Boltzman, an atomist, whose scientific work and research prepared the ground for Einsteins work on the quantum-structure of electromagnetic radiation or the discovery of the photoelectric effect. (nowak)
Multi-component Lattice Boltzmann simulation of the hydrodynamics in drip emitters
Directory of Open Access Journals (Sweden)
Giacomo Falcucci
2017-09-01
Full Text Available In this paper, we propose a fast and efficient numerical technique based on the Lattice Boltzmann method (LBM to model the flow through a reference drip emitter geometry. The aim of the study is to demonstrate the applicability of the LBM as a reliable simulation tool for the hydraulic optimisation of irrigation systems. Results for the water flow through a rectangular drip emitter are in good agreement with literature numerical and experimental data. Furthermore, we demonstrate the feasibility of the proposed model to simulate a multi-component flow that could be used to simulate the presence of additives, contaminants, and suspended particles.
The linearized Boltzmann equation: a concise and accurate solution of the temperature-jump problem
Siewert, C E
2003-01-01
Polynomial expansion procedures, along with an analytical discrete-ordinates method, are used to solve the temperature-jump problem based on a rigorous version of the linearized Boltzmann equation for rigid-sphere interactions. In particular, the temperature and density perturbations and the temperature-jump coefficient are obtained (essentially) analytically in terms of a modern version of the discrete-ordinates method. The developed algorithms are implemented for general values of the accommodation coefficient to yield numerical results that can be considered a new standard of reference.
Lattice Boltzmann method for simulation of compressible flows on standard lattices.
Prasianakis, Nikolaos I; Karlin, Iliya V
2008-07-01
The recently introduced lattice Boltzmann model for thermal flow simulation on a standard lattice [Prasianakis and Karlin, Phys. Rev. E 76, 016702 (2007)] is studied numerically in the case where compressibility effects are essential. It is demonstrated that the speed of sound and shock propagation are described correctly in a wide temperature range, and that it is possible to take into account additional physics such as heat sources and sinks. A remarkable simplicity of the model makes it viable for engineering applications in subsonic flows with large temperature and density variations.
Ahrenholz, Benjamin
2009-01-01
Die vorliegende Dissertation gibt im Wesentlichen die Arbeiten wieder, die im Rahmen des FIMOTUM Projektes durchgeführt worden sind, welches sich vornehmlich auf die Untersuchung von Transporteigenschaften in ungesättigten porösen Medien fokussierte. Hierfür wurde ein Software-Prototyp auf Basis der Gitter Boltzmann Methode (LBM) entwickelt und ausführlich validiert. Die vorgestellte LB-Methode basiert auf dem Multiple-Relaxation-Time (MRT) Modell und verwendet Fluid/Wand Randbedingungen mit ...
International Nuclear Information System (INIS)
Kawashima, S.; Matsumara, A.; Nishida, T.
1979-01-01
The compressible and heat-conductive Navier-Stokes equation obtained as the second approximation of the formal Chapman-Enskog expansion is investigated on its relations to the original nonlinear Boltzmann equation and also to the incompressible Navier-Stokes equation. The solutions of the Boltzmann equation and the incompressible Navier-Stokes equation for small initial data are proved to be asymptotically equivalent (mod decay rate tsup(-5/4)) as t → + infinitely to that of the compressible Navier-Stokes equation for the corresponding initial data. (orig.) 891 HJ/orig. 892 MKO
Ginzburg, Irina; Steiner, Konrad
2002-03-15
The filling process of viscoplastic metal alloys and plastics in expanding cavities is modelled using the lattice Boltzmann method in two and three dimensions. These models combine the regularized Bingham model for viscoplastic fluids with a free-interface algorithm. The latter is based on a modified immiscible lattice Boltzmann model in which one species is the fluid and the other one is considered to be a vacuum. The boundary conditions at the curved liquid-vacuum interface are met without any geometrical front reconstruction from a first-order Chapman-Enskog expansion. The numerical results obtained with these models are found in good agreement with available theoretical and numerical analysis.
Directory of Open Access Journals (Sweden)
Yeou-Jiunn Chen
2014-01-01
Full Text Available Proper anesthesia is very important for patients to get through surgery without pain and then avoid some other problems. By monitoring the depth of sedation for a patient, it could help a clinician to provide a suitable amount of anesthetic and other clinical treatment. In hospital, a patient is usually monitored by different types of biological systems. To predict the depth of sedation from biological signals is able to ease patient monitoring services. In this study, continuous restricted Boltzmann machines based neural network is proposed to predict the depth of sedation. The biological signals including heart rate, blood pressure, peripheral capillary oxygen saturation, and body weight are selected as analytic features. To improve the accuracy, the signals related to the state of anesthesia including fractional anesthetic concentration, end-tidal carbon dioxide, fraction inspiration carbon dioxide, and minimum alveolar concentration are also adopted in this study. Using minimizing contrastive divergence, a continuous restricted Boltzmann machine is trained and then used to predict the depth of sedation. The experimental results showed that the proposed approach outperforms feed-forward neural network and modular neural network. Besides, it would be able to ease patient monitoring services by using biological systems and promote healthcare quality.
Tian, Fang-Bao; Luo, Haoxiang; Zhu, Luoding; Liao, James C.; Lu, Xi-Yun
2012-01-01
We have introduced a modified penalty approach into the flow-structure interaction solver that combines an immersed boundary method (IBM) and a multi-block lattice Boltzmann method (LBM) to model an incompressible flow and elastic boundaries with finite mass. The effect of the solid structure is handled by the IBM in which the stress exerted by the structure on the fluid is spread onto the collocated grid points near the boundary. The fluid motion is obtained by solving the discrete lattice Boltzmann equation. The inertial force of the thin solid structure is incorporated by connecting this structure through virtual springs to a ghost structure with the equivalent mass. This treatment ameliorates the numerical instability issue encountered in this type of problems. Thanks to the superior efficiency of the IBM and LBM, the overall method is extremely fast for a class of flow-structure interaction problems where details of flow patterns need to be resolved. Numerical examples, including those involving multiple solid bodies, are presented to verify the method and illustrate its efficiency. As an application of the present method, an elastic filament flapping in the Kármán gait and the entrainment regions near a cylinder is studied to model fish swimming in these regions. Significant drag reduction is found for the filament, and the result is consistent with the metabolic cost measured experimentally for the live fish. PMID:23564971
Directory of Open Access Journals (Sweden)
Jin Su
2017-11-01
Full Text Available Elastic instabilities could happen in viscoelastic flows as the Weissenberg number is enlarged, and this phenomenon makes the numerical simulation of viscoelastic fluids more difficult. In this study, we introduce a coupled lattice Boltzmann method to solve the equations of viscoelastic fluids, which has a great capability of simulating the high Weissenberg number problem. Different from some traditional methods, two kinds of distribution functions are defined respectively for the evolution of the momentum and stress tensor equations. We mainly aim to investigate some key factors of the symmetry-breaking transition induced by elastic instability of viscoelastic fluids using this numerical coupled lattice Boltzmann method. In the results, we firstly find that the ratio of kinematical viscosity has an important influence on the transition of the elastic instability; the transition between the single stationary and cycling dominant vortex can be controlled via changing the ratio of kinematical viscosity in a periodic extensional flow. Finally, we can also observe a new transition state of instability for the flow showing the banded structure at higher Weissenberg number.
Quadratic inner element subgrid scale discretisation of the Boltzmann transport equation
International Nuclear Information System (INIS)
Baker, C.M.J.; Buchan, A.G.; Pain, C.C.; Tollit, B.; Eaton, M.D.; Warner, P.
2012-01-01
This paper explores the application of the inner element subgrid scale method to the Boltzmann transport equation using quadratic basis functions. Previously, only linear basis functions for both the coarse scale and the fine scale were considered. This paper, therefore, analyses the advantages of using different coarse and subgrid basis functions for increasing the accuracy of the subgrid scale method. The transport of neutral particle radiation may be described by the Boltzmann transport equation (BTE) which, due to its 7 dimensional phase space, is computationally expensive to resolve. Multi-scale methods offer an approach to efficiently resolve the spatial dimensions of the BTE by separating the solution into its coarse and fine scales and formulating a solution whereby only the computationally efficient coarse scales need to be solved. In previous work an inner element subgrid scale method was developed that applied a linear continuous and discontinuous finite element method to represent the solution’s coarse and fine scale components. This approach was shown to generate efficient and stable solutions, and so this article continues its development by formulating higher order quadratic finite element expansions over the continuous and discontinuous scales. Here it is shown that a solution’s convergence can be improved significantly using higher order basis functions. Furthermore, by using linear finite elements to represent coarse scales in combination with quadratic fine scales, convergence can also be improved with only a modest increase in computational expense.
Effects of Nanoparticles on Melting Process with Phase-Change Using the Lattice Boltzmann Method
Ibrahem, Ahmed M.
2017-05-04
In this work, the problem of nanoparticles dispersion effects on coupled heat transfer and solid-liquid phase change has been studied. The lattice Boltzmann method (LBM) enthalpy-based is employed. The collision model of lattice Bhatangar-Gross-Krook (LBGK) is used to solve the problem of 1D melting by conduction. On the other hand, we use the model of multi-distribution functions (MDF) to calculate the density, the velocity and the temperature for the problem of 2D melting by free convection, associated with different boundary conditions. In these simulations, the volume fractions of copper nanoparticles (0-2%) added to water-base fluid and Rayleigh numbers of 103to105. We use the Chapman-Enskog expansion to derive the governing macroscopic quantities from the mesoscopic lattice Boltzmann equation. The results obtained by these models have been compared to an analytical solution or other numerical methods. The effects of nanoparticles on conduction and natural convection during the melting process have been investigated. Moreover, the influences of nanoparticles on moving of the phase change front, the thermal conductivity and the latent heat of fusion are also studied.
Gray free-energy multiphase lattice Boltzmann model with effective transport and wetting properties
Zalzale, Mohamad; Ramaioli, M.; Scrivener, K. L.; McDonald, P. J.
2016-11-01
The paper shows that it is possible to combine the free-energy lattice Boltzmann approach to multiphase modeling of fluids involving both liquid and vapor with the partial bounce back lattice Boltzmann approach to modeling effective media. Effective media models are designed to mimic the properties of porous materials with porosity much finer than the scale of the simulation lattice. In the partial bounce-back approach, an effective media parameter or bounce-back fraction controls fluid transport. In the combined model, a wetting potential is additionally introduced that controls the wetting properties of the fluid with respect to interfaces between free space (white nodes), effective media (gray nodes), and solids (black nodes). The use of the wetting potential combined with the bounce-back parameter gives the model the ability to simulate transport and sorption of a wide range of fluid in material systems. Results for phase separation, permeability, contact angle, and wicking in gray media are shown. Sorption is explored in small sections of model multiscale porous systems to demonstrate two-step desorption, sorption hysteresis, and the ink-bottle effect.
Boltzmann equation and Monte Carlo studies of electron transport in resistive plate chambers
International Nuclear Information System (INIS)
Bošnjaković, D; Petrović, Z Lj; Dujko, S; White, R D
2014-01-01
A multi term theory for solving the Boltzmann equation and Monte Carlo simulation technique are used to investigate electron transport in Resistive Plate Chambers (RPCs) that are used for timing and triggering purposes in many high energy physics experiments at CERN and elsewhere. Using cross sections for electron scattering in C 2 H 2 F 4 , iso-C 4 H 10 and SF 6 as an input in our Boltzmann and Monte Carlo codes, we have calculated data for electron transport as a function of reduced electric field E/N in various C 2 H 2 F 4 /iso-C 4 H 10 /SF 6 gas mixtures used in RPCs in the ALICE, CMS and ATLAS experiments. Emphasis is placed upon the explicit and implicit effects of non-conservative collisions (e.g. electron attachment and/or ionization) on the drift and diffusion. Among many interesting and atypical phenomena induced by the explicit effects of non-conservative collisions, we note the existence of negative differential conductivity (NDC) in the bulk drift velocity component with no indication of any NDC for the flux component in the ALICE timing RPC system. We systematically study the origin and mechanisms for such phenomena as well as the possible physical implications which arise from their explicit inclusion into models of RPCs. Spatially-resolved electron transport properties are calculated using a Monte Carlo simulation technique in order to understand these phenomena. (paper)
International Nuclear Information System (INIS)
Stoltz, G; Lazzeri, M; Mauri, F
2009-01-01
We present a study of the phononic thermal conductivity of isotopically disordered carbon nanotubes. In particular, the behaviour of the thermal conductivity as a function of the system length is investigated, using Green's function techniques to compute the transmission across the system. The method is implemented using linear scaling algorithms, which allow us to reach systems of lengths up to L = 2.5 μm (with up to 200 000 atoms). As for 1D systems, it is observed that the conductivity diverges with the system size L. We also observe a dramatic decrease of the thermal conductance for systems of experimental sizes (roughly 80% at room temperature for L = 2.5 μm), when a large fraction of isotopic disorder is introduced. The results obtained with Green's function techniques are compared to results obtained with a Boltzmann description of thermal transport. There is a good agreement between both approaches for systems of experimental sizes, even in the presence of Anderson localization. This is particularly interesting since the computation of the transmission using Boltzmann's equation is much less computationally expensive, so that larger systems may be studied with this method.
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Stuart Bartlett
2017-08-01
Full Text Available The lattice Boltzmann method is an efficient computational fluid dynamics technique that can accurately model a broad range of complex systems. As well as single-phase fluids, it can simulate thermohydrodynamic systems and passive scalar advection. In recent years, it also gained attention as a means of simulating chemical phenomena, as interest in self-organization processes increased. This paper will present a widely-used and versatile lattice Boltzmann model that can simultaneously incorporate fluid dynamics, heat transfer, buoyancy-driven convection, passive scalar advection, chemical reactions and enthalpy changes. All of these effects interact in a physically accurate framework that is simple to code and readily parallelizable. As well as a complete description of the model equations, several example systems will be presented in order to demonstrate the accuracy and versatility of the method. New simulations, which analyzed the effect of a reversible reaction on the transport properties of a convecting fluid, will also be described in detail. This extra chemical degree of freedom was utilized by the system to augment its net heat flux. The numerical method outlined in this paper can be readily deployed for a vast range of complex flow problems, spanning a variety of scientific disciplines.
Botello-Smith, Wesley M; Luo, Ray
2015-10-26
Continuum solvent models have been widely used in biomolecular modeling applications. Recently much attention has been given to inclusion of implicit membranes into existing continuum Poisson-Boltzmann solvent models to extend their applications to membrane systems. Inclusion of an implicit membrane complicates numerical solutions of the underlining Poisson-Boltzmann equation due to the dielectric inhomogeneity on the boundary surfaces of a computation grid. This can be alleviated by the use of the periodic boundary condition, a common practice in electrostatic computations in particle simulations. The conjugate gradient and successive over-relaxation methods are relatively straightforward to be adapted to periodic calculations, but their convergence rates are quite low, limiting their applications to free energy simulations that require a large number of conformations to be processed. To accelerate convergence, the Incomplete Cholesky preconditioning and the geometric multigrid methods have been extended to incorporate periodicity for biomolecular applications. Impressive convergence behaviors were found as in the previous applications of these numerical methods to tested biomolecules and MMPBSA calculations.
Nanoscale roughness effect on Maxwell-like boundary conditions for the Boltzmann equation
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Brull, S., E-mail: Stephane.Brull@math.u-bordeaux.fr; Charrier, P., E-mail: Pierre.Charrier@math.u-bordeaux.fr; Mieussens, L., E-mail: Luc.Mieussens@math.u-bordeaux.fr [University of Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, F-33400 Talence (France)
2016-08-15
It is well known that the roughness of the wall has an effect on microscale gas flows. This effect can be shown for large Knudsen numbers by using a numerical solution of the Boltzmann equation. However, when the wall is rough at a nanometric scale, it is necessary to use a very small mesh size which is much too expansive. An alternative approach is to incorporate the roughness effect in the scattering kernel of the boundary condition, such as the Maxwell-like kernel introduced by the authors in a previous paper. Here, we explain how this boundary condition can be implemented in a discrete velocity approximation of the Boltzmann equation. Moreover, the influence of the roughness is shown by computing the structure scattering pattern of mono-energetic beams of the incident gas molecules. The effect of the angle of incidence of these molecules, of their mass, and of the morphology of the wall is investigated and discussed in a simplified two-dimensional configuration. The effect of the azimuthal angle of the incident beams is shown for a three-dimensional configuration. Finally, the case of non-elastic scattering is considered. All these results suggest that our approach is a promising way to incorporate enough physics of gas-surface interaction, at a reasonable computing cost, to improve kinetic simulations of micro- and nano-flows.
Poisson-Boltzmann thermodynamics of counterions confined by curved hard walls
Šamaj, Ladislav; Trizac, Emmanuel
2016-01-01
We consider a set of identical mobile pointlike charges (counterions) confined to a domain with curved hard walls carrying a uniform fixed surface charge density, the system as a whole being electroneutral. Three domain geometries are considered: a pair of parallel plates, the cylinder, and the sphere. The particle system in thermal equilibrium is assumed to be described by the nonlinear Poisson-Boltzmann theory. While the effectively one-dimensional plates and the two-dimensional cylinder have already been solved, the three-dimensional sphere problem is not integrable. It is shown that the contact density of particles at the charged surface is determined by a first-order Abel differential equation of the second kind which is a counterpart of Enig's equation in the critical theory of gravitation and combustion or explosion. This equation enables us to construct the exact series solutions of the contact density in the regions of small and large surface charge densities. The formalism provides, within the mean-field Poisson-Boltzmann framework, the complete thermodynamics of counterions inside a charged sphere (salt-free system).
Lattice Boltzmann simulations of the time evolution of living multicellular systems.
Cristea, Artur; Neagu, Adrian; Sofonea, Victor
2011-01-01
Embryonic tissues and multicellular aggregates of adult cells mimic the behavior of highly viscous liquids. The liquid analogy helps to understand morphogenetic phenomena, such as cell sorting and tissue fusion, observed in developmental biology and tissue engineering. Tissue fusion is vital in tissue printing, an emergent technique based on computer-controlled deposition of tissue fragments and biocompatible materials. Computer simulations proved useful in predicting post-printing shape changes of tissue constructs. The simulation methods available to date, however, are unable to describe the time evolution of living systems made of millions of cells. The Lattice Boltzmann (LB) approach allows the implementation of interaction forces between the constituents of the system and yields time evolution in terms of distribution functions. With tissue engineering applications in mind, we have developed a finite difference Lattice Boltzmann model of a multicellular system and applied it to simulate the sidewise fusion of two contiguous cylinders made of cohesive cells and embedded in a medium (hydrogel). We have identified a biologically relevant range of model parameters. The proposed LB model may be extended to describe the time evolution of more complex multicellular structures such as sheets or tubes produced by tissue printing. © 2011 – IOS Press and the authors. All rights reserved
An efficient numerical method for solving the Boltzmann equation in multidimensions
Dimarco, Giacomo; Loubère, Raphaël; Narski, Jacek; Rey, Thomas
2018-01-01
In this paper we deal with the extension of the Fast Kinetic Scheme (FKS) (Dimarco and Loubère, 2013 [26]) originally constructed for solving the BGK equation, to the more challenging case of the Boltzmann equation. The scheme combines a robust and fast method for treating the transport part based on an innovative Lagrangian technique supplemented with conservative fast spectral schemes to treat the collisional operator by means of an operator splitting approach. This approach along with several implementation features related to the parallelization of the algorithm permits to construct an efficient simulation tool which is numerically tested against exact and reference solutions on classical problems arising in rarefied gas dynamic. We present results up to the 3 D × 3 D case for unsteady flows for the Variable Hard Sphere model which may serve as benchmark for future comparisons between different numerical methods for solving the multidimensional Boltzmann equation. For this reason, we also provide for each problem studied details on the computational cost and memory consumption as well as comparisons with the BGK model or the limit model of compressible Euler equations.
A Combined MPI-CUDA Parallel Solution of Linear and Nonlinear Poisson-Boltzmann Equation
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José Colmenares
2014-01-01
Full Text Available The Poisson-Boltzmann equation models the electrostatic potential generated by fixed charges on a polarizable solute immersed in an ionic solution. This approach is often used in computational structural biology to estimate the electrostatic energetic component of the assembly of molecular biological systems. In the last decades, the amount of data concerning proteins and other biological macromolecules has remarkably increased. To fruitfully exploit these data, a huge computational power is needed as well as software tools capable of exploiting it. It is therefore necessary to move towards high performance computing and to develop proper parallel implementations of already existing and of novel algorithms. Nowadays, workstations can provide an amazing computational power: up to 10 TFLOPS on a single machine equipped with multiple CPUs and accelerators such as Intel Xeon Phi or GPU devices. The actual obstacle to the full exploitation of modern heterogeneous resources is efficient parallel coding and porting of software on such architectures. In this paper, we propose the implementation of a full Poisson-Boltzmann solver based on a finite-difference scheme using different and combined parallel schemes and in particular a mixed MPI-CUDA implementation. Results show great speedups when using the two schemes, achieving an 18.9x speedup using three GPUs.
Lattice Boltzmann methods for the simulation of heat transfer in particle suspensions
International Nuclear Information System (INIS)
McCullough, J.W.S.; Leonardi, C.R.; Jones, B.D.; Aminossadati, S.M.; Williams, J.R.
2016-01-01
Highlights: • Development of a lattice Boltzmann heat transfer model for curved boundaries. • Thermodynamic coupling aims to ensure continuity of both temperature and heat flux. • Good correlation found in transient comparison of results to analytical solutions. • Illustration of the developed model applied to a moving particle test case. - Abstract: This study examines the use of a lattice Boltzmann method framework to study heat transfer behaviours within particle suspensions. This has been done through the use of an adapted interface condition to attempt to resolve the required continuity of temperature and flux at the boundary between the solid and fluid phases. The proposed method is tested against analytical solutions for layered media in both a 1D bar and a radial layout. These tests showed that the model was able to generate results with first order convergence towards the analytical outcomes. The model was then used to examine the behaviour of two moving particles travelling along a channel to illustrate its potential for resolving complex suspension flows.
International Nuclear Information System (INIS)
Bartolomaeus, G.; Wilhelm, J.
1982-01-01
In the kinetic theory a great variety of physical systems is investigated by means of Boltzmann-like equations. This approach is used for neutral gases, neutron as well as radiation transport, plasmas etc. For many problems the knowledge of the properties of the collision operators is of great importance, especially if eigenvalue problems occur. The paper presents an investigation of the properties of the collision operators of the Boltzmann equation covering elastic, exciting and deexciting processes in a weakly ionized plasma. First, a short survey of the importance of eigenfunctions and eigenvalues in the kinetic theory of various systems is given. Then, properties of the outscattering operator as dependent on the course of the differential cross section are considered. Finally, for the inscattering operator such properties as selfadjointness and rotational invariance are investigated in detail. These considerations provide the basis for the proof of compactness and for first conclusions on the spectral properties of the collision operators in the second part of this paper. (author)
Topologically Ordered Feature Extraction Based on Sparse Group Restricted Boltzmann Machines
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Zhong Chen
2015-01-01
Full Text Available How to extract topologically ordered features efficiently from high-dimensional data is an important problem of unsupervised feature learning domains for deep learning. To address this problem, we propose a new type of regularization for Restricted Boltzmann Machines (RBMs. Adding two extra terms in the log-likelihood function to penalize the group weights and topologically ordered factors, this type of regularization extracts topologically ordered features based on sparse group Restricted Boltzmann Machines (SGRBMs. Therefore, it encourages an RBM to learn a much smoother probability distribution because its formulations turn out to be a combination of the group weight-decay and topologically ordered factor regularizations. We apply this proposed regularization scheme to image datasets of natural images and Flying Apsara images in the Dunhuang Grotto Murals at four different historical periods. The experimental results demonstrate that the combination of these two extra terms in the log-likelihood function helps to extract more discriminative features with much sparser and more aggregative hidden activation probabilities.
Knotted Vortices: Entropic Lattice Boltzmann Method for Simulation of Vortex dynamics
Boesch, Fabian; Chikatamarla, Shyam; Karlin, Ilya
2013-11-01
Knotted and interlinked vortex structures in real fluids are conjectured to play a major role in hydrodynamic flow dissipation. Much interest lies in determining their temporal stability and the mechanism through which knots dissolve. Kleckner and Irvine recently have shown the existence of such knotted vortices experimentally by accelerating hydrofoils in water. In the present work we employ the entropic lattice Boltzmann method (ELBM) to perform DNS simulations of the creation and dynamics of knotted vortex rings inspired by the experimental setup in. ELBM renders LBM scheme unconditionally stable by restoring the second law of thermodynamics (the Boltzmann H-theorem), and thus enables simulations of large domains and high Reynolds numbers with DNS quality. The results presented in this talk provide an in-depth study of the dynamics of knotted vortices and vortex reconnection events and confirm the existence of trefoil knots in silicio for the first time. This work was supported by a grant from the Swiss National Supercomputing Centre (CSCS) under project ID s347.
Lattice Boltzmann simulation of endothermal catalytic reaction in catalyst porous media
International Nuclear Information System (INIS)
Li Xunfeng; Cai Jun; Xin Fang; Huai Xiulan; Guo Jiangfeng
2013-01-01
Gas catalytic reaction in a fixed bed reactor is a general process in chemical industry. The chemical reaction process involves the complex multi-component flow, heat and mass transfer coupling chemical reaction in the catalyst porous structure. The lattice Boltzmann method is developed to simulate the complex process of the surface catalytic reaction in the catalyst porous media. The non-equilibrium extrapolation method is used to treat the boundaries. The porous media is structured by Sierpinski carpet fractal structure. The velocity correction is adopted on the reaction surface. The flow, temperature and concentration fields calculated by the lattice Boltzmann method are compared with those computed by the CFD software. The effects of the inlet velocity, porosity and inlet components ratio on the conversion are also studied. Highlights: ► LBM is developed to simulate the surface catalytic reaction. ► The Sierpinski carpet structure is used to construct the porous media. ► The LBM results are in agreement with the CFD predictions. ► Velocity, temperature and concentration fields are obtained. ► Effects of the velocity, porosity and concentration on conversion are analyzed.
Porter, M. L.; Wildenschild, D.; Schaap, M. G.
2007-12-01
The interface that exists between immiscible fluids plays an important role in multiphase flow and transport in subsurface environments. In this study interfacial area per volume was investigated using computed microtomographic image data and lattice-Boltzmann simulations. A multicomponent lattice-Boltzmann model was used to simulate air-water drainage and imbibition experiments. The pore geometry for the simulations was generated using computed microtomographic image data from the experiments. Based on analysis of the Reynolds, Capillary and Bond number it was determined that capillarity was the dominating force in the experiments, thus gravity, viscous and inertial forces were not taken into account in the simulations. Both pressure and flux boundary conditions were investigated with the simulations. The flux boundary conditions reflect the conditions in the experiments. The pressure boundary conditions are consistent with the more traditional methods for measuring capillary pressure - saturation curves. Simulations with both boundary conditions are in good agreement for the capillary pressure saturation curves. Comparisons between experimental and simulated capillary pressure - saturation curves show relatively good agreement. A preliminary comparison between nonwetting - wetting phase interfacial area per volume estimates indicates good agreement for drainage, however, the simulated interfacial area estimates for imbibition were significantly higher than those obtained in the experiments. The exact cause of the high estimates during imbibition is currently under investigation.
An improved lattice Boltzmann scheme for multiphase fluid with multi-range interactions
Energy Technology Data Exchange (ETDEWEB)
Maquignon, Nicolas; Duchateau, Julien; Roussel, Gilles; Rousselle, François; Renaud, Christophe [Laboratoire Informatique Signal et Image de la Côte d' Opale, 50 rue Ferdinand Buisson, 62100 Calais (France); Université du Littoral Côte d' Opale, 1 place de l' Yser, 59140, Dunkerque (France); Association INNOCOLD, MREI 1, 145 (France)
2014-10-06
Modeling of fluids with liquid to gas phase transition has become important for understanding many environmental or industrial processes. Such simulations need new techniques, because traditional solvers are often limited. The Lattice Boltzmann Model (LBM) allows simulate complex fluids, because its mesoscopic nature gives possibility to incorporate additional physics in comparison to usual methods. In this work, an improved lattice Boltzmann model for phase transition flow will be introduced. First, the state of art for Shan and Chen (SC) type of LBM will be reminded. Then, link to real thermodynamics will be established with Maxwell equal areas construction. Convergence to isothermal liquid vapor equilibrium will be shown and discussed. Inclusion of an equation of state for real fluid and better incorporation of force term is presented. Multi-range interactions have been used for SC model, but it hasn't been yet applied to real fluid with non-ideal equation of state. In this work, we evaluate this model when it is applied to real liquid-vapor equilibrium. We show that important differences are found for evaluation of gas density. In order to recover thermodynamic consistency, we use a new scheme for calculation of force term, which is a combination of multi range model and numerical weighting used by Gong and Cheng. We show the superiority of our new model by studying convergence to equilibrium values over a large temperature range. We prove that spurious velocities remaining at equilibrium are decreased.
Transition point prediction in a multicomponent lattice Boltzmann model: Forcing scheme dependencies
Küllmer, Knut; Krämer, Andreas; Joppich, Wolfgang; Reith, Dirk; Foysi, Holger
2018-02-01
Pseudopotential-based lattice Boltzmann models are widely used for numerical simulations of multiphase flows. In the special case of multicomponent systems, the overall dynamics are characterized by the conservation equations for mass and momentum as well as an additional advection diffusion equation for each component. In the present study, we investigate how the latter is affected by the forcing scheme, i.e., by the way the underlying interparticle forces are incorporated into the lattice Boltzmann equation. By comparing two model formulations for pure multicomponent systems, namely the standard model [X. Shan and G. D. Doolen, J. Stat. Phys. 81, 379 (1995), 10.1007/BF02179985] and the explicit forcing model [M. L. Porter et al., Phys. Rev. E 86, 036701 (2012), 10.1103/PhysRevE.86.036701], we reveal that the diffusion characteristics drastically change. We derive a generalized, potential function-dependent expression for the transition point from the miscible to the immiscible regime and demonstrate that it is shifted between the models. The theoretical predictions for both the transition point and the mutual diffusion coefficient are validated in simulations of static droplets and decaying sinusoidal concentration waves, respectively. To show the universality of our analysis, two common and one new potential function are investigated. As the shift in the diffusion characteristics directly affects the interfacial properties, we additionally show that phenomena related to the interfacial tension such as the modeling of contact angles are influenced as well.
A pore-scale approach to colloid-surface interaction in liquid using lattice Boltzmann models.
Larsen, J. D.; Schaap, M. G.
2016-12-01
Knowledge of colloid transport and collection efficiency is important for understanding the transport of some contaminants of emerging concern (CEC) and for developing environmental remediation systems such as geologic filters. The interaction forces between colloids and soil materials are central to colloid transport and retention or immobilization. In this study a physical modeling approach to represent colloidal transport through porous media has been developed, using the lattice Boltzmann methodology. Lattice Boltzmann models have the uncanny ability to represent pore scale fluid flow through complex structures such as geological material. A cellular approach to computing colloid forces is applied for computational efficiency, and colloids are tracked continuously through the model. Grid refinement effects are quantified to balance computational efficiency with discretization effects. Representation of physical forces including DLVO create a natural fluid solid boundary condition for colloid transport. Collector efficiencies of geologic materials and colloid distribution curves can be produced. The present work focuses on simple porous media with a single wetting fluid phase, but the approach can be extended to heterogeneous geologic materials and multiphase systems.
Conditioning and Robustness of RNA Boltzmann Sampling under Thermodynamic Parameter Perturbations.
Rogers, Emily; Murrugarra, David; Heitsch, Christine
2017-07-25
Understanding how RNA secondary structure prediction methods depend on the underlying nearest-neighbor thermodynamic model remains a fundamental challenge in the field. Minimum free energy (MFE) predictions are known to be "ill conditioned" in that small changes to the thermodynamic model can result in significantly different optimal structures. Hence, the best practice is now to sample from the Boltzmann distribution, which generates a set of suboptimal structures. Although the structural signal of this Boltzmann sample is known to be robust to stochastic noise, the conditioning and robustness under thermodynamic perturbations have yet to be addressed. We present here a mathematically rigorous model for conditioning inspired by numerical analysis, and also a biologically inspired definition for robustness under thermodynamic perturbation. We demonstrate the strong correlation between conditioning and robustness and use its tight relationship to define quantitative thresholds for well versus ill conditioning. These resulting thresholds demonstrate that the majority of the sequences are at least sample robust, which verifies the assumption of sampling's improved conditioning over the MFE prediction. Furthermore, because we find no correlation between conditioning and MFE accuracy, the presence of both well- and ill-conditioned sequences indicates the continued need for both thermodynamic model refinements and alternate RNA structure prediction methods beyond the physics-based ones. Copyright © 2017. Published by Elsevier Inc.
Discrete Boltzmann Method with Maxwell-Type Boundary Condition for Slip Flow
Zhang, Yu-Dong; Xu, Ai-Guo; Zhang, Guang-Cai; Chen, Zhi-Hua
2018-01-01
The rarefied effect of gas flow in microchannel is significant and cannot be well described by traditional hydrodynamic models. It has been known that discrete Boltzmann model (DBM) has the potential to investigate flows in a relatively wider range of Knudsen number because of its intrinsic kinetic nature inherited from Boltzmann equation. It is crucial to have a proper kinetic boundary condition for DBM to capture the velocity slip and the flow characteristics in the Knudsen layer. In this paper, we present a DBM combined with Maxwell-type boundary condition model for slip flow. The tangential momentum accommodation coefficient is introduced to implement a gas-surface interaction model. Both the velocity slip and the Knudsen layer under various Knudsen numbers and accommodation coefficients can be well described. Two kinds of slip flows, including Couette flow and Poiseuille flow, are simulated to verify the model. To dynamically compare results from different models, the relation between the definition of Knudsen number in hard sphere model and that in BGK model is clarified. Support of National Natural Science Foundation of China under Grant Nos. 11475028, 11772064, and 11502117 Science Challenge Project under Grant Nos. JCKY2016212A501 and TZ2016002
Discrete Boltzmann modeling of Rayleigh-Taylor instability in two-component compressible flows
Lin, Chuandong; Xu, Aiguo; Zhang, Guangcai; Luo, Kai Hong; Li, Yingjun
2017-11-01
A discrete Boltzmann model (DBM) is proposed to probe the Rayleigh-Taylor instability (RTI) in two-component compressible flows. Each species has a flexible specific-heat ratio and is described by one discrete Boltzmann equation (DBE). Independent discrete velocities are adopted for the two DBEs. The collision and force terms in the DBE account for the molecular collision and external force, respectively. Two types of force terms are exploited. In addition to recovering the modified Navier-Stokes equations in the hydrodynamic limit, the DBM has the capability of capturing detailed nonequilibrium effects. Furthermore, we use the DBM to investigate the dynamic process of the RTI. The invariants of tensors for nonequilibrium effects are presented and studied. For low Reynolds numbers, both global nonequilibrium manifestations and the growth rate of the entropy of mixing show three stages (i.e., the reducing, increasing, and then decreasing trends) in the evolution of the RTI. On the other hand, the early reducing tendency is suppressed and even eliminated for high Reynolds numbers. Relevant physical mechanisms are analyzed and discussed.
GPU accelerated study of heat transfer and fluid flow by lattice Boltzmann method on CUDA
Ren, Qinlong
Lattice Boltzmann method (LBM) has been developed as a powerful numerical approach to simulate the complex fluid flow and heat transfer phenomena during the past two decades. As a mesoscale method based on the kinetic theory, LBM has several advantages compared with traditional numerical methods such as physical representation of microscopic interactions, dealing with complex geometries and highly parallel nature. Lattice Boltzmann method has been applied to solve various fluid behaviors and heat transfer process like conjugate heat transfer, magnetic and electric field, diffusion and mixing process, chemical reactions, multiphase flow, phase change process, non-isothermal flow in porous medium, microfluidics, fluid-structure interactions in biological system and so on. In addition, as a non-body-conformal grid method, the immersed boundary method (IBM) could be applied to handle the complex or moving geometries in the domain. The immersed boundary method could be coupled with lattice Boltzmann method to study the heat transfer and fluid flow problems. Heat transfer and fluid flow are solved on Euler nodes by LBM while the complex solid geometries are captured by Lagrangian nodes using immersed boundary method. Parallel computing has been a popular topic for many decades to accelerate the computational speed in engineering and scientific fields. Today, almost all the laptop and desktop have central processing units (CPUs) with multiple cores which could be used for parallel computing. However, the cost of CPUs with hundreds of cores is still high which limits its capability of high performance computing on personal computer. Graphic processing units (GPU) is originally used for the computer video cards have been emerged as the most powerful high-performance workstation in recent years. Unlike the CPUs, the cost of GPU with thousands of cores is cheap. For example, the GPU (GeForce GTX TITAN) which is used in the current work has 2688 cores and the price is only 1
A multi scale approximation solution for the time dependent Boltzmann-transport equation
International Nuclear Information System (INIS)
Merk, B.
2004-03-01
The basis of all transient simulations for nuclear reactor cores is the reliable calculation of the power production. The local power distribution is generally calculated by solving the space, time, energy and angle dependent neutron transport equation known as Boltzmann equation. The computation of exact solutions of the Boltzmann equation is very time consuming. For practical numerical simulations approximated solutions are usually unavoidable. The objective of this work is development of an effective multi scale approximation solution for the Boltzmann equation. Most of the existing methods are based on separation of space and time. The new suggested method is performed without space-time separation. This effective approximation solution is developed on the basis of an expansion for the time derivative of different approximations to the Boltzmann equation. The method of multiple scale expansion is used for the expansion of the time derivative, because the problem of the stiff time behaviour can't be expressed by standard expansion methods. This multiple scale expansion is used in this work to develop approximation solutions for different approximations of the Boltzmann equation, starting from the expansion of the point kinetics equations. The resulting analytic functions are used for testing the applicability and accuracy of the multiple scale expansion method for an approximation solution with 2 delayed neutron groups. The results are tested versus the exact analytical results for the point kinetics equations. Very good agreement between both solutions is obtained. The validity of the solution with 2 delayed neutron groups to approximate the behaviour of the system with 6 delayed neutron groups is demonstrated in an additional analysis. A strategy for a solution with 4 delayed neutron groups is described. A multiple scale expansion is performed for the space-time dependent diffusion equation for one homogenized cell with 2 delayed neutron groups. The result is
On Equivalence of Nonequilibrium Thermodynamic and Statistical Entropies
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Purushottam D. Gujrati
2015-02-01
Full Text Available We review the concept of nonequilibrium thermodynamic entropy and observables and internal variables as state variables, introduced recently by us, and provide a simple first principle derivation of additive statistical entropy, applicable to all nonequilibrium states by treating thermodynamics as an experimental science. We establish their numerical equivalence in several cases, which includes the most important case when the thermodynamic entropy is a state function. We discuss various interesting aspects of the two entropies and show that the number of microstates in the Boltzmann entropy includes all possible microstates of non-zero probabilities even if the system is trapped in a disjoint component of the microstate space. We show that negative thermodynamic entropy can appear from nonnegative statistical entropy.
Inverse statistical physics of protein sequences: a key issues review
Cocco, Simona; Feinauer, Christoph; Figliuzzi, Matteo; Monasson, Rémi; Weigt, Martin
2018-03-01
In the course of evolution, proteins undergo important changes in their amino acid sequences, while their three-dimensional folded structure and their biological function remain remarkably conserved. Thanks to modern sequencing techniques, sequence data accumulate at unprecedented pace. This provides large sets of so-called homologous, i.e. evolutionarily related protein sequences, to which methods of inverse statistical physics can be applied. Using sequence data as the basis for the inference of Boltzmann distributions from samples of microscopic configurations or observables, it is possible to extract information about evolutionary constraints and thus protein function and structure. Here we give an overview over some biologically important questions, and how statistical-mechanics inspired modeling approaches can help to answer them. Finally, we discuss some open questions, which we expect to be addressed over the next years.
Statistical thermodynamics of a two-dimensional relativistic gas.
Montakhab, Afshin; Ghodrat, Malihe; Barati, Mahmood
2009-03-01
In this paper we study a fully relativistic model of a two-dimensional hard-disk gas. This model avoids the general problems associated with relativistic particle collisions and is therefore an ideal system to study relativistic effects in statistical thermodynamics. We study this model using molecular-dynamics simulation, concentrating on the velocity distribution functions. We obtain results for x and y components of velocity in the rest frame (Gamma) as well as the moving frame (Gamma;{'}) . Our results confirm that Jüttner distribution is the correct generalization of Maxwell-Boltzmann distribution. We obtain the same "temperature" parameter beta for both frames consistent with a recent study of a limited one-dimensional model. We also address the controversial topic of temperature transformation. We show that while local thermal equilibrium holds in the moving frame, relying on statistical methods such as distribution functions or equipartition theorem are ultimately inconclusive in deciding on a correct temperature transformation law (if any).
The q-exponential family in statistical physics
International Nuclear Information System (INIS)
Naudts, Jan
2010-01-01
The Boltzmann-Gibbs probability distribution, seen as a statistical model, belongs to the exponential family. Recently, the latter concept has been generalized. The q-exponential family has been shown to be relevant for the statistical description of small isolated systems. Two main applications are reviewed: 1. The distribution of the momentum of a single particle is a q-Gaussian, the distribution of its velocity is a deformed Maxwellian; 2. The configurational density distribution belongs to the q-exponential family. The definition of the temperature of small isolated systems is discussed. It depends on defining the thermodynamic entropy of a microcanonical ensemble in a suitable manner. The simple example of non-interacting harmonic oscillators shows that Renyi's entropy functional leads to acceptable results.
Systematic comparison of Tsallis statistics for charged pions produced in pp collisions
Energy Technology Data Exchange (ETDEWEB)
Parvan, A.S. [Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics, Dubna (Russian Federation); Horia Hulubei National Institute of Physics and Nuclear Engineering, Department of Theoretical Physics, Bucharest-Magurele (Romania); Teryaev, O.V. [Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics, Dubna (Russian Federation); Cleymans, J. [University of Cape Town, UCT-CERN Research Centre and Department of Physics, Cape Town (South Africa)
2017-05-15
The energy dependence of Tsallis statistics parameters is presented for charged pions produced at beam energies ranging between 6.3 GeV and 7 TeV. It is found that deviations from Boltzmann statistics are monotonically growing with beam energy. The energy dependence of the parameters T and q of the negatively charged pions in the energy range 6.3 < √(s) < 7000 GeV reveals that the deviation of the transverse momentum distribution from the exponential becomes more and more pronounced as the beam energy is increased. The parameter q increases with beam energy while the temperature T slowly decreases. (orig.)
Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states
International Nuclear Information System (INIS)
Gamba, Irene M.; Tharkabhushanam, Sri Harsha
2009-01-01
We propose a new spectral Lagrangian based deterministic solver for the non-linear Boltzmann transport equation (BTE) in d-dimensions for variable hard sphere (VHS) collision kernels with conservative or non-conservative binary interactions. The method is based on symmetries of the Fourier transform of the collision integral, where the complexity in its computation is reduced to a separate integral over the unit sphere S d-1 . The conservation of moments is enforced by Lagrangian constraints. The resulting scheme, implemented in free space, is very versatile and adjusts in a very simple manner to several cases that involve energy dissipation due to local micro-reversibility (inelastic interactions) or elastic models of slowing down process. Our simulations are benchmarked with available exact self-similar solutions, exact moment equations and analytical estimates for the homogeneous Boltzmann equation, both for elastic and inelastic VHS interactions. Benchmarking of the simulations involves the selection of a time self-similar rescaling of the numerical distribution function which is performed using the continuous spectrum of the equation for Maxwell molecules as studied first in Bobylev et al. [A.V. Bobylev, C. Cercignani, G. Toscani, Proof of an asymptotic property of self-similar solutions of the Boltzmann equation for granular materials, Journal of Statistical Physics 111 (2003) 403-417] and generalized to a wide range of related models in Bobylev et al. [A.V. Bobylev, C. Cercignani, I.M. Gamba, On the self-similar asymptotics for generalized non-linear kinetic Maxwell models, Communication in Mathematical Physics, in press. URL: ( )]. The method also produces accurate results in the case of inelastic diffusive Boltzmann equations for hard spheres (inelastic collisions under thermal bath), where overpopulated non-Gaussian exponential tails have been conjectured in computations by stochastic methods [T.V. Noije, M. Ernst, Velocity distributions in homogeneously
... Standards Act and Program MQSA Insights MQSA National Statistics Share Tweet Linkedin Pin it More sharing options ... but should level off with time. Archived Scorecard Statistics 2018 Scorecard Statistics 2017 Scorecard Statistics 2016 Scorecard ...
Résumé : La méthode Lattice Boltzmann (LBM)et la celle des ...
African Journals Online (AJOL)
JOSLIN
Lattice Boltzmann Method (LBM ) , Finite Difference Explicit ( DFE ) , Hybrid combines the two previous methods and Finite ... In addition, simulations using four methods show that for Ra = 105 the system is damped oscillating, it is oscillating periodic ..... convection heat transfert in a horizontal concentric annulus".Computer ...
Czech Academy of Sciences Publication Activity Database
Ziaja, B.; Saxena, V.; Son, S.-K.; Medvedev, N.; Barbrel, B.; Woloncewicz, B.; Stránský, Michal
2016-01-01
Roč. 93, č. 5 (2016), 1-6, č. článku 053210. ISSN 2470-0045 R&D Projects: GA MŠk(CZ) LG13029 Institutional support: RVO:68378271 Keywords : X-ray * Boltzmann equation Subject RIV: BL - Plasma and Gas Discharge Physics Impact factor: 2.366, year: 2016
Harting, Jens; Venturoli, Maddalena; Coveney, Peter V
2004-08-15
Well-designed lattice Boltzmann codes exploit the essentially embarrassingly parallel features of the algorithm and so can be run with considerable efficiency on modern supercomputers. Such scalable codes permit us to simulate the behaviour of increasingly large quantities of complex condensed matter systems. In the present paper, we present some preliminary results on the large-scale three-dimensional lattice Boltzmann simulation of binary immiscible fluid flows through a porous medium, derived from digitized X-ray micro-tomographic data of Bentheimer sandstone, and from the study of the same fluids under shear. Simulations on such scales can benefit considerably from the use of computational steering, and we describe our implementation of steering within the lattice Boltzmann code, called LB3D, making use of the RealityGrid steering library. Our large-scale simulations benefit from the new concept of capability computing, designed to prioritize the execution of big jobs on major supercomputing resources. The advent of persistent computational grids promises to provide an optimal environment in which to deploy these mesoscale simulation methods, which can exploit the distributed nature of computer, visualization and storage resources to reach scientific results rapidly; we discuss our work on the grid-enablement of lattice Boltzmann methods in this context. Copyright 2004 The Royal Society
Salomons, E.M.; Lohman, W.J.A.; Zhou, H.
2016-01-01
Propagation of sound waves in air can be considered as a special case of fluid dynamics. Consequently, the lattice Boltzmann method (LBM) for fluid flow can be used for simulating sound propagation. In this article application of the LBM to sound propagation is illustrated for various cases:
Wu, J.; Shu, C.
2010-07-01
The recently proposed boundary condition-enforced immersed boundary-lattice Boltzmann method (IB-LBM) [14] is improved in this work to simulate three-dimensional incompressible viscous flows. In the conventional IB-LBM, the restoring force is pre-calculated, and the non-slip boundary condition is not enforced as compared to body-fitted solvers. As a result, there is a flow penetration to the solid boundary. This drawback was removed by the new version of IB-LBM [14], in which the restoring force is considered as unknown and is determined in such a way that the non-slip boundary condition is enforced. Since Eulerian points are also defined inside the solid boundary, the computational domain is usually regular and the Cartesian mesh is used. On the other hand, to well capture the boundary layer and in the meantime, to save the computational effort, we often use non-uniform mesh in IB-LBM applications. In our previous two-dimensional simulations [14], the Taylor series expansion and least squares-based lattice Boltzmann method (TLLBM) was used on the non-uniform Cartesian mesh to get the flow field. The final expression of TLLBM is an algebraic formulation with some weighting coefficients. These coefficients could be computed in advance and stored for the following computations. However, this way may become impractical for 3D cases as the memory requirement often exceeds the machine capacity. The other way is to calculate the coefficients at every time step. As a result, extra time is consumed significantly. To overcome this drawback, in this study, we propose a more efficient approach to solve lattice Boltzmann equation on the non-uniform Cartesian mesh. As compared to TLLBM, the proposed approach needs much less computational time and virtual storage. Its good accuracy and efficiency are well demonstrated by its application to simulate the 3D lid-driven cubic cavity flow. To valid the combination of proposed approach with the new version of IBM [14] for 3D flows
Sampling, Probability Models and Statistical Reasoning Statistical ...
Indian Academy of Sciences (India)
Home; Journals; Resonance – Journal of Science Education; Volume 1; Issue 5. Sampling, Probability Models and Statistical Reasoning Statistical Inference. Mohan Delampady V R Padmawar. General Article Volume 1 Issue 5 May 1996 pp 49-58 ...
Sampling, Probability Models and Statistical Reasoning Statistical
Indian Academy of Sciences (India)
Home; Journals; Resonance – Journal of Science Education; Volume 1; Issue 5. Sampling, Probability Models and Statistical Reasoning Statistical Inference. Mohan Delampady V R Padmawar. General Article Volume 1 Issue 5 May 1996 pp 49-58 ...
Energy Technology Data Exchange (ETDEWEB)
Tripathy, Sushanta; Khuntia, Arvind; Tiwari, Swatantra Kumar; Sahoo, Raghunath [Indian Institute of Technology Indore, Discipline of Physics, School of Basic Sciences, Indore (India)
2017-05-15
In the continuation of our previous work, the transverse-momentum (p{sub T}) spectra and nuclear modification factor (R{sub AA}) are derived using the relaxation time approximation of Boltzmann Transport Equation (BTE). The initial p{sub T}-distribution used to describe p + p collisions has been studied with the perturbative-Quantum Chromodynamics (pQCD) inspired power-law distribution, Hagedorn's empirical formula and with the Tsallis non-extensive statistical distribution. The non-extensive Tsallis distribution is observed to describe the complete range of the transverse-momentum spectra. The Boltzmann-Gibbs Blast Wave (BGBW) distribution is used as the equilibrium distribution in the present formalism, to describe the p{sub T}-distribution and nuclear modification factor in nucleus-nucleus collisions. The experimental data for Pb+Pb collisions at √(s{sub NN}) = 2.76 TeV at the Large Hadron Collider at CERN have been analyzed for pions, kaons, protons, K{sup *0} and φ. It is observed that the present formalism while explaining the transverse-momentum spectra up to 5 GeV/c, explains the nuclear modification factor very well up to 8 GeV/c in p{sub T} for all these particles except for protons. R{sub AA} is found to be independent of the degree of non-extensivity, q{sub pp} after p{sub T} ∝ 8 GeV/c. (orig.)
Statistics for dental researchers: descriptive statistics
Mohammad Reza Baneshi PhD; Amir Reza Ghassemi DDS; Arash Shahravan DDS, MS
2012-01-01
Descriptive statistics is the process of summarizing gathered raw data from a research and creating useful statistics,which help the better understanding of data. According to the types of variables, which consist of qualitative andquantitative variables, some descriptive statistics have been introduced. Frequency percentage is used in qualitativedata, and mean, median, mode, standard deviation, standard error, variance, and range are some of the statistics whichare used in quantitative data....
Lattice Boltzmann simulation of droplet formation in T-junction geometries
Busuioc, Sergiu; Ambruş, Victor E.; Sofonea, Victor
2017-01-01
The formation of droplets in T-junction configurations is investigated using a two-dimensional Lattice Boltzmann model for liquid-vapor systems. We use an expansion of the equilibrium distribution function with respect to Hermite polynomials and an off-lattice velocity set. To evolve the distribution functions we use the second order corner transport upwind numerical scheme and a third order scheme is used to compute the gradient operators in the force term. The droplet formation successfully recovers the squeezing, dripping and jetting regimes. We find that the droplet length decreases proportionally with the flow rate of the continuous phase and increases with the flow rate of the dispersed phase in all simulation configurations and has a linear dependency on the surface tension parameter κ.
Lattice Boltzmann simulations of bubble formation in a microfluidic T-junction.
Amaya-Bower, Luz; Lee, Taehun
2011-06-28
A lattice Boltzmann equation method based on the Cahn-Hilliard diffuse interface theory is developed to investigate the bubble formation process in a microchannel with T-junction mixing geometry. The bubble formation process has different regimes, namely, squeezing, dripping and jetting regimes, which correspond to the primary forces acting on the system. Transition from regime to regime is generally dictated by the capillary number Ca, volumetric flow ratio Q and viscosity ratio λ. A systematic analysis is performed to evaluate these effects. The computations are performed in the range of 10(-4)
Kaehler, G; Wagner, A J
2013-06-01
Current implementations of fluctuating ideal-gas descriptions with the lattice Boltzmann methods are based on a fluctuation dissipation theorem, which, while greatly simplifying the implementation, strictly holds only for zero mean velocity and small fluctuations. We show how to derive the fluctuation dissipation theorem for all k, which was done only for k=0 in previous derivations. The consistent derivation requires, in principle, locally velocity-dependent multirelaxation time transforms. Such an implementation is computationally prohibitively expensive but, with a small computational trick, it is feasible to reproduce the correct FDT without overhead in computation time. It is then shown that the previous standard implementations perform poorly for non vanishing mean velocity as indicated by violations of Galilean invariance of measured structure factors. Results obtained with the method introduced here show a significant reduction of the Galilean invariance violations.
Self-consistent relativistic Boltzmann-Uehling-Uhlenbeck equation for the Δ distribution function
International Nuclear Information System (INIS)
Mao, G.; Li, Z.; Zhuo, Y.
1996-01-01
We derive the self-consistent relativistic Boltzmann-Uehling-Uhlenbeck (RBUU) equation for the delta distribution function within the framework which we have done for nucleon close-quote s. In our approach, the Δ isobars are treated in essentially the same way as nucleons. Both mean field and collision terms of Δ close-quote s RBUU equation are derived from the same effective Lagrangian and presented analytically. We calculate the in-medium NΔ elastic and inelastic scattering cross sections up to twice nuclear matter density and the results show that the in-medium cross sections deviate substantially from Cugnon close-quote s parametrization that is commonly used in the transport model. copyright 1996 The American Physical Society
Influence of thermal fluctuations on ligament break-up: a fluctuating lattice Boltzmann study
Xue, Xiao; Biferale, Luca; Sbragaglia, Mauro; Toschi, Federico
2017-11-01
Thermal fluctuations are essential ingredients in a nanoscale system, driving Brownian motion of particles and capillary waves at non-ideal interfaces. Here we study the influence of thermal fluctuations on the breakup of liquid ligaments at the nanoscale. We offer quantitative characterization of the effects of thermal fluctuations on the Plateau-Rayleigh mechanism that drives the breakup process of ligaments. Due to thermal fluctuations, the droplet sizes after break-up need to be analyzed in terms of their distribution over an ensemble made of repeated experiments. To this aim, we make use of numerical simulations based on the fluctuating lattice Boltzmann method (FLBM) for multicomponent mixtures. The method allows an accurate and efficient simulation of the fluctuating hydrodynamics equations of a binary mixture, where both stochastic viscous stresses and diffusion fluxes are introduced. This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No 642069.
Modeling of flow of particles in a non-Newtonian fluid using lattice Boltzmann method
DEFF Research Database (Denmark)
Skocek, Jan; Svec, Oldrich; Spangenberg, Jon
2011-01-01
To predict correctly the castings process of self compacting concrete a numerical model capable of simulating flow patterns at the structural scale and at the same time the impact of the varying volume fraction of aggregates and other phenomena at the scale of aggregates on the flow evolution...... is necessary. In this contribution, the model at the scale of aggregates is introduced. The conventional lattice Boltzmann method for fluid flow is enriched with the immersed boundary method with direct forcing to simulate the flow of rigid particles in a non- Newtonian liquid. Basic ingredients of the model...... are presented and discussed with the emphasis on a newly developed algorithm for the dynamics of particles whose interactions strongly depend on velocities of particles. The application of the model is demonstrated by a parametric study with varying volume fractions of aggregates and speed of shearing used...
Hybrid Lattice Boltzmann Method for the Simulation of Blending Process in Static Mixers
Latt, Jonas; Kontaxakis, Dimitrios; Chatagny, Laurent; Muggli, Felix; Chopard, Bastien
2013-12-01
A lattice Boltzmann method is proposed to simulate the blending of two fluids in static, laminar mixers. The method uses a mesh-based algorithm to solve for the fluid flow, and a meshless technique to trace the interface between the blended fluids. This hybrid approach is highly accurate, because the position of the interface can be traced beyond the resolution of the grid. The numerical diffusion is negligible in this model, and it is possible to reproduce mixing patterns that contain more than one hundred striations with high fidelity. The implementation of this method in the massively parallel library Palabos is presented, and simulation results are compared with experimental data to emphasize the accuracy of the results.
A numerical solution of the linear Boltzmann equation using cubic B-splines.
Khurana, Saheba; Thachuk, Mark
2012-03-07
A numerical method using cubic B-splines is presented for solving the linear Boltzmann equation. The collision kernel for the system is chosen as the Wigner-Wilkins kernel. A total of three different representations for the distribution function are presented. Eigenvalues and eigenfunctions of the collision matrix are obtained for various mass ratios and compared with known values. Distribution functions, along with first and second moments, are evaluated for different mass and temperature ratios. Overall it is shown that the method is accurate and well behaved. In particular, moments can be predicted with very few points if the representation is chosen well. This method produces sparse matrices, can be easily generalized to higher dimensions, and can be cast into efficient parallel algorithms. © 2012 American Institute of Physics
A dynamic mesh refinement technique for Lattice Boltzmann simulations on octree-like grids
Neumann, Philipp
2012-04-27
In this contribution, we present our new adaptive Lattice Boltzmann implementation within the Peano framework, with special focus on nanoscale particle transport problems. With the continuum hypothesis not holding anymore on these small scales, new physical effects - such as Brownian fluctuations - need to be incorporated. We explain the overall layout of the application, including memory layout and access, and shortly review the adaptive algorithm. The scheme is validated by different benchmark computations in two and three dimensions. An extension to dynamically changing grids and a spatially adaptive approach to fluctuating hydrodynamics, allowing for the thermalisation of the fluid in particular regions of interest, is proposed. Both dynamic adaptivity and adaptive fluctuating hydrodynamics are validated separately in simulations of particle transport problems. The application of this scheme to an oscillating particle in a nanopore illustrates the importance of Brownian fluctuations in such setups. © 2012 Springer-Verlag.
Multiscale Lattice Boltzmann method for flow simulations in highly heterogenous porous media
Li, Jun
2013-01-01
A lattice Boltzmann method (LBM) for flow simulations in highly heterogeneous porous media at both pore and Darcy scales is proposed in the paper. In the pore scale simulations, flow of two phases (e.g., oil and gas) or two immiscible fluids (e.g., water and oil) are modeled using cohesive or repulsive forces, respectively. The relative permeability can be computed using pore-scale simulations and seamlessly applied for intermediate and Darcy-scale simulations. A multiscale LBM that can reduce the computational complexity of existing LBM and transfer the information between different scales is implemented. The results of coarse-grid, reduced-order, simulations agree very well with the averaged results obtained using fine grid.
Boltzmann sampling for an XY model using a non-degenerate optical parametric oscillator network
Takeda, Y.; Tamate, S.; Yamamoto, Y.; Takesue, H.; Inagaki, T.; Utsunomiya, S.
2018-01-01
We present an experimental scheme of implementing multiple spins in a classical XY model using a non-degenerate optical parametric oscillator (NOPO) network. We built an NOPO network to simulate a one-dimensional XY Hamiltonian with 5000 spins and externally controllable effective temperatures. The XY spin variables in our scheme are mapped onto the phases of multiple NOPO pulses in a single ring cavity and interactions between XY spins are implemented by mutual injections between NOPOs. We show the steady-state distribution of optical phases of such NOPO pulses is equivalent to the Boltzmann distribution of the corresponding XY model. Estimated effective temperatures converged to the setting values, and the estimated temperatures and the mean energy exhibited good agreement with the numerical simulations of the Langevin dynamics of NOPO phases.
Rahman Prize Lecture: Lattice Boltzmann simulation of complex states of flowing matter
Succi, Sauro
Over the last three decades, the Lattice Boltzmann (LB) method has gained a prominent role in the numerical simulation of complex flows across an impressively broad range of scales, from fully-developed turbulence in real-life geometries, to multiphase flows in micro-fluidic devices, all the way down to biopolymer translocation in nanopores and lately, even quark-gluon plasmas. After a brief introduction to the main ideas behind the LB method and its historical developments, we shall present a few selected applications to complex flow problems at various scales of motion. Finally, we shall discuss prospects for extreme-scale LB simulations of outstanding problems in the physics of fluids and its interfaces with material sciences and biology, such as the modelling of fluid turbulence, the optimal design of nanoporous gold catalysts and protein folding/aggregation in crowded environments.
Phase-field-lattice Boltzmann studies for dendritic growth with natural convection
Takaki, Tomohiro; Rojas, Roberto; Sakane, Shinji; Ohno, Munekazu; Shibuta, Yasushi; Shimokawabe, Takashi; Aoki, Takayuki
2017-09-01
Simulating dendritic growth with natural convection is challenging because of the size of the computational domain required when compared to the dendrite scale. In this study, a phase-field-lattice Boltzmann model was used to simulate dendritic growth in the presence of natural convection due to a difference in solute concentration. To facilitate and accelerate the large-scale simulation, a parallel computing code with multiple graphics processing units was developed. The effects of the computational domain size as well as those of gravity on the dendritic morphologies were examined by performing two-dimensional free dendritic growth simulations with natural convection. The effects of the gravity direction on the dendrite spacing and morphology were also investigated by simulating unidirectional solidification from multiple seeds.
Lattice Boltzmann model for free-surface flow and its application to filling process in casting
Ginzburg, I
2003-01-01
A generalized lattice Boltzmann model to simulate free-surface is constructed in both two and three dimensions. The proposed model satisfies the interfacial boundary conditions accurately. A distinctive feature of the model is that the collision processes is carried out only on the points occupied partially or fully by the fluid. To maintain a sharp interfacial front, the method includes an anti-diffusion algorithm. The unknown distribution functions at the interfacial region are constructed according to the first-order Chapman-Enskog analysis. The interfacial boundary conditions are satisfied exactly by the coefficients in the Chapman-Enskog expansion. The distribution functions are naturally expressed in the local interfacial coordinates. The macroscopic quantities at the interface are extracted from the least-square solutions of a locally linearized system obtained from the known distribution functions. The proposed method does not require any geometric front construction and is robust for any interfacial ...
International Nuclear Information System (INIS)
Ehnder, A.Ya.; Ehnder, I.A.
1999-01-01
A new approach to develop nonlinear moment method to solve the Boltzmann equation is presented. This approach is based on the invariance of collision integral as to the selection of the base functions. The Sonin polynomials with the Maxwell weighting function are selected to serve as the base functions. It is shown that for the arbitrary cross sections of the interaction the matrix elements corresponding to the moments from the nonlinear integral of collisions are bound by simple recurrent bonds enabling to express all nonlinear matrix elements in terms of the linear ones. As a result, high-efficiency numerical pattern to calculate nonlinear matrix elements is obtained. The presented approach offers possibilities both to calculate relaxation processes within high speed range and to some more complex kinetic problems [ru
Application of the lattice Boltzmann method to transition in oscillatory channel flow
Cosgrove, J A; Tonge, S J; Munro, C G; Greated, C A; Campbell, D M
2003-01-01
In this study the applicability of the lattice Boltzmann method to oscillatory channel flow with a zero mean velocity has been evaluated. The model has been compared to exact analytical solutions in the laminar case (Re subdelta < 100, where Re subdelta is the Reynolds number based on the Stokes layer) for the Womersley parameter 1 < alpha < 31. In this regime, there was good agreement between numerical and exact analytical solutions. The model was then applied to study the primary instability of oscillatory channel flow with a zero mean velocity. For these transitionary flows the parameters were varied in the range 400 < Re subdelta < 1000 and 4 < alpha < 16. Disturbances superimposed on the numerical solution triggered the two-dimensional primary instability. This phenomenon has not been numerically evaluated over the range of alpha or Re subdelta currently investigated. The results are consistent with quasi-steady linear stability theories and previous numerical investigations.
Peng, Y.; Shu, C.; Chew, Y. T.; Qiu, J.
2003-03-01
An alternative new method called lattice Boltzmann method (LBM) is applied in this work to simulate the flows in Czochralski crystal growth, which is one of the widely used prototypical systems for melt-crystal growth. The standard LBM can only be used in Cartesian coordinate system and we extend it to be applicable to this axisymmetric thermal flow problem, avoiding the use of three-dimensional LBM on Cartesian coordinate system. The extension is based on the following idea. By inserting position and time dependent source terms into the evolution equation of standard LBM, the continuity and NS equations on the cylindrical coordinate system [1] can be recovered. Our extension is validated by its application to the benchmark problem suggested by Wheeler [2].
Peng, Y; Chew, Y T; Qiu, J
2003-01-01
An alternative new method called lattice Boltzmann method (LBM) is applied in this work to simulate the flows in Czochralski crystal growth, which is one of the widely used prototypical systems for melt-crystal growth. The standard LBM can only be used in Cartesian coordinate system and we extend it to be applicable to this axisymmetric thermal flow problem, avoiding the use of three-dimensional LBM on Cartesian coordinate system. The extension is based on the following idea. By inserting position and time dependent source terms into the evolution equation of standard LBM, the continuity and NS equations on the cylindrical coordinate system can be recovered. Our extension is validated by its application to the benchmark problem suggested by Wheeler .
International Nuclear Information System (INIS)
Peng, Y.; Shu, C.; Chew, Y.T.; Qiu, J.
2003-01-01
An alternative new method called lattice Boltzmann method (LBM) is applied in this work to simulate the flows in Czochralski crystal growth, which is one of the widely used prototypical systems for melt-crystal growth. The standard LBM can only be used in Cartesian coordinate system and we extend it to be applicable to this axisymmetric thermal flow problem, avoiding the use of three-dimensional LBM on Cartesian coordinate system. The extension is based on the following idea. By inserting position and time dependent source terms into the evolution equation of standard LBM, the continuity and NS equations on the cylindrical coordinate system can be recovered. Our extension is validated by its application to the benchmark problem suggested by Wheeler
Study of nonequilibrium work distributions from a fluctuating lattice Boltzmann model.
Nasarayya Chari, S Siva; Murthy, K P N; Inguva, Ramarao
2012-04-01
A system of ideal gas is switched from an initial equilibrium state to a final state not necessarily in equilibrium, by varying a macroscopic control variable according to a well-defined protocol. The distribution of work performed during the switching process is obtained. The equilibrium free energy difference, ΔF, is determined from the work fluctuation relation. Some of the work values in the ensemble shall be less than ΔF. We term these as ones that "violate" the second law of thermodynamics. A fluctuating lattice Boltzmann model has been employed to carry out the simulation of the switching experiment. Our results show that the probability of violation of the second law increases with the increase of switching time (τ) and tends to one-half in the reversible limit of τ→∞.
Sparse dynamical Boltzmann machine for reconstructing complex networks with binary dynamics
Chen, Yu-Zhong; Lai, Ying-Cheng
2018-03-01
Revealing the structure and dynamics of complex networked systems from observed data is a problem of current interest. Is it possible to develop a completely data-driven framework to decipher the network structure and different types of dynamical processes on complex networks? We develop a model named sparse dynamical Boltzmann machine (SDBM) as a structural estimator for complex networks that host binary dynamical processes. The SDBM attains its topology according to that of the original system and is capable of simulating the original binary dynamical process. We develop a fully automated method based on compressive sensing and a clustering algorithm to construct the SDBM. We demonstrate, for a variety of representative dynamical processes on model and real world complex networks, that the equivalent SDBM can recover the network structure of the original system and simulates its dynamical behavior with high precision.
International Nuclear Information System (INIS)
Ozaki, Hideaki
2004-01-01
Using the closed-time-path formalism, we construct perturbative frameworks, in terms of quasiparticle picture, for studying quasiuniform relativistic quantum field systems near equilibrium and non-equilibrium quasistationary systems. We employ the derivative expansion and take in up to the second-order term, i.e., one-order higher than the gradient approximation. After constructing self-energy resumed propagator, we formulated two kinds of mutually equivalent perturbative frameworks: The first one is formulated on the basis of the 'bare' number density function, and the second one is formulated on the basis of 'physical' number density function. In the course of construction of the second framework, the generalized Boltzmann equations directly come out, which describe the evolution of the system. (author)
Cumulant solution of the elastic Boltzmann transport equation in an infinite uniform medium
International Nuclear Information System (INIS)
Cai, W.; Lax, M.; Alfano, R. R.
2000-01-01
We consider an analytical solution of the time-dependent elastic Boltzmann transport equation in an infinite uniform isotropic medium with an arbitrary phase function. We obtain (1) the exact distribution in angle, (2) the exact first and second spatial cumulants at any angle, and (3) an approximate combined distribution in position and angle and a spatial distribution whose central position and half-width of spread are always exact. The resulting Gaussian distribution has a center that advances in time, and an ellipsoidal contour that grows and changes shape providing a clear picture of the time evolution of the particle migration from near ballistic, through snakelike and into the final diffusive regime. (c) 2000 The American Physical Society
Multispeed Lattice Boltzmann Model with Space-Filling Lattice for Transcritical Shallow Water Flows
Directory of Open Access Journals (Sweden)
Y. Peng
2017-01-01
Full Text Available Inspired by the recent success of applying multispeed lattice Boltzmann models with a non-space-filling lattice for simulating transcritical shallow water flows, the capabilities of their space-filling counterpart are investigated in this work. Firstly, two lattice models with five integer discrete velocities are derived by using the method of matching hydrodynamics moments and then tested with two typical 1D problems including the dam-break flow over flat bed and the steady flow over bump. In simulations, the derived space-filling multispeed models, together with the stream-collision scheme, demonstrate better capability in simulating flows with finite Froude number. However, the performance is worse than the non-space-filling model solved by finite difference scheme. The stream-collision scheme with second-order accuracy may be the reason since a numerical scheme with second-order accuracy is prone to numerical oscillations at discontinuities, which is worthwhile for further study.
Simulation of Thermomagnetic Convection in a Cavity Using the Lattice Boltzmann Model
Directory of Open Access Journals (Sweden)
Mahshid Hadavand
2011-01-01
Full Text Available Thermomagnetic convection in a differentially heated square cavity with an infinitely long third dimension is numerically simulated using the single relaxation time lattice Boltzmann method (LBM. This problem is of considerable interest when dealing with cooling of microelectronic devices, in situations where natural convection does not meet the cooling requirements, and forced convection is not viable due to the difficulties associated with pumping a ferrofluid. Therefore, circulation is achieved by imposing a magnetic field, which is created and controlled by placing a dipole at the bottom of the enclosure. The magnitude of the magnetic force is controlled by changing the electrical current through the dipole. In this study, the effects of combined natural convection and magnetic convection, which is commonly known as “thermomagnetic convection,” are analysed in terms of the flow modes and heat transfer characteristics of a magnetic fluid.
Inamuro, Takaji; Hayashi, Hirofumi; Koshiyama, Masahiro
2008-11-01
The lattice Boltzmann method (LBM) for multicomponent immiscible fluids is applied to the simulations of solid-fluid mixture flows including spherical and nonspherical particles in a square pipe. A spherical solid particle is modeled by a droplet with strong interfacial tension and large viscosity, and consequently there is no need to track the moving solid-liquid boundary explicitly. Nonspherical (discoid and biconcave discoid) solid particles are made by applying artificial forces to the spherical droplet. It is found that spherical particles move around stable positions between the wall and the center of the pipe. On the other hand, a biconcave discoid particle moves along a helical path around the center of the pipe with periodic oscillations in its orientation. The radius of the helical path and the polar angle of the orientation increase as the hollow of the concave becomes larger.
DEFF Research Database (Denmark)
van Tulder, Gijs; de Bruijne, Marleen
2016-01-01
outperform these standard filter banks because they learn a feature description directly from the training data. Like many other representation learning methods, restricted Boltzmann machines are unsupervised and are trained with a generative learning objective; this allows them to learn representations from...... for describing the training data and for classification. We present experiments with feature learning for lung texture classification and airway detection in CT images. In both applications, a combination of learning objectives outperformed purely discriminative or generative learning, increasing, for instance......, the lung tissue classification accuracy by 1 to 8 percentage points. This shows that discriminative learning can help an otherwise unsupervised feature learner to learn filters that are optimized for classification....
Directory of Open Access Journals (Sweden)
J. Alinejad
2016-01-01
Full Text Available The purpose of this paper is to investigate the EGM method and the behavior of a solid particle suspended in a twodimensional rectangular cavity due to conjugate natural convection. A thermal lattice Boltzmann BGK model is implemented to simulate the two dimensional natural convection and the particle phase was modeled using the Lagrangian–Lagrangian approach where the solid particles are treated as points moving in the computational domain as a result of the fluid motion. Entropy generation due to heat transfer irreversibility, isotherms, streamlines and Nusselt numbers were obtained and discussed. Total entropy generations in various cases are also reported and optimum case is presented based on minimum entropy generation.
Extended Lattice Boltzmann Method with Application to Predict Aerodynamic Loads of Long Span Bridge
Liu, Tiancheng; Liu, Gao; Li, Yi; Ge, Yaojun
2010-05-01
The lattice Boltzmann (LB) method, a new conceptual approach to solve the fluid dynamics problem, is presented at first. The turbulence model is incorporated into the normal LB equation to simulate turbulence flow in the form of turbulence relaxation time determined by the nonequilibrium particle distribution function and Smagorinsky model. The total relaxation time is defined as the contribution of molecule viscosity and turbulence eddy viscosity. The aerodynamic forces on bridge girders are predicted by present LB method and the analysis of flow state is performed. The validity of LB method is verified through comparing the present results with the available experimental data and those obtained from the solutions of Navier-Stockes equation like Reynolds averaged Navier-Stokes (RANS) and discrete vortex method (DVM).
Balanced growth path solutions of a Boltzmann mean field game model for knowledge growth
Burger, Martin
2016-11-18
In this paper we study balanced growth path solutions of a Boltzmann mean field game model proposed by Lucas and Moll [15] to model knowledge growth in an economy. Agents can either increase their knowledge level by exchanging ideas in learning events or by producing goods with the knowledge they already have. The existence of balanced growth path solutions implies exponential growth of the overall production in time. We prove existence of balanced growth path solutions if the initial distribution of individuals with respect to their knowledge level satisfies a Pareto-tail condition. Furthermore we give first insights into the existence of such solutions if in addition to production and knowledge exchange the knowledge level evolves by geometric Brownian motion.
Lattice Boltzmann modeling of transport phenomena in fuel cells and flow batteries
Xu, Ao; Shyy, Wei; Zhao, Tianshou
2017-06-01
Fuel cells and flow batteries are promising technologies to address climate change and air pollution problems. An understanding of the complex multiscale and multiphysics transport phenomena occurring in these electrochemical systems requires powerful numerical tools. Over the past decades, the lattice Boltzmann (LB) method has attracted broad interest in the computational fluid dynamics and the numerical heat transfer communities, primarily due to its kinetic nature making it appropriate for modeling complex multiphase transport phenomena. More importantly, the LB method fits well with parallel computing due to its locality feature, which is required for large-scale engineering applications. In this article, we review the LB method for gas-liquid two-phase flows, coupled fluid flow and mass transport in porous media, and particulate flows. Examples of applications are provided in fuel cells and flow batteries. Further developments of the LB method are also outlined.
Energy Technology Data Exchange (ETDEWEB)
Bu, W.; Vaknin, D.; Travesset, A. (Iowa State)
2010-07-13
Surface sensitive synchrotron-x-ray scattering studies reveal the distributions of monovalent ions next to highly charged interfaces. A lipid phosphate (dihexadecyl hydrogen phosphate) was spread as a monolayer at the air-water interface, containing CsI at various concentrations. Using anomalous reflectivity off and at the L{sub 3} Cs{sup +} resonance, we provide spatial counterion distributions (Cs{sup +}) next to the negatively charged interface over a wide range of ionic concentrations. We argue that at low salt concentrations and for pure water the enhanced concentration of hydroniums H{sub 3}O{sup +} at the interface leads to proton transfer back to the phosphate group by a high contact potential, whereas high salt concentrations lower the contact potential resulting in proton release and increased surface charge density. The experimental ionic distributions are in excellent agreement with a renormalized-surface-charge Poisson-Boltzmann theory without fitting parameters or additional assumptions.
Bu, Wei; Vaknin, David; Travesset, Alex
2005-12-01
Surface sensitive synchrotron-x-ray scattering studies reveal the distributions of monovalent ions next to highly charged interfaces. A lipid phosphate (dihexadecyl hydrogen phosphate) was spread as a monolayer at the air-water interface, containing CsI at various concentrations. Using anomalous reflectivity off and at the L3 Cs+ resonance, we provide spatial counterion distributions (Cs+) next to the negatively charged interface over a wide range of ionic concentrations. We argue that at low salt concentrations and for pure water the enhanced concentration of hydroniums H3O+ at the interface leads to proton transfer back to the phosphate group by a high contact potential, whereas high salt concentrations lower the contact potential resulting in proton release and increased surface charge density. The experimental ionic distributions are in excellent agreement with a renormalized-surface-charge Poisson-Boltzmann theory without fitting parameters or additional assumptions.
Hybrid modeling of CO2 sequestration processes using the lattice-Boltzmann method and PFLOTRAN
Porter, M. L.; Coon, E. T.; Kang, Q.; Lichtner, P. C.; Carey, J. W.
2011-12-01
Successful CO2 injection and sequestration requires fundamental understanding of many complex processes encountered in multiphase flow and reactive transport through porous media. Although these processes are inherently governed by microscopic interfacial phenomena, they must be described at much larger scales for many practical engineering applications. In this work we present a parallel hybrid modeling scheme that couples a lattice-Boltzmann (LB) simulator for porescale multiphase flow to PFLOTRAN for continuum (Darcy) scale multiphase flow and both continuum and porescale reactive transport. We discuss details regarding the LB method, PFLOTRAN, and the coupling of the two simulators. In addition, we present a number of simulations that validate and highlight both the porescale and hybrid modeling capabilities for applications involving CO2 sequestration.
Learning features for tissue classification with the classification restricted Boltzmann machine
DEFF Research Database (Denmark)
van Tulder, Gijs; de Bruijne, Marleen
2014-01-01
Performance of automated tissue classification in medical imaging depends on the choice of descriptive features. In this paper, we show how restricted Boltzmann machines (RBMs) can be used to learn features that are especially suited for texture-based tissue classification. We introduce...... the convolutional classification RBM, a combination of the existing convolutional RBM and classification RBM, and use it for discriminative feature learning. We evaluate the classification accuracy of convolutional and non-convolutional classification RBMs on two lung CT problems. We find that RBM-learned features...... outperform conventional RBM-based feature learning, which is unsupervised and uses only a generative learning objective, as well as often-used filter banks. We show that a mixture of generative and discriminative learning can produce filters that give a higher classification accuracy....
Lattice-Boltzmann-based simulations of diffusiophoresis of colloids and cells
Kreft Pearce, Jennifer; Castigliego, Joshua
Increasing environmental degradation due to plastic pollutants requires innovative solutions that facilitate the extraction of pollutants without harming local biota. We present results from a lattice-Boltzmann-base Brownian Dynamics simulation on diffusiophoresis and the separation of particles within the system. A gradient in viscosity that simulates a concentration gradient in a dissolved polymer allows us to separate various types of particles based on their deformability. As seen in previous experiments, simulated particles that have a higher deformability react differently to the polymer matrix than those with a lower deformability. Therefore, the particles can be separated from each other. The system described above was simulated with various concentration gradients as well as various Soret coefficients in order to optimize the separation of the particles. This simulation, in particular, was intended to model an oceanic system where the particles of interest were motile and nonmotile plankton and microplastics. The separation of plankton from the microplastics was achieved.
Lid-driven cavity flow using a discrete velocity method for solving the Boltzmann equation
Sekaran, Aarthi; Varghese, Philip; Estes, Samuel; Goldstein, David
2016-11-01
We extend the discrete velocity method for solving the Boltzmann equation previously used for one-dimensional problems to two spatial dimensions. The collision integral is computed using collisions between velocity classes selected randomly using a Monte Carlo method. Arbitrary post-collision velocities are mapped back onto the grid using a projection scheme which conserves mass, momentum, and energy. In addition, a variance reduction scheme is implemented to decrease noise and further reduce computational effort. The convection part of the equation is computed using first order upwind finite differences. We apply this discrete velocity scheme to the 2D lid-driven square cavity flow problem with Ar as the fluid medium and explore the effect of the additional flexibility available in this quasi-particle based stochastic method on the accuracy and noise level in the solutions obtained.
Simulating gas-liquid flow in a micro-channel with the lattice Boltzmann method
Shi, Grace; Lazouskaya, Volha; Jin, Yan; Wang, Lian-Ping
2007-11-01
The flows of water in natural soil porous media with air-water interface are important to colloid-facilitated transport of contaminants and other phenomena with groundwater as the carrier. These flows are complex in terms of the geometrical feature and physical and chemical forces involved. As first step, we here demonstrate that a gas-liquid interfacial viscous flow in a 3D micro-channel with a square cross-section can be simulated using the lattice Boltzmann method. The talk will cover the detailed ingredients of the two-phase LBE model including the proper equation of state, surface tension, and the triple-phase boundary conditions. Methods to improve the stability of the code such as using multiple relaxation times will be tested. Preliminary results will be presented and compared to parallel experimental observations using confocal laser scanning microscopy.
International Nuclear Information System (INIS)
Darquie, B.; Mejri, S.; Sow, P. L. T.; Lemarchand, C.; Triki, M.; Tokunaga, S. K.; Borde, C. J.; Chardonnet, C.; Daussy, C.
2013-01-01
Accurate molecular spectroscopy in the mid-infrared region allows precision measurements of fundamental constants. For instance, measuring the linewidth of an isolated Doppler-broadened absorption line of ammonia around 10 μm enables a determination of the Boltzmann constant k B . We report on our latest measurements. By fitting this lineshape to several models which include Dicke narrowing or speed-dependent collisional effects, we find that a determination of k B with an uncertainty of a few ppm is reachable. This is comparable to the best current uncertainty obtained using acoustic methods and would make a significant contribution to any new value of k B determined by the CODATA. Furthermore, having multiple independent measurements at these accuracies opens the possibility of defining the kelvin by fixing k B , an exciting prospect considering the upcoming redefinition of the International System of Units. (authors)
Phase-field-lattice Boltzmann study for lamellar eutectic growth in a natural convection melt
Directory of Open Access Journals (Sweden)
Ang Zhang
2017-11-01
Full Text Available In the present study, the influence of natural convection on the lamellar eutectic growth is determined by a phase-field-lattice Boltzmann study for Al-Cu eutectic alloy. The mass difference resulting from concentration difference led to the fluid flow, and a robust parallel and adaptive mesh refinement algorithm was employed to improve the computational efficiency without any compromising accuracy. Results show that the existence of natural convection would affect the growth undercooling and thus control the interface shape by adjusting the lamellar width. In particular, by alternating the magnitude of the solute expansion coefficient, the strength of the natural convection is changed. Corresponding microstructure patterns are discussed and compared with those under no-convection conditions.
Cole-Hopf Transformation Based Lattice Boltzmann Model for One-dimensional Burgers’ Equation
Qi, Xiao-Tong; Shi, Bao-Chang; Chai, Zhen-Hua
2018-03-01
In this paper, we present a Cole-Hopf transformation based lattice Boltzmann (LB) model for solving one-dimensional Burgers’ equation, and compared to available LB models, the effect of nonlinear convection term can be eliminated. Through Chapman-Enskog analysis, it can be found that the converted diffusion equation based on the Cole-Hopf transformation can be recovered correctly from present LB model. Some numerical tests are also performed to validate the present LB model, and the numerical results show that, similar to previous LB models, the present model also has a second-order convergence rate in space, but it is more accurate than the previous ones. Supported by the National Natural Science Foundation of China under Grant No. 51576079
Lattice Boltzmann simulations of immiscible displacement process with large viscosity ratios
Rao, Parthib; Schaefer, Laura
2017-11-01
Immiscible displacement is a key physical mechanism involved in enhanced oil recovery and carbon sequestration processes. This multiphase flow phenomenon involves a complex interplay of viscous, capillary, inertial and wettability effects. The lattice Boltzmann (LB) method is an accurate and efficient technique for modeling and simulating multiphase/multicomponent flows especially in complex flow configurations and media. In this presentation we present numerical simulation results of displacement process in thin long channels. The results are based on a new psuedo-potential multicomponent LB model with multiple relaxation time collision (MRT) model and explicit forcing scheme. We demonstrate that the proposed model is capable of accurately simulating the displacement process involving fluids with a wider range of viscosity ratios (>100) and which also leads to viscosity-independent interfacial tension and reduction of some important numerical artifacts.
Lattice Boltzmann simulations for wall-flow dynamics in porous ceramic diesel particulate filters
Lee, Da Young; Lee, Gi Wook; Yoon, Kyu; Chun, Byoungjin; Jung, Hyun Wook
2018-01-01
Flows through porous filter walls of wall-flow diesel particulate filter are investigated using the lattice Boltzmann method (LBM). The microscopic model of the realistic filter wall is represented by randomly overlapped arrays of solid spheres. The LB simulation results are first validated by comparison to those from previous hydrodynamic theories and constitutive models for flows in porous media with simple regular and random solid-wall configurations. We demonstrate that the newly designed randomly overlapped array structures of porous walls allow reliable and accurate simulations for the porous wall-flow dynamics in a wide range of solid volume fractions from 0.01 to about 0.8, which is beyond the maximum random packing limit of 0.625. The permeable performance of porous media is scrutinized by changing the solid volume fraction and particle Reynolds number using Darcy's law and Forchheimer's extension in the laminar flow region.
Directory of Open Access Journals (Sweden)
G Boroni
2017-03-01
Full Text Available Lattice Boltzmann Method (LBM has shown great potential in fluid simulations, but performance issues and difficulties to manage complex boundary conditions have hindered a wider application. The upcoming of Graphic Processing Units (GPU Computing offered a possible solution for the performance issue, and methods like the Immersed Boundary (IB algorithm proved to be a flexible solution to boundaries. Unfortunately, the implicit IB algorithm makes the LBM implementation in GPU a non-trivial task. This work presents a fully parallel GPU implementation of LBM in combination with IB. The fluid-boundary interaction is implemented via GPU kernels, using execution configurations and data structures specifically designed to accelerate each code execution. Simulations were validated against experimental and analytical data showing good agreement and improving the computational time. Substantial reductions of calculation rates were achieved, lowering down the required time to execute the same model in a CPU to about two magnitude orders.
Spherical harmonics and energy polynomial solution of the Boltzmann equation for neutrons, 1
International Nuclear Information System (INIS)
Toledo, P.S. de
1974-01-01
The approximate solution of the source-free energy-dependent Boltzmann transport equation for neutrons in plane geometry and isotropic scattering case was given by Leonard and Ferziger using a truncated development in a series of energy-polynomials for the energy dependent neutron flux and solving exactly for the angular dependence. The presence in the general solution of eigenfunctions belonging to a continuous spectrum gives rise to difficult analytical problems in the application of their method even to simple problems. To avoid such difficulties, the angular dependence is treated by a spherical harmonics method and a general solution of the energy-dependent transport equation in plane geometry and isotropic scattering is obtained, in spite of the appearance of matrices as argument of the angular polynomials [pt
Fluid-Structure Interaction based on Lattice Boltzmann and p-FEM
Ahrenholz, Benjamin; Geller, Sebastian; Krafczyk, Manfred
2010-03-01
Over the last decade the Lattice Boltzmann Method (LBM) has matured as an efficient method for solving the Navier-Stokes equations. The p-version of the Finite Element Method (p-FEM) has proved to be highly efficient for a variety of problems in the field of structural mechanics. The focus of this contribution is to investigate the validity and efficiency of the coupling of two completely different numerical methods to simulate transient bidirectional Fluid-Structure Interaction (FSI) problems with very large structural deflections. In this contribution the treatment of moving boundaries in the fluid solver is presented, the computation of tractions and displacements on the boundary as well as the explicit coupling algorithm itself. In addition, efficiency aspects of the two approaches for two- and three-dimensional laminar flow examples at intermediate Reynolds numbers are discussed. Finally we give an outlook on modeling turbulent FSI problems.
Lattice Boltzmann method used to simulate particle motion in a conduit
Directory of Open Access Journals (Sweden)
Dolanský Jindřich
2017-06-01
Full Text Available A three-dimensional numerical simulation of particle motion in a pipe with a rough bed is presented. The simulation based on the Lattice Boltzmann Method (LBM employs the hybrid diffuse bounce-back approach to model moving boundaries. The bed of the pipe is formed by stationary spherical particles of the same size as the moving particles. Particle movements are induced by gravitational and hydrodynamic forces. To evaluate the hydrodynamic forces, the Momentum Exchange Algorithm is used. The LBM unified computational frame makes it possible to simulate both the particle motion and the fluid flow and to study mutual interactions of the carrier liquid flow and particles and the particle–bed and particle–particle collisions. The trajectories of simulated and experimental particles are compared. The Particle Tracking method is used to track particle motion. The correctness of the applied approach is assessed.
National Statistical Commission and Indian Official Statistics*
Indian Academy of Sciences (India)
IAS Admin
existence in July. 2006, is mandated, among its functions, to exercise statistical co-ordination between. Ministries,. Departments and other agencies of the. Central government; ... tween the Directorate General of Commercial Intelligence and. Statistics ... in some states do not play a nodal role in the coordination of statistical ...
National Statistical Commission and Indian Official Statistics*
Indian Academy of Sciences (India)
IAS Admin
Advanced Institute of. Maths, Stats and Com- puter Science, UoH. Campus, Hyderabad. His research interests include theory and practice of sample surveys .... other agencies of the. Central government; and to exercise statistical audit over the statistical activities to ensure quality and integrity of the statistical products.
Nagakura, Hiroki; Iwakami, Wakana; Furusawa, Shun; Okawa, Hirotada; Harada, Akira; Sumiyoshi, Kohsuke; Yamada, Shoichi; Matsufuru, Hideo; Imakura, Akira
2018-02-01
We present the first results of our spatially axisymmetric core-collapse supernova simulations with full Boltzmann neutrino transport, which amount to a time-dependent five-dimensional (two in space and three in momentum space) problem. Special relativistic effects are fully taken into account with a two-energy-grid technique. We performed two simulations for a progenitor of 11.2 M ⊙, employing different nuclear equations of state (EOSs): Lattimer and Swesty’s EOS with the incompressibility of K = 220 MeV (LS EOS) and Furusawa’s EOS based on the relativistic mean field theory with the TM1 parameter set (FS EOS). In the LS EOS, the shock wave reaches ∼700 km at 300 ms after bounce and is still expanding, whereas in the FS EOS it stalled at ∼200 km and has started to recede by the same time. This seems to be due to more vigorous turbulent motions in the former during the entire postbounce phase, which leads to higher neutrino-heating efficiency in the neutrino-driven convection. We also look into the neutrino distributions in momentum space, which is the advantage of the Boltzmann transport over other approximate methods. We find nonaxisymmetric angular distributions with respect to the local radial direction, which also generate off-diagonal components of the Eddington tensor. We find that the rθ component reaches ∼10% of the dominant rr component and, more importantly, it dictates the evolution of lateral neutrino fluxes, dominating over the θθ component, in the semitransparent region. These data will be useful to further test and possibly improve the prescriptions used in the approximate methods.
Progress in lattice Boltzmann methods for magnetohydrodynamic flows relevant to fusion applications
International Nuclear Information System (INIS)
Pattison, M.J.; Premnath, K.N.; Morley, N.B.; Abdou, M.A.
2008-01-01
In this paper, an approach to simulating magnetohydrodynamic (MHD) flows based on the lattice Boltzmann method (LBM) is presented. The dynamics of the flow are simulated using a so-called multiple relaxation time (MRT) lattice Boltzmann equation (LBE), in which a source term is included for the Lorentz force. The evolution of the magnetic induction is represented by introducing a vector distribution function and then solving an appropriate lattice kinetic equation for this function. The solution of both distribution functions are obtained through a simple, explicit, and computationally efficient stream-and-collide procedure. The use of the MRT collision term enhances the numerical stability over that of a single relaxation time approach. To apply the methodology to solving practical problems, a new extrapolation-based method for imposing magnetic boundary conditions is introduced and a technique for simulating steady-state flows with low magnetic Prandtl number is developed. In order to resolve thin layers near the walls arising in the presence of high magnetic fields, a non-uniform gridding strategy is introduced through an interpolated-streaming step applied to both distribution functions. These advances are particularly important for applications in fusion engineering where liquid metal flows with low magnetic Prandtl numbers and high Hartmann numbers are introduced. A number of MHD benchmark problems, under various physical and geometrical conditions are presented, including 3-D MHD lid driven cavity flow, high Hartmann number flows and turbulent MHD flows, with good agreement with prior data. Due to the local nature of the method, the LBM also demonstrated excellent performance on parallel machines, with almost linear scaling up to 128 processors for a MHD flow problem
On-the-fly Numerical Surface Integration for Finite-Difference Poisson-Boltzmann Methods.
Cai, Qin; Ye, Xiang; Wang, Jun; Luo, Ray
2011-11-01
Most implicit solvation models require the definition of a molecular surface as the interface that separates the solute in atomic detail from the solvent approximated as a continuous medium. Commonly used surface definitions include the solvent accessible surface (SAS), the solvent excluded surface (SES), and the van der Waals surface. In this study, we present an efficient numerical algorithm to compute the SES and SAS areas to facilitate the applications of finite-difference Poisson-Boltzmann methods in biomolecular simulations. Different from previous numerical approaches, our algorithm is physics-inspired and intimately coupled to the finite-difference Poisson-Boltzmann methods to fully take advantage of its existing data structures. Our analysis shows that the algorithm can achieve very good agreement with the analytical method in the calculation of the SES and SAS areas. Specifically, in our comprehensive test of 1,555 molecules, the average unsigned relative error is 0.27% in the SES area calculations and 1.05% in the SAS area calculations at the grid spacing of 1/2Å. In addition, a systematic correction analysis can be used to improve the accuracy for the coarse-grid SES area calculations, with the average unsigned relative error in the SES areas reduced to 0.13%. These validation studies indicate that the proposed algorithm can be applied to biomolecules over a broad range of sizes and structures. Finally, the numerical algorithm can also be adapted to evaluate the surface integral of either a vector field or a scalar field defined on the molecular surface for additional solvation energetics and force calculations.
Schaap, M. G.; Porter, M. L.; Wildenschild, D.
2008-12-01
Capillary pressure plays an important role in multiphase flow through porous media. At the microscale, capillary pressure is defined by Laplace's law, which takes into account interfacial phenomena such as surface tension, interfacial curvature and the contact angle. However, at the macroscale, capillary pressure is taken to be a function of the wetting phase saturation alone and the pressures are measured separately in each phase, typically outside the porous medium. It has been hypothesized that in addition to saturation, capillary pressure should be a function of interfacial area per volume. In this study a comparison between interfacial areas obtained from microtomographic image data and lattice-Boltzmann simulations for drainage and imbibition processes is presented. It is shown that there is good agreement between the measured and simulated capillary pressure curves. In addition, there is agreement between the interfacial area estimates for drainage, however, interfacial area estimates for imbibition are higher in the simulations than in the experiments. Image analysis indicates that during imbibition there is substantially more flow via films in the lattice-Boltzmann simulations than in the experiments, which may explain the higher interfacial areas. Scanning curves for imbibition and drainage were also simulated and a surface was fit to the capillary pressure - saturation interfacial area data. The surface indicates that the additional dependence of capillary pressure on interfacial area may provide insights into the hysteretic nature of the capillary pressure- saturation relationship. Furthermore, this study suggests that interfacial area per volume is dependent upon the dominant flow mechanism (i.e. piston or finger) within the system, as well as, the connectedness of the wetting phase, thus providing valuable information that can not be obtained from the capillary pressure - saturation relationship alone.
Evaluation of ion binding to DNA duplexes using a size-modified Poisson-Boltzmann theory.
Chu, Vincent B; Bai, Yu; Lipfert, Jan; Herschlag, Daniel; Doniach, Sebastian
2007-11-01
Poisson-Boltzmann (PB) theory is among the most widely applied electrostatic theories in biological and chemical science. Despite its reasonable success in explaining a wide variety of phenomena, it fails to incorporate two basic physical effects, ion size and ion-ion correlations, into its theoretical treatment. Recent experimental work has shown significant deviations from PB theory in competitive monovalent and divalent ion binding to a DNA duplex. The experimental data for monovalent binding are consistent with a hypothesis that attributes these deviations to counterion size. To model the observed differences, we have generalized an existing size-modified Poisson-Boltzmann (SMPB) theory and developed a new numerical implementation that solves the generalized theory around complex, atomistic representations of biological molecules. The results of our analysis show that good agreement to data at monovalent ion concentrations up to approximately 150 mM can be attained by adjusting the ion-size parameters in the new size-modified theory. SMPB calculations employing calibrated ion-size parameters predict experimental observations for other nucleic acid structures and salt conditions, demonstrating that the theory is predictive. We are, however, unable to model the observed deviations in the divalent competition data with a theory that only accounts for size but neglects ion-ion correlations, highlighting the need for theoretical descriptions that further incorporate ion-ion correlations. The accompanying numerical solver has been released publicly, providing the general scientific community the ability to compute SMPB solutions around a variety of different biological structures with only modest computational resources.
Boltzmann equation for a mixture of gases with non-conservative processes
International Nuclear Information System (INIS)
Martiarena, M.L.
1989-01-01
The nonlinear and non-isotropic Boltzmann equation (NLBE) including several molecular species, non-conservative channels and external forces. The general solution of that equation is obtained for a spatially homogeneous mixture of L gases, consisting of Maxwell particles, as a Generalized Laguerre expansion, within a Hilbert space. Removal and self-generation effects are included in presence of a time-dependent external force. An exact particular solution is studied generalizing the well-known BKW-mode for a mixture of L gases with inelastic processes. An homogeneous gas of test particles, in d dimension, is considered which interacts with a background host medium in the presence of an external space and time dependent force. Scattering, removal and self-generation collisions are included. The inhomogeneous Boltzmann equation for this system to an homogeneous one is reduced without background or external forces, using a generalized Nilkoskii transform. It is shown that a background of field particles can confine the test gas, even in absence of external forces. Furthermore, the solution of NLBE with non-isotropic singular initial conditions, is analyzed. The NLBE is transformed into an integral equation which is solved iteratively. The evolution of delta and step singularities in the distribution function is discussed during the initial layer and compared with the isotropic case. As an application of the methods abovementioned, the collision of a beam of ions or neutral atoms with a carbon-foil is considered. The electron experimental spectra from a transport equation is described. It is supposed that convoy electron may be produced inside the solid by single ion-atom collisions as ELC or ECC. The produced electrons lost energy by collision with the atoms of the material, which are considered at rest. The electron distribution function is numerically calculated. The ratio between the intrinsic convoy electron peak height to the background electron intensity
Solution of the Boltzmann-Fokker-Planck transport equation using exponential nodal schemes
International Nuclear Information System (INIS)
Ortega J, R.; Valle G, E. del
2003-01-01
There are carried out charge and energy calculations deposited due to the interaction of electrons with a plate of a certain material, solving numerically the electron transport equation for the Boltzmann-Fokker-Planck approach of first order in plate geometry with a computer program denominated TEOD-NodExp (Transport of Electrons in Discreet Ordinates, Nodal Exponentials), using the proposed method by the Dr. J. E. Morel to carry out the discretization of the variable energy and several spatial discretization schemes, denominated exponentials nodal. It is used the Fokker-Planck equation since it represents an approach of the Boltzmann transport equation that is been worth whenever it is predominant the dispersion of small angles, that is to say, resulting dispersion in small dispersion angles and small losses of energy in the transport of charged particles. Such electrons could be those that they face with a braking plate in a device of thermonuclear fusion. In the present work its are considered electrons of 1 MeV that impact isotropically on an aluminum plate. They were considered three different thickness of plate that its were designated as problems 1, 2 and 3. In the calculations it was used the discrete ordinate method S 4 with expansions of the dispersion cross sections until P 3 order. They were considered 25 energy groups of uniform size between the minimum energy of 0.1 MeV and the maximum of 1.0 MeV; the one spatial intervals number it was considered variable and it was assigned the values of 10, 20 and 30. (Author)
Statistics for dental researchers: descriptive statistics
Directory of Open Access Journals (Sweden)
Mohammad Reza Baneshi PhD
2012-09-01
Full Text Available Descriptive statistics is the process of summarizing gathered raw data from a research and creating useful statistics,which help the better understanding of data. According to the types of variables, which consist of qualitative andquantitative variables, some descriptive statistics have been introduced. Frequency percentage is used in qualitativedata, and mean, median, mode, standard deviation, standard error, variance, and range are some of the statistics whichare used in quantitative data. In health sciences, the majority of continuous variables follow a normal distribution.skewness and kurtosis are two statistics which help to compare a given distribution with the normal distribution.
Li, Chuan; Sánchez, René-Vinicio; Zurita, Grover; Cerrada, Mariela; Cabrera, Diego
2016-06-17
Fault diagnosis is important for the maintenance of rotating machinery. The detection of faults and fault patterns is a challenging part of machinery fault diagnosis. To tackle this problem, a model for deep statistical feature learning from vibration measurements of rotating machinery is presented in this paper. Vibration sensor signals collected from rotating mechanical systems are represented in the time, frequency, and time-frequency domains, each of which is then used to produce a statistical feature set. For learning statistical features, real-value Gaussian-Bernoulli restricted Boltzmann machines (GRBMs) are stacked to develop a Gaussian-Bernoulli deep Boltzmann machine (GDBM). The suggested approach is applied as a deep statistical feature learning tool for both gearbox and bearing systems. The fault classification performances in experiments using this approach are 95.17% for the gearbox, and 91.75% for the bearing system. The proposed approach is compared to such standard methods as a support vector machine, GRBM and a combination model. In experiments, the best fault classification rate was detected using the proposed model. The results show that deep learning with statistical feature extraction has an essential improvement potential for diagnosing rotating machinery faults.
Algebraic statistics computational commutative algebra in statistics
Pistone, Giovanni; Wynn, Henry P
2000-01-01
Written by pioneers in this exciting new field, Algebraic Statistics introduces the application of polynomial algebra to experimental design, discrete probability, and statistics. It begins with an introduction to Gröbner bases and a thorough description of their applications to experimental design. A special chapter covers the binary case with new application to coherent systems in reliability and two level factorial designs. The work paves the way, in the last two chapters, for the application of computer algebra to discrete probability and statistical modelling through the important concept of an algebraic statistical model.As the first book on the subject, Algebraic Statistics presents many opportunities for spin-off research and applications and should become a landmark work welcomed by both the statistical community and its relatives in mathematics and computer science.
Adrenal Gland Tumors: Statistics
... Gland Tumor: Statistics Request Permissions Adrenal Gland Tumor: Statistics Approved by the Cancer.Net Editorial Board , 03/ ... primary adrenal gland tumor is very uncommon. Exact statistics are not available for this type of tumor ...
Neuroendocrine Tumor: Statistics
... Tumor > Neuroendocrine Tumor: Statistics Request Permissions Neuroendocrine Tumor: Statistics Approved by the Cancer.Net Editorial Board , 11/ ... the body. It is important to remember that statistics on the survival rates for people with a ...
State transportation statistics 2009
2009-01-01
The Bureau of Transportation Statistics (BTS), a part of DOTs Research and Innovative Technology Administration (RITA), presents State Transportation Statistics 2009, a statistical profile of transportation in the 50 states and the District of Col...
BTS statistical standards manual
2005-10-01
The Bureau of Transportation Statistics (BTS), like other federal statistical agencies, establishes professional standards to guide the methods and procedures for the collection, processing, storage, and presentation of statistical data. Standards an...
Savage, Leonard J
1972-01-01
Classic analysis of the foundations of statistics and development of personal probability, one of the greatest controversies in modern statistical thought. Revised edition. Calculus, probability, statistics, and Boolean algebra are recommended.
... Childhood Cancer Statistics Childhood Cancer Statistics – Graphs and Infographics Number of Diagnoses Incidence Rates Over Time Cancer Deaths Per Year 5-Year Survival Rate Infographics Childhood Cancer Statistics – Important Facts Each year, the ...
Statistical Mechanics of Japanese Labor Markets
Chen, He
We introduce a probabilistic model to analyze job-matching processes of recent Japanese labor markets, in particular, for university graduates by means of statistical physics. To make a model of the market efficiently, we take into account several hypotheses. Namely, each company fixes the (business year independent) number of opening positions for newcomers. The ability of gathering newcomers depends on the result of job matching process in past business years. This fact means that the ability of the company is weakening if the company did not make their quota or the company gathered applicants too much over the quota. All university graduates who are looking for their jobs can access the public information about the ranking of companies. By assuming the above essential key points, we construct the local energy function of each company and describe the probability that an arbitrary company gets students at each business year by a Boltzmann-Gibbs distribution. We evaluate the relevant physical quantities such as the employment rate and Gini index. We discuss social inequalities in labor markets, and provide some ways to improve these situations, such as the informal job offer rate, the job-worker mismatch between students and companies. Graduate School of Information Science and Technology.
Wang, Peng; Wang, Lian-Ping; Guo, Zhaoli
2016-10-01
The main objective of this work is to perform a detailed comparison of the lattice Boltzmann equation (LBE) and the recently developed discrete unified gas-kinetic scheme (DUGKS) methods for direct numerical simulation (DNS) of the decaying homogeneous isotropic turbulence and the Kida vortex flow in a periodic box. The flow fields and key statistical quantities computed by both methods are compared with those from the pseudospectral method at both low and moderate Reynolds numbers. The results show that the LBE is more accurate and efficient than the DUGKS, but the latter has a superior numerical stability, particularly for high Reynolds number flows. In addition, we conclude that the DUGKS can adequately resolve the flow when the minimum spatial resolution parameter k_{max}η>3, where k_{max} is the maximum resolved wave number and η is the flow Kolmogorov length. This resolution requirement can be contrasted with the requirements of k_{max}η>1 for the pseudospectral method and k_{max}η>2 for the LBE. It should be emphasized that although more validations should be conducted before the DUGKS can be called a viable tool for DNS of turbulent flows, the present work contributes to the overall assessment of the DUGKS, and it provides a basis for further applications of DUGKS in studying the physics of turbulent flows.
Asinari, Pietro
2010-10-01
The homogeneous isotropic Boltzmann equation (HIBE) is a fundamental dynamic model for many applications in thermodynamics, econophysics and sociodynamics. Despite recent hardware improvements, the solution of the Boltzmann equation remains extremely challenging from the computational point of view, in particular by deterministic methods (free of stochastic noise). This work aims to improve a deterministic direct method recently proposed [V.V. Aristov, Kluwer Academic Publishers, 2001] for solving the HIBE with a generic collisional kernel and, in particular, for taking care of the late dynamics of the relaxation towards the equilibrium. Essentially (a) the original problem is reformulated in terms of particle kinetic energy (exact particle number and energy conservation during microscopic collisions) and (b) the computation of the relaxation rates is improved by the DVM-like correction, where DVM stands for Discrete Velocity Model (ensuring that the macroscopic conservation laws are exactly satisfied). Both these corrections make possible to derive very accurate reference solutions for this test case. Moreover this work aims to distribute an open-source program (called HOMISBOLTZ), which can be redistributed and/or modified for dealing with different applications, under the terms of the GNU General Public License. The program has been purposely designed in order to be minimal, not only with regards to the reduced number of lines (less than 1000), but also with regards to the coding style (as simple as possible). Program summaryProgram title: HOMISBOLTZ Catalogue identifier: AEGN_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEGN_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU General Public License No. of lines in distributed program, including test data, etc.: 23 340 No. of bytes in distributed program, including test data, etc.: 7 635 236 Distribution format: tar
Sailfish: A flexible multi-GPU implementation of the lattice Boltzmann method
Januszewski, M.; Kostur, M.
2014-09-01
We present Sailfish, an open source fluid simulation package implementing the lattice Boltzmann method (LBM) on modern Graphics Processing Units (GPUs) using CUDA/OpenCL. We take a novel approach to GPU code implementation and use run-time code generation techniques and a high level programming language (Python) to achieve state of the art performance, while allowing easy experimentation with different LBM models and tuning for various types of hardware. We discuss the general design principles of the code, scaling to multiple GPUs in a distributed environment, as well as the GPU implementation and optimization of many different LBM models, both single component (BGK, MRT, ELBM) and multicomponent (Shan-Chen, free energy). The paper also presents results of performance benchmarks spanning the last three NVIDIA GPU generations (Tesla, Fermi, Kepler), which we hope will be useful for researchers working with this type of hardware and similar codes. Catalogue identifier: AETA_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AETA_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU Lesser General Public License, version 3 No. of lines in distributed program, including test data, etc.: 225864 No. of bytes in distributed program, including test data, etc.: 46861049 Distribution format: tar.gz Programming language: Python, CUDA C, OpenCL. Computer: Any with an OpenCL or CUDA-compliant GPU. Operating system: No limits (tested on Linux and Mac OS X). RAM: Hundreds of megabytes to tens of gigabytes for typical cases. Classification: 12, 6.5. External routines: PyCUDA/PyOpenCL, Numpy, Mako, ZeroMQ (for multi-GPU simulations), scipy, sympy Nature of problem: GPU-accelerated simulation of single- and multi-component fluid flows. Solution method: A wide range of relaxation models (LBGK, MRT, regularized LB, ELBM, Shan-Chen, free energy, free surface) and boundary conditions within the lattice
Song, Wenyu; Zhang, Yaning; Li, Bingxi; Xu, Fei; Fu, Zhongbin
2017-06-01
In the current study, a macroscopic lattice Boltzmann model for simulating the heat and moisture transport phenomenon in unsaturated porous media during the freezing process was proposed. The proposed model adopted percolation threshold to reproduce the extra resistance in frozen fringe during the freezing process. The freezing process in Kanagawa sandy loam soil was demonstrated by the proposed model. The numerical result showed good agreement with the experimental result. The proposed model also offered higher computational efficiency and better agreement with the experimental result than the existing numerical models. Lattice Boltzmann method is suitable for simulating complex heat and mass transfer process in porous media at macroscopic scale under proper dimensionless criterion, which makes it a potentially powerful tool for engineering application.
International Nuclear Information System (INIS)
Baracca, A.; Renn, Juergen
2008-01-01
It will argue in this talk that the main Planck's merit in 1900 was the overcoming of a mechanist and reductionist horizon, while the effective introduction of the 'quantum', as a physical entity, is very doubtful. On the one hand, Planck maintained an absolute conception of the laws of thermodynamics, and did not accept Boltzmann's probabilistic, and reductionist view (as he declared in his Authobiography, and indirectly appears from Zermelo's paper). On the other hand, the formula known as 'Rayleigh's law' for the spectrum of the cavity radiation was not proposed at all in 1900 (nor later on, as a general expression of the spectrum), and in any case Rayliegh's had absolutely no influence on Planck. We will discuss the early 1900 purely thermodynamical derivation of Planck's law, and his subsequent exploitment of Boltzmann's probabilistic treatment, with the 'discretization' procedure, in the 'act of desesperation' in order to justify his result
Liu, Qing; He, Ya-Ling
2015-11-01
In this paper, a double multiple-relaxation-time lattice Boltzmann model is developed for simulating transient solid-liquid phase change problems in porous media at the representative elementary volume scale. The model uses two different multiple-relaxation-time lattice Boltzmann equations, one for the flow field and the other for the temperature field with nonlinear latent heat source term. The model is based on the generalized non-Darcy formulation, and the solid-liquid interface is traced through the liquid fraction which is determined by the enthalpy-based method. The present model is validated by numerical simulations of conduction melting in a semi-infinite space, solidification in a semi-infinite corner, and convection melting in a square cavity filled with porous media. The numerical results demonstrate the efficiency and accuracy of the present model for simulating transient solid-liquid phase change problems in porous media.
Nitz, D. E.; Curry, J. J.; Buuck, M.; DeMann, A.; Mitchell, N.; Shull, W.
2018-02-01
We report radiative transition probabilities for 5029 emission lines of neutral cerium within the wavelength range 417-1110 nm. Transition probabilities for only 4% of these lines have been previously measured. These results are obtained from a Boltzmann analysis of two high resolution Fourier transform emission spectra used in previous studies of cerium, obtained from the digital archives of the National Solar Observatory at Kitt Peak. The set of transition probabilities used for the Boltzmann analysis are those published by Lawler et al (2010 J. Phys. B: At. Mol. Opt. Phys. 43 085701). Comparisons of branching ratios and transition probabilities for lines common to the two spectra provide important self-consistency checks and test for the presence of self-absorption effects. Estimated 1σ uncertainties for our transition probability results range from 10% to 18%.
Energy Technology Data Exchange (ETDEWEB)
Mendes, Albert C.R., E-mail: albert@fisica.ufjf.br [Departamento de Física, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora - MG (Brazil); Takakura, Flavio I., E-mail: takakura@fisica.ufjf.br [Departamento de Física, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora - MG (Brazil); Abreu, Everton M.C., E-mail: evertonabreu@ufrrj.br [Grupo de Física Teórica e Matemática Física, Departamento de Física, Universidade Federal Rural do Rio de Janeiro, 23890-971, Seropédica - RJ (Brazil); Departamento de Física, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora - MG (Brazil); Neto, Jorge Ananias, E-mail: jorge@fisica.ufjf.br [Departamento de Física, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora - MG (Brazil)
2017-05-15
In this work we have obtained a higher-derivative Lagrangian for a charged fluid coupled with the electromagnetic fluid and the Dirac’s constraints analysis was discussed. A set of first-class constraints fixed by noncovariant gauge condition were obtained. The path integral formalism was used to obtain the partition function for the corresponding higher-derivative Hamiltonian and the Faddeev–Popov ansatz was used to construct an effective Lagrangian. Through the partition function, a Stefan–Boltzmann type law was obtained. - Highlights: • Higher-derivative Lagrangian for a charged fluid. • Electromagnetic coupling and Dirac’s constraint analysis. • Partition function through path integral formalism. • Stefan–Boltzmann-kind law through the partition function.
The Phase Space Elementary Cell in Classical and Generalized Statistics
Directory of Open Access Journals (Sweden)
Piero Quarati
2013-10-01
Full Text Available In the past, the phase-space elementary cell of a non-quantized system was set equal to the third power of the Planck constant; in fact, it is not a necessary assumption. We discuss how the phase space volume, the number of states and the elementary-cell volume of a system of non-interacting N particles, changes when an interaction is switched on and the system becomes or evolves to a system of correlated non-Boltzmann particles and derives the appropriate expressions. Even if we assume that nowadays the volume of the elementary cell is equal to the cube of the Planck constant, h3, at least for quantum systems, we show that there is a correspondence between different values of h in the past, with important and, in principle, measurable cosmological and astrophysical consequences, and systems with an effective smaller (or even larger phase-space volume described by non-extensive generalized statistics.
ADAPTIVE FINITE ELEMENT MODELING TECHNIQUES FOR THE POISSON-BOLTZMANN EQUATION
HOLST, MICHAEL; MCCAMMON, JAMES ANDREW; YU, ZEYUN; ZHOU, YOUNGCHENG; ZHU, YUNRONG
2011-01-01
We consider the design of an effective and reliable adaptive finite element method (AFEM) for the nonlinear Poisson-Boltzmann equation (PBE). We first examine the two-term regularization technique for the continuous problem recently proposed by Chen, Holst, and Xu based on the removal of the singular electrostatic potential inside biomolecules; this technique made possible the development of the first complete solution and approximation theory for the Poisson-Boltzmann equation, the first provably convergent discretization, and also allowed for the development of a provably convergent AFEM. However, in practical implementation, this two-term regularization exhibits numerical instability. Therefore, we examine a variation of this regularization technique which can be shown to be less susceptible to such instability. We establish a priori estimates and other basic results for the continuous regularized problem, as well as for Galerkin finite element approximations. We show that the new approach produces regularized continuous and discrete problems with the same mathematical advantages of the original regularization. We then design an AFEM scheme for the new regularized problem, and show that the resulting AFEM scheme is accurate and reliable, by proving a contraction result for the error. This result, which is one of the first results of this type for nonlinear elliptic problems, is based on using continuous and discrete a priori L∞ estimates to establish quasi-orthogonality. To provide a high-quality geometric model as input to the AFEM algorithm, we also describe a class of feature-preserving adaptive mesh generation algorithms designed specifically for constructing meshes of biomolecular structures, based on the intrinsic local structure tensor of the molecular surface. All of the algorithms described in the article are implemented in the Finite Element Toolkit (FETK), developed and maintained at UCSD. The stability advantages of the new regularization scheme
Calculation of the permeability in porous media using the lattice Boltzmann method
International Nuclear Information System (INIS)
Eshghinejadfard, Amir; Daróczy, László; Janiga, Gábor; Thévenin, Dominique
2016-01-01
Highlights: • Lattice Boltzmann simulation of fluid flow in porous media delivers a high accuracy. • Domain size, relaxation time and force scheme affect the calculated permeability. • Multiple relaxation time model shows very low viscosity dependence as compared to single relaxation time. • The choice of relaxation time and force scheme is a trade-off between the required accuracy and computational cost. - Abstract: In this paper, the lattice Boltzmann method (LBM) is used to simulate three-dimensional laminar flows in porous media and to calculate the associated permeability. An in-house, parallelized code using the message passing interface technique is employed for the study. Three different flow configurations are studied: first, by manually specifying solid cells in a face-centered cube (FCC); then, doing the same in a body-centered cube (BCC); and finally by reading the solid cells for a real 3D geometry from a set of experimental 2D computed tomography images. In all simulations, the Reynolds number is kept well below 1. It was found that the current LBM simulations yield good estimates for the permeability value. The impact of the employed force scheme and single- or multiple-relaxation time (SRT, MRT) was also studied. Although each force scheme (Guo-SRT, Guo-MRT and Shan-Chen-SRT) may show better results in some regions, the strong dependency of SRT models on relaxation time suggests that the proper choice of the force scheme, relaxation time and domain resolution is a compromise between the required accuracy and computational cost. First, higher resolutions lead as expected to increasingly accurate results but requires more computational cost and time. Second, the MRT model shows a lower viscosity dependence in comparison with SRT models but is somewhat slower. Also, the results are more sensitive to the relaxation time value for coarser domains. Furthermore, lower relaxation times necessitate a higher number of iterations to reach the steady
Corre , Samuel; Belmiloudi , Aziz
2016-01-01
International audience; In this work, a modified coupling Lattice Boltzmann Model (LBM) in simulation of cardiac electrophysiology is developed in order to capture the detailed activities of macro- to micro-scale transport processes. The propagation of electrical activity in the human heart through torso is mathematically modeled by bidomain type systems. As transmembrane potential evolves, we take into account domain anisotropical properties using intracellular and extracellular conductivity...
Verschaeve, Joris C G
2011-06-13
By means of the continuity equation of the incompressible Navier-Stokes equations, additional physical arguments for the derivation of a formulation of the no-slip boundary condition for the lattice Boltzmann method for straight walls at rest are obtained. This leads to a boundary condition that is second-order accurate with respect to the grid spacing and conserves mass. In addition, the boundary condition is stable for relaxation frequencies close to two.