Stochastic particle approximations for generalized Boltzmann models and convergence estimates
Graham, Carl; Méléard, Sylvie
1997-01-01
We specify the Markov process corresponding to a generalized mollified Boltzmann equation with general motion between collisions and nonlinear bounded jump (collision) operator, and give the nonlinear martingale problem it solves. We consider various linear interacting particle systems in order to approximate this nonlinear process. We prove propagation of chaos, in variation norm on path space with a precise rate of convergence, using coupling and interaction graph techniqu...
Approximate Message Passing with Restricted Boltzmann Machine Priors
Tramel, Eric W; Krzakala, Florent
2015-01-01
Approximate Message Passing (AMP) has been shown to be an excellent statistical approach to signal inference and compressed sensing problem. The AMP framework provides modularity in the choice of signal prior; here we propose a hierarchical form of the Gauss-Bernouilli prior which utilizes a Restricted Boltzmann Machine (RBM) trained on the signal support to push reconstruction performance beyond that of simple iid priors for signals whose support can be well represented by a trained binary RBM. We present and analyze two methods of RBM factorization and demonstrate how these affect signal reconstruction performance within our proposed algorithm. Finally, using the MNIST handwritten digit dataset, we show experimentally that using an RBM allows AMP to approach oracle-support performance.
Approximate message passing with restricted Boltzmann machine priors
Tramel, Eric W.; Drémeau, Angélique; Krzakala, Florent
2016-07-01
Approximate message passing (AMP) has been shown to be an excellent statistical approach to signal inference and compressed sensing problems. The AMP framework provides modularity in the choice of signal prior; here we propose a hierarchical form of the Gauss-Bernoulli prior which utilizes a restricted Boltzmann machine (RBM) trained on the signal support to push reconstruction performance beyond that of simple i.i.d. priors for signals whose support can be well represented by a trained binary RBM. We present and analyze two methods of RBM factorization and demonstrate how these affect signal reconstruction performance within our proposed algorithm. Finally, using the MNIST handwritten digit dataset, we show experimentally that using an RBM allows AMP to approach oracle-support performance.
The Nonclassical Diffusion Approximation to the Nonclassical Linear Boltzmann Equation
Vasques, Richard
2015-01-01
We show that, by correctly selecting the probability distribution function $p(s)$ for a particle's distance-to-collision, the nonclassical diffusion equation can be represented exactly by the nonclassical linear Boltzmann equation for an infinite homogeneous medium. This choice of $p(s)$ preserves the $true$ mean-squared free path of the system, which sheds new light on the results obtained in previous work.
Energy Technology Data Exchange (ETDEWEB)
Liu, Jian; Miller, William H.
2006-09-06
The thermal Gaussian approximation (TGA) recently developed by Mandelshtam et al has been demonstrated to be a practical way for approximating the Boltzmann operator exp(-{beta}H) for multidimensional systems. In this paper the TGA is combined with semiclassical (SC) initial value representations (IVRs) for thermal time correlation functions. Specifically, it is used with the linearized SC-IVR (LSC-IVR, equivalent to the classical Wigner model), and the 'forward-backward semiclassical dynamics' (FBSD) approximation developed by Makri et al. Use of the TGA with both of these approximate SC-IVRs allows the oscillatory part of the IVR to be integrated out explicitly, providing an extremely simple result that is readily applicable to large molecular systems. Calculation of the force-force autocorrelation for a strongly anharmonic oscillator demonstrates its accuracy, and of the velocity autocorrelation function (and thus the diffusion coefficient) of liquid neon demonstrates its applicability.
Radtke, Gregg A; Hadjiconstantinou, Nicolas G
2009-05-01
We present an efficient variance-reduced particle simulation technique for solving the linearized Boltzmann transport equation in the relaxation-time approximation used for phonon, electron, and radiative transport, as well as for kinetic gas flows. The variance reduction is achieved by simulating only the deviation from equilibrium. We show that in the limit of small deviation from equilibrium of interest here, the proposed formulation achieves low relative statistical uncertainty that is also independent of the magnitude of the deviation from equilibrium, in stark contrast to standard particle simulation methods. Our results demonstrate that a space-dependent equilibrium distribution improves the variance reduction achieved, especially in the collision-dominated regime where local equilibrium conditions prevail. We also show that by exploiting the physics of relaxation to equilibrium inherent in the relaxation-time approximation, a very simple collision algorithm with a clear physical interpretation can be formulated. PMID:19518597
Weak and strong coupling limits of the Boltzmann equation in the relaxation-time approximation
Jaiswal, Amaresh; Redlich, Krzysztof
2016-01-01
We consider a momentum dependent relaxation time for the Boltzmann equation in the relaxation time approximation. We employ a power law parametrization for the momentum dependence of the relaxation time, and calculate the shear and bulk viscosity, as well as, the charge and heat conductivity. We show, that for the two popular parametrizations, referred to as the linear and quadratic ansatz, one can obtain transport coefficients which corresponds to the weak and strong coupling regimes, respectively. We also show that, for a system of massless particles with vanishing chemical potential, the off-equilibrium corrections to the phase-space distribution function calculated with the quadratic ansatz are identical with those of the Grad's 14-moment method.
A New 2D-Transport, 1D-Diffusion Approximation of the Boltzmann Transport equation
Energy Technology Data Exchange (ETDEWEB)
Larsen, Edward
2013-06-17
The work performed in this project consisted of the derivation, implementation, and testing of a new, computationally advantageous approximation to the 3D Boltz- mann transport equation. The solution of the Boltzmann equation is the neutron flux in nuclear reactor cores and shields, but solving this equation is difficult and costly. The new “2D/1D” approximation takes advantage of a special geometric feature of typical 3D reactors to approximate the neutron transport physics in a specific (ax- ial) direction, but not in the other two (radial) directions. The resulting equation is much less expensive to solve computationally, and its solutions are expected to be sufficiently accurate for many practical problems. In this project we formulated the new equation, discretized it using standard methods, developed a stable itera- tion scheme for solving the equation, implemented the new numerical scheme in the MPACT code, and tested the method on several realistic problems. All the hoped- for features of this new approximation were seen. For large, difficult problems, the resulting 2D/1D solution is highly accurate, and is calculated about 100 times faster than a 3D discrete ordinates simulation.
Numerical approximation of the Euler-Poisson-Boltzmann model in the quasineutral limit
Degond, Pierre; Liu, Hailiang; Savelief, Dominique; Vignal, Marie-Hélène
2012-01-01
International audience This paper analyzes various schemes for the Euler-Poisson-Boltzmann (EPB) model of plasma physics. This model consists of the pressureless gas dynamics equations coupled with the Poisson equation and where the Boltzmann relation relates the potential to the electron density. If the quasi-neutral assumption is made, the Poisson equation is replaced by the constraint of zero local charge and the model reduces to the Isothermal Compressible Euler (ICE) model. We compare...
International Nuclear Information System (INIS)
Purpose: To investigate the use of the linear Boltzmann transport equation as a dose calculation tool which can account for interface effects, while still having faster computation times than Monte Carlo methods. In particular, we introduce a forward scattering approximation, in hopes of improving calculation time without a significant hindrance to accuracy. Methods: Two coupled Boltzmann transport equations were constructed, one representing the fluence of photons within the medium, and the other, the fluence of electrons. We neglect the scattering term within the electron transport equation, resulting in an extreme forward scattering approximation to reduce computational complexity. These equations were then solved using a numerical technique for solving partial differential equations, known as a finite difference scheme, where the fluence at each discrete point in space is calculated based on the fluence at the previous point in the particle's path. Using this scheme, it is possible to develop a solution to the Boltzmann transport equations by beginning with boundary conditions and iterating across the entire medium. The fluence of electrons can then be used to find the dose at any point within the medium. Results: Comparisons with Monte Carlo simulations indicate that even simplistic techniques for solving the linear Boltzmann transport equation yield expected interface effects, which many popular dose calculation algorithms are not capable of predicting. Implementation of a forward scattering approximation does not appear to drastically reduce the accuracy of this algorithm. Conclusion: Optimized implementations of this algorithm have been shown to be very accurate when compared with Monte Carlo simulations, even in build up regions where many models fail. Use of a forward scattering approximation could potentially give a reasonably accurate dose distribution in a shorter amount of time for situations where a completely accurate dose distribution is not
Numerical approximation of the Euler-Poisson-Boltzmann model in the quasineutral limit
Degond, Pierre; Savelief, Dominique; Vignal, Marie-Hélène
2010-01-01
This paper analyzes various schemes for the Euler-Poisson-Boltzmann (EPB) model of plasma physics. This model consists of the pressureless gas dynamics equations coupled with the Poisson equation and where the Boltzmann relation relates the potential to the electron density. If the quasi-neutral assumption is made, the Poisson equation is replaced by the constraint of zero local charge and the model reduces to the Isothermal Compressible Euler (ICE) model. We compare a numerical strategy based on the EPB model to a strategy using a reformulation (called REPB formulation). The REPB scheme captures the quasi-neutral limit more accurately.
Relaxation rate, diffusion approximation and Fick's law for inelastic scattering Boltzmann models
Lods, Bertrand; Mouhot, Clément; Toscani, Giuseppe
2008-01-01
We consider the linear dissipative Boltzmann equation describing inelastic interactions of particles with a fixed background. For the simplified model of Maxwell molecules first, we give a complete spectral analysis, and deduce from it the optimal rate of exponential convergence to equilibrium. Moreover we show the convergence to the heat equation in the diffusive limit and compute explicitely the diffusivity. Then for the physical model of hard spheres we use a suitable entropy functional fo...
Boltzmann's Concept of Reality
Ribeiro, Marcelo B.; Videira, Antonio A. P.
2007-01-01
In this article we describe and analyze the concept of reality developed by the Austrian theoretical physicist Ludwig Boltzmann. It is our thesis that Boltzmann was fully aware that reality could, and actually was, described by different points of view. In spite of this, Boltzmann did not renounce the idea that reality is real. We also discuss his main motivations to be strongly involved with philosophy of science, as well as further developments made by Boltzmann himself of his main philosop...
Matrix-valued Quantum Lattice Boltzmann Method
Mendl, Christian B
2013-01-01
We develop a numerical framework for the quantum analogue of the "classical" lattice Boltzmann method (LBM), with the Maxwell-Boltzmann distribution replaced by the Fermi-Dirac function. To accommodate the spin density matrix, the distribution functions become 2x2-matrix valued. We show that the efficient, commonly used BGK approximation of the collision operator is valid in the present setting. The framework could leverage the principles of LBM for simulating complex spin systems, with applications to spintronics.
Quantum corrections for Boltzmann equation
Institute of Scientific and Technical Information of China (English)
M.; Levy; PETER
2008-01-01
We present the lowest order quantum correction to the semiclassical Boltzmann distribution function,and the equation satisfied by this correction is given. Our equation for the quantum correction is obtained from the conventional quantum Boltzmann equation by explicitly expressing the Planck constant in the gradient approximation,and the quantum Wigner distribution function is expanded in pow-ers of Planck constant,too. The negative quantum correlation in the Wigner dis-tribution function which is just the quantum correction terms is naturally singled out,thus obviating the need for the Husimi’s coarse grain averaging that is usually done to remove the negative quantum part of the Wigner distribution function. We also discuss the classical limit of quantum thermodynamic entropy in the above framework.
Sman, van der R.G.M.
2006-01-01
In the special case of relaxation parameter = 1 lattice Boltzmann schemes for (convection) diffusion and fluid flow are equivalent to finite difference/volume (FD) schemes, and are thus coined finite Boltzmann (FB) schemes. We show that the equivalence is inherent to the homology of the Maxwell-Bolt
Pruning Boltzmann networks and hidden Markov models
DEFF Research Database (Denmark)
Pedersen, Morten With; Stork, D.
1996-01-01
We present sensitivity-based pruning algorithms for general Boltzmann networks. Central to our methods is the efficient calculation of a second-order approximation to the true weight saliencies in a cross-entropy error. Building upon previous work which shows a formal correspondence between linea...... and thus the proper weight is pruned at each pruning step. In all our experiments in small problems, pruning reduces the generalization error; in most cases the pruned networks facilitate interpretation as well......We present sensitivity-based pruning algorithms for general Boltzmann networks. Central to our methods is the efficient calculation of a second-order approximation to the true weight saliencies in a cross-entropy error. Building upon previous work which shows a formal correspondence between linear...... 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...
Ludwig Boltzmann: Atomic genius
International Nuclear Information System (INIS)
On the centenary of the death of Ludwig Boltzmann, Carlo Cercignani examines the immense contributions of the man who pioneered our understanding of the atomic nature of matter. The man who first gave a convincing explanation of the irreversibility of the macroscopic world and the symmetry of the laws of physics was the Austrian physicist Ludwig Boltzmann, who tragically committed suicide 100 years ago this month. One of the key figures in the development of the atomic theory of matter, Boltzmann's fame will be forever linked to two fundamental contributions to science. The first was his interpretation of 'entropy' as a mathematically well-defined measure of the disorder of atoms. The second was his derivation of what is now known as the Boltzmann equation, which describes the statistical properties of a gas as made up of molecules. The equation, which described for the first time how a probability can evolve with time, allowed Boltzmann to explain why macroscopic phenomena are irreversible. The key point is that while microscopic objects like atoms can behave reversibly, we never see broken coffee cups reforming because it would involve a long series of highly improbable interactions - and not because it is forbidden by the laws of physics. (U.K.)
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...
Advanced Mean Field Theory of Restricted Boltzmann Machine
Huang, Haiping; Toyoizumi, Taro
2015-01-01
Learning in restricted Boltzmann machine is typically hard due to the computation of gradients of log-likelihood function. To describe the network state statistics of the restricted Boltzmann machine, we develop an advanced mean field theory based on the Bethe approximation. Our theory provides an efficient message passing based method that evaluates not only the partition function (free energy) but also its gradients without requiring statistical sampling. The results are compared with those...
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.
On Boltzmann's genius and thermodynamics
Gyftopoulos, Elias P
2007-01-01
A recent essay [1] reminds us of how richly Boltzmann deserves to be admiringly commemorated for the originality of his ideas on the occasion of his 150th birthday. Without any doubt, the scientific community owes Boltzmann a great debt of gratitude for his ingenious and pathfinding contributions. However, the essay chooses to illustrate this important memorial by statements and inferences that perhaps are questionable today even to Boltzmann himself. I will comment only on three issues.
Crystallographic Lattice Boltzmann Method.
Namburi, Manjusha; Krithivasan, Siddharth; Ansumali, Santosh
2016-01-01
Current approaches to Direct Numerical Simulation (DNS) are computationally quite expensive for most realistic scientific and engineering applications of Fluid Dynamics such as automobiles or atmospheric flows. The Lattice Boltzmann Method (LBM), with its simplified kinetic descriptions, has emerged as an important tool for simulating hydrodynamics. In a heterogeneous computing environment, it is often preferred due to its flexibility and better parallel scaling. However, direct simulation of realistic applications, without the use of turbulence models, remains a distant dream even with highly efficient methods such as LBM. In LBM, a fictitious lattice with suitable isotropy in the velocity space is considered to recover Navier-Stokes hydrodynamics in macroscopic limit. The same lattice is mapped onto a cartesian grid for spatial discretization of the kinetic equation. In this paper, we present an inverted argument of the LBM, by making spatial discretization as the central theme. We argue that the optimal spatial discretization for LBM is a Body Centered Cubic (BCC) arrangement of grid points. We illustrate an order-of-magnitude gain in efficiency for LBM and thus a significant progress towards feasibility of DNS for realistic flows. PMID:27251098
Crystallographic Lattice Boltzmann Method
Namburi, Manjusha; Krithivasan, Siddharth; Ansumali, Santosh
2016-06-01
Current approaches to Direct Numerical Simulation (DNS) are computationally quite expensive for most realistic scientific and engineering applications of Fluid Dynamics such as automobiles or atmospheric flows. The Lattice Boltzmann Method (LBM), with its simplified kinetic descriptions, has emerged as an important tool for simulating hydrodynamics. In a heterogeneous computing environment, it is often preferred due to its flexibility and better parallel scaling. However, direct simulation of realistic applications, without the use of turbulence models, remains a distant dream even with highly efficient methods such as LBM. In LBM, a fictitious lattice with suitable isotropy in the velocity space is considered to recover Navier-Stokes hydrodynamics in macroscopic limit. The same lattice is mapped onto a cartesian grid for spatial discretization of the kinetic equation. In this paper, we present an inverted argument of the LBM, by making spatial discretization as the central theme. We argue that the optimal spatial discretization for LBM is a Body Centered Cubic (BCC) arrangement of grid points. We illustrate an order-of-magnitude gain in efficiency for LBM and thus a significant progress towards feasibility of DNS for realistic flows.
Tracy, C. A.; Widom, H.
1997-01-01
Using exact results from the theory of completely integrable systems of the Painleve/Toda type, we examine the consequences for the theory of polyelectrolytes in the (nonlinear) Poisson-Boltzmann approximation.
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.
Heavy Flavor Suppression: Boltzmann vs Langevin
Das, Santosh K; Plumari, Salvatore; Greco, Vincenzo
2013-01-01
The propagation of heavy flavor through the quark gluon plasma has been treated commonly within the framework of Langevin dynamics, i.e. assuming the heavy flavor momentum transfer is much smaller than the light one. On the other hand a similar suppression factor $R_{AA}$ has been observed experimentally for light and heavy flavors. We present a thorough study of the approximations involved by Langevin equation by mean of a direct comparison with the full collisional integral within the framework of Boltzmann transport equation. We have compared the results obtained in both approaches which can differ substantially for charm quark leading to quite different values extracted for the heavy quark diffusion coefficient. In the case of bottom quark the approximation appears to be quite reasonable.
Joint Training of Deep Boltzmann Machines
Goodfellow, Ian; Courville, Aaron; Bengio, Yoshua
2012-01-01
We introduce a new method for training deep Boltzmann machines jointly. Prior methods require an initial learning pass that trains the deep Boltzmann machine greedily, one layer at a time, or do not perform well on classifi- cation tasks.
DEFF Research Database (Denmark)
Svec, Oldrich; Skoček, Jan
2013-01-01
The ability of the Lattice Boltzmann method, as the fluid dynamics solver, to properly simulate macroscopic Navier’s slip boundary condition is investigated. An approximate equation relating the Lattice Boltzmann variable slip boundary condition with the macroscopic Navier’s slip boundary conditi...
Ludwig Boltzmann A Pioneer of Modern Physics
Flamm, D
1997-01-01
In two respects Ludwig Boltzmann was a pioneer of quantum mechanics. First because in his statistical interpretation of the second law of thermodynamics he introduced the theory of probability into a fundamental law of physics and thus broke with the classical prejudice, that fundamental laws have to be strictly deterministic. Even Max Planck had not been ready to accept Boltzmann's statistical methods until 1900. With Boltzmann's pioneering work the probabilistic interpretation of quantum mechanics had already a precedent. In fact in a paper in 1897 Boltzmann had already suggested to Planck to use his statistical methods for the treatment of black body radiation. The second pioneering step towards quantum mechanics was Boltzmann's introduction of discrete energy levels. Boltzmann used this method already in his 1872 paper on the H-theorem. One may ask whether Boltzmann considered this procedure only as a mathematical device or whether he attributed physical significance to it. In this connection Ostwald repo...
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.)
Multiphase cascaded lattice Boltzmann method
Lycett-Brown, D.; Luo, K. H.
2014-01-01
To improve the stability of the lattice Boltzmann method (LBM) at high Reynolds number the cascaded LBM has recently been introduced. As in the multiple relaxation time (MRT) method the cascaded LBM introduces additional relaxation times into the collision operator, but does so in a co-moving reference frame. This has been shown to significantly increase stability at low viscosity in the single phase case. Here the cascaded LBM is further developed to include multiphase flow. For this the for...
Student understanding of the Boltzmann factor
Smith, Trevor I; Thompson, John R
2015-01-01
We present results of our investigation into student understanding of the physical significance and utility of the Boltzmann factor in several simple models. We identify various justifications, both correct and incorrect, that students use when answering written questions that require application of the Boltzmann factor. Results from written data as well as teaching interviews suggest that many students can neither recognize situations in which the Boltzmann factor is applicable, nor articulate the physical significance of the Boltzmann factor as an expression for multiplicity, a fundamental quantity of statistical mechanics. The specific student difficulties seen in the written data led us to develop a guided-inquiry tutorial activity, centered around the derivation of the Boltzmann factor, for use in undergraduate statistical mechanics courses. We report on the development process of our tutorial, including data from teaching interviews and classroom observations on student discussions about the Boltzmann f...
High-order hydrodynamics via lattice Boltzmann methods.
Colosqui, Carlos E
2010-02-01
In this work, closure of the Boltzmann-Bhatnagar-Gross-Krook (Boltzmann-BGK) moment hierarchy is accomplished via projection of the distribution function f onto a space H(N) spanned by N-order Hermite polynomials. While successive order approximations retain an increasing number of leading-order moments of f , the presented procedure produces a hierarchy of (single) N-order partial-differential equations providing exact analytical description of the hydrodynamics rendered by ( N-order) lattice Boltzmann-BGK (LBBGK) simulation. Numerical analysis is performed with LBBGK models and direct simulation Monte Carlo for the case of a sinusoidal shear wave (Kolmogorov flow) in a wide range of Weissenberg number Wi=taunuk(2) (i.e., Knudsen number Kn=lambdak=square root Wi); k is the wave number, [corrected] tau is the relaxation time of the system, and lambda approximately tauc(s) is the mean-free path, where c(s) is the speed of sound. The present results elucidate the applicability of LBBGK simulation under general nonequilibrium conditions.
Lattice Boltzmann method with the cell-population equilibrium
Institute of Scientific and Technical Information of China (English)
Zhou Xiao-Yang; Cheng Bing; Shi Bao-Chang
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 nonnegative 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.
SIMPLE LATTICE BOLTZMANN MODEL FOR TRAFFIC FLOWS
Institute of Scientific and Technical Information of China (English)
Yan Guangwu; Hu Shouxin
2000-01-01
A lattice Boltzmann model with 5-bit lattice for traffic flows is proposed.Using the Chapman-Enskog expansion and multi-scale technique,we obtain the higher-order moments of equilibrium distribution function.A simple traffic light problem is simulated by using the present lattice Boltzmann model,and the result agrees well with analytical solution.
Lattice Boltzmann scheme for relativistic fluids
Mendoza, M.; B. Boghosian; 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.
The Einstein-Boltzmann system and positivity
Lee, Ho
2012-01-01
The Einstein-Boltzmann system is studied, with particular attention to the non-negativity of the solution of the Boltzmann equation. A new parametrization of post-collisional momenta in general relativity is introduced and then used to simplify the conditions on the collision cross-section given by Bancel and Choquet-Bruhat. The non-negativity of solutions of the Boltzmann equation on a given curved spacetime has been studied by Bichteler and by Tadmon. By examining to what extent the results of these authors apply in the framework of Bancel and Choquet-Bruhat, the non-negativity problem for the Einstein-Boltzmann system is resolved for a certain class of scattering kernels. It is emphasized that it is a challenge to extend the existing theory of the Cauchy problem for the Einstein-Boltzmann system so as to include scattering kernels which are physically well-motivated.
Non-linear effects in the Boltzmann equation
International Nuclear Information System (INIS)
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.)
Thermal cascaded lattice Boltzmann method
Fei, Linlin
2016-01-01
In this paper, a thermal cascaded lattice Boltzmann method (TCLBM) is developed in combination with the double-distribution-function (DDF) approach. A density distribution function relaxed by the cascaded scheme is employed to solve the flow field, and a total energy distribution function relaxed by the BGK scheme is used to solve temperature field, where two distribution functions are coupled naturally. The forcing terms are incorporated by means of central moments, which is consistent with the previous force scheme [Premnath \\emph{et al.}, Phys. Rev. E \\textbf{80}, 036702 (2009)] but the derivation is more intelligible and the evolution process is simpler. In the method, the viscous heat dissipation and compression work are taken into account, the Prandtl number and specific-heat ratio are adjustable, the external force is considered directly without the Boussinesq assumption, and the low-Mach number compressible flows can also be simulated. The forcing scheme is tested by simulating a steady Taylor-Green f...
Lattice Boltzmann algorithm for continuum multicomponent flow.
Halliday, I; Hollis, A P; Care, C M
2007-08-01
We present a multicomponent lattice Boltzmann simulation for continuum fluid mechanics, paying particular attention to the component segregation part of the underlying algorithm. In the principal result of this paper, the dynamics of a component index, or phase field, is obtained for a segregation method after U. D'Ortona [Phys. Rev. E 51, 3718 (1995)], due to Latva-Kokko and Rothman [Phys. Rev. E 71 056702 (2005)]. The said dynamics accord with a simulation designed to address multicomponent flow in the continuum approximation and underwrite improved simulation performance in two main ways: (i) by reducing the interfacial microcurrent activity considerably and (ii) by facilitating simulational access to regimes of flow with a low capillary number and drop Reynolds number [I. Halliday, R. Law, C. M. Care, and A. Hollis, Phys. Rev. E 73, 056708 (2006)]. The component segregation method studied, used in conjunction with Lishchuk's method [S. V. Lishchuk, C. M. Care, and I. Halliday, Phys. Rev. E 67, 036701 (2003)], produces an interface, which is distributed in terms of its component index; however, the hydrodynamic boundary conditions which emerge are shown to support the notion of a sharp, unstructured, continuum interface. PMID:17930175
Multiphase lattice Boltzmann methods theory and application
Huang, Haibo; Lu, Xiyun
2015-01-01
Theory and Application of Multiphase Lattice Boltzmann Methods presents a comprehensive review of all popular multiphase Lattice Boltzmann Methods developed thus far and is aimed at researchers and practitioners within relevant Earth Science disciplines as well as Petroleum, Chemical, Mechanical and Geological Engineering. Clearly structured throughout, this book will be an invaluable reference on the current state of all popular multiphase Lattice Boltzmann Methods (LBMs). The advantages and disadvantages of each model are presented in an accessible manner to enable the reader to choose the
Dyatko, Nikolay; Donko, Zoltan
2015-01-01
At low reduced electric fields the electron energy distribution function in heavy noble gases can take two distinct shapes. This "bistability effect" - in which electron-electron (Coulomb) collisions play an essential role - is analyzed here for Xe with a Boltzmann equation approach and with a first principles particle simulation method. The solution of the Boltzmann equation adopts the usual approximations of (i) searching for the distribution function in the form of two terms ("two-term app...
Relativistic Boltzmann theory for a plasma
International Nuclear Information System (INIS)
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.)
Relativistic Entropy and Related Boltzmann Kinetics
Kaniadakis, G
2009-01-01
It is well known that the particular form of the two-particle correlation function, in the collisional integral of the classical Boltzmman equation, fix univocally the entropy of the system, which turn out to be the Boltzmann-Gibbs-Shannon entropy. In the ordinary relativistic Boltzmann equation, some standard generalizations, with respect its classical version, imposed by the special relativity, are customarily performed. The only ingredient of the equation, which tacitely remains in its original classical form, is the two-particle correlation function, and this fact imposes that also the relativistic kinetics is governed by the Boltzmann-Gibbs-Shannon entropy. Indeed the ordinary relativistic Boltzmann equation admits as stationary stable distribution, the exponential Juttner distribution. Here, we show that the special relativity laws and the maximum entropy principle, suggest a relativistic generalization also of the two-particle correlation function and then of the entropy. The so obtained, fully relativ...
Ergodicity, ensembles, irreversibility in Boltzmann and beyond
Gallavotti, G
1994-01-01
The implications of the original misunderstanding of the etymology of the word "ergodic" are discussed, and the contents of a not too well known paper by Boltzmann are critically examined. The connection with the modern theory of Ruelle is attempted
Ergodicity, ensembles, irreversibility in Boltzmann and beyond
Gallavotti, Giovanni
1995-03-01
The contents of a not too well-known paper by Boltzmann are critically examined. The etymology of the word ergodic and its implications are discussed. A connection with the modern theory of Ruelle is attempted.
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
Thermal Lattice Boltzmann Model for Compressible Fluid
Institute of Scientific and Technical Information of China (English)
SUN Cheng-Hai
2000-01-01
We formulate a new thermal lattice Boltzmann model to simulate compressible flows with a high Mach number.The main difference from the standard lattice Boltzmann models is that the particle velocities are no longer a constant, varying with the mean velocity and internal energy. The proper heat conduction term in the energy equation is recovered by modification of the fluctuating kinetic energy transported by particles. The simulation of Couette flow is in good agreement with the analytical solutions.
A Viscosity Adaptive Lattice Boltzmann Method
Conrad, Daniel
2015-01-01
The present thesis describes the development and validation of a viscosity adaption method for the numerical simulation of non-Newtonian fluids on the basis of the Lattice Boltzmann Method (LBM), as well as the development and verification of the related software bundle SAM-Lattice. By now, Lattice Boltzmann Methods are established as an alternative approach to classical computational fluid dynamics methods. The LBM has been shown to be an accurate and efficient tool for the numerical...
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. PMID:26565365
Student understanding of the Boltzmann factor
Smith, Trevor I.; Mountcastle, Donald B.; Thompson, John R.
2015-12-01
[This paper is part of the Focused Collection on Upper Division Physics Courses.] We present results of our investigation into student understanding of the physical significance and utility of the Boltzmann factor in several simple models. We identify various justifications, both correct and incorrect, that students use when answering written questions that require application of the Boltzmann factor. Results from written data as well as teaching interviews suggest that many students can neither recognize situations in which the Boltzmann factor is applicable nor articulate the physical significance of the Boltzmann factor as an expression for multiplicity, a fundamental quantity of statistical mechanics. The specific student difficulties seen in the written data led us to develop a guided-inquiry tutorial activity, centered around the derivation of the Boltzmann factor, for use in undergraduate statistical mechanics courses. We report on the development process of our tutorial, including data from teaching interviews and classroom observations of student discussions about the Boltzmann factor and its derivation during the tutorial development process. This additional information informed modifications that improved students' abilities to complete the tutorial during the allowed class time without sacrificing the effectiveness as we have measured it. These data also show an increase in students' appreciation of the origin and significance of the Boltzmann factor during the student discussions. Our findings provide evidence that working in groups to better understand the physical origins of the canonical probability distribution helps students gain a better understanding of when the Boltzmann factor is applicable and how to use it appropriately in answering relevant questions.
Bazow, D; Heinz, U; Martinez, M; Noronha, J
2016-01-01
The dissipative dynamics of an expanding massless gas with constant cross section in a spatially flat Friedmann-Lema\\^itre-Robertson-Walker (FLRW) universe is studied. The mathematical problem of solving the full nonlinear relativistic Boltzmann equation is recast into an infinite set of nonlinear ordinary differential equations for the moments of the one-particle distribution function. Momentum-space resolution is determined by the number of non-hydrodynamic modes included in the moment hierarchy, i.e., by the truncation order. We show that in the FLRW spacetime the non-hydrodynamic modes decouple completely from the hydrodynamic degrees of freedom. This results in the system flowing as an ideal fluid while at the same time producing entropy. The solutions to the nonlinear Boltzmann equation exhibit transient tails of the distribution function with nontrivial momentum dependence. The evolution of this tail is not correctly captured by the relaxation time approximation nor by the linearized Boltzmann equation...
Niven, Ivan
2008-01-01
This self-contained treatment originated as a series of lectures delivered to the Mathematical Association of America. It covers basic results on homogeneous approximation of real numbers; the analogue for complex numbers; basic results for nonhomogeneous approximation in the real case; the analogue for complex numbers; and fundamental properties of the multiples of an irrational number, for both the fractional and integral parts.The author refrains from the use of continuous fractions and includes basic results in the complex case, a feature often neglected in favor of the real number discuss
2006-01-01
This interactive tutorial presents the following concepts of Approximation Techniques: Methods of Weighted Residual (MWR), Weak Formulatioin, Piecewise Continuous Function, Galerkin Finite Element FormulationExplanations especially for mathematical statements are provided using mouseover the highlight equations. ME4613 Finite Element Methods
Phantom cosmology and Boltzmann brains problem
Astashenok, Artyom V; Yurov, Valerian V
2013-01-01
We consider the well-known Boltzmann brains problem in frames of simple phantom energy models with little rip, big rip and big freeze singularity. It is showed that these models (i) satisfy to observational data and (ii) may be free from Boltzmann brains problem. The human observers in phantom models can exist only in during for a certain period $t
Fast lattice Boltzmann solver for relativistic hydrodynamics.
Mendoza, M; Boghosian, B M; Herrmann, H J; Succi, S
2010-07-01
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
Grid refinement for entropic lattice Boltzmann models
Dorschner, B; Chikatamarla, S S; Karlin, I V
2016-01-01
We propose a novel multi-domain grid refinement technique with extensions to entropic incompressible, thermal and compressible lattice Boltzmann models. Its validity and accuracy are accessed by comparison to available direct numerical simulation and experiment for the simulation of isothermal, thermal and viscous supersonic flow. In particular, we investigate the advantages of grid refinement for the set-ups of turbulent channel flow, flow past a sphere, Rayleigh-Benard convection as well as the supersonic flow around an airfoil. Special attention is payed to analyzing the adaptive features of entropic lattice Boltzmann models for multi-grid simulations.
Fast lattice Boltzmann solver for relativistic hydrodynamics.
Mendoza, M; Boghosian, B M; Herrmann, H J; Succi, S
2010-07-01
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. PMID:20867451
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.
Test of Information Theory on the Boltzmann Equation
Hyeon-Deuk, Kim; Hayakawa, Hisao
2002-01-01
We examine information theory using the steady-state Boltzmann equation. In a nonequilibrium steady-state system under steady heat conduction, the thermodynamic quantities from information theory are calculated and compared with those from the steady-state Boltzmann equation. We have found that information theory is inconsistent with the steady-state Boltzmann equation.
Test of Information Theory on the Boltzmann Equation
Kim, Hyeon-Deuk; Hayakawa, Hisao
2003-01-01
We examine information theory using the steady-state Boltzmann equation. In a nonequilibrium steady-state system under steady heat conduction, the thermodynamic quantities from information theory are calculated and compared with those from the steady-state Boltzmann equation. We have found that information theory is inconsistent with the steady-state Boltzmann equation.
A Fluctuating Lattice Boltzmann Method for the Diffusion Equation
Wagner, Alexander J
2016-01-01
We derive a fluctuating lattice Boltzmann method for the diffusion equation. The derivation removes several shortcomings of previous derivations for fluctuating lattice Boltzmann methods for hydrodynamic systems. The comparative simplicity of this diffusive system highlights the basic features of this first exact derivation of a fluctuating lattice Boltzmann method.
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. T
Derivation of anisotropic dissipative fluid dynamics from the Boltzmann equation
Molnár, Etele; Niemi, Harri; Rischke, Dirk H.
2016-06-01
Fluid-dynamical equations of motion can be derived from the Boltzmann equation in terms of an expansion around a single-particle distribution function which is in local thermodynamical equilibrium, i.e., isotropic in momentum space in the rest frame of a fluid element. However, in situations where the single-particle distribution function is highly anisotropic in momentum space, such as the initial stage of heavy-ion collisions at relativistic energies, such an expansion is bound to break down. Nevertheless, one can still derive a fluid-dynamical theory, called anisotropic dissipative fluid dynamics, in terms of an expansion around a single-particle distribution function, f^0 k, which incorporates (at least parts of) the momentum anisotropy via a suitable parametrization. We construct such an expansion in terms of polynomials in energy and momentum in the direction of the anisotropy and of irreducible tensors in the two-dimensional momentum subspace orthogonal to both the fluid velocity and the direction of the anisotropy. From the Boltzmann equation we then derive the set of equations of motion for the irreducible moments of the deviation of the single-particle distribution function from f^0 k. Truncating this set via the 14-moment approximation, we obtain the equations of motion of anisotropic dissipative fluid dynamics.
Philippi, P C; Surmas, R; Philippi, Paulo Cesar; Santos, Luis Orlando Emerich dos; Surmas, Rodrigo
2005-01-01
The particles model, the collision model, the polynomial development used for the equilibrium distribution, the time discretization and the velocity discretization are factors that let the lattice Boltzmann framework (LBM) far away from its conceptual support: the continuous Boltzmann equation (BE). Most collision models are based on the BGK, single parameter, relaxation-term leading to constant Prandtl numbers. The polynomial expansion used for the equilibrium distribution introduces an upper-bound in the local macroscopic speed. Most widely used time discretization procedures give an explicit numerical scheme with second-order time step errors. In thermal problems, quadrature did not succeed in giving discrete velocity sets able to generate multi-speed regular lattices. All these problems, greatly, difficult the numerical simulation of LBM based algorithms. In present work, the systematic derivation of lattice-Boltzmann models from the continuous Boltzmann equation is discussed. The collision term in the li...
Stefan-Boltzmann Law for Massive Photons
Moreira, E. S.; Ribeiro, T. G.
2016-08-01
This paper generalizes the Stefan-Boltzmann law to include massive photons. A crucial ingredient to obtain the correct formula for the radiance is to realize that a massive photon does not travel at the speed of (massless) light. It follows that, contrary to what could be expected, the radiance is not proportional to the energy density times the speed of light.
Lattice Boltzmann Models for Complex Fluids
Flekkoy, E. G.; Herrmann, H. J.
1993-01-01
We present various Lattice Boltzmann Models which reproduce the effects of rough walls, shear thinning and granular flow. We examine the boundary layers generated by the roughness of the walls. Shear thinning produces plug flow with a sharp density contrast at the boundaries. Density waves are spontaneously generated when the viscosity has a nonlinear dependence on density which characterizes granular flow.
THREE WAY DECOMPOSITION FOR THE BOLTZMANN EQUATION
Institute of Scientific and Technical Information of China (English)
Ilgis Ibragimov; Sergej Rjasanow
2009-01-01
The initial value problem for the spatially homogeneous Boltzmann equation is considered. A deterministic numerical scheme for this problem is developed by the use of the three way decomposition of the unknown function as well as of the collision integral. On this way, almost linear complexity of the algorithm is achieved. Some numerical examples are presented.
Approximate Representations and Approximate Homomorphisms
Moore, Cristopher
2010-01-01
Approximate algebraic structures play a defining role in arithmetic combinatorics and have found remarkable applications to basic questions in number theory and pseudorandomness. Here we study approximate representations of finite groups: functions f:G -> U_d such that Pr[f(xy) = f(x) f(y)] is large, or more generally Exp_{x,y} ||f(xy) - f(x)f(y)||^2$ is small, where x and y are uniformly random elements of the group G and U_d denotes the unitary group of degree d. We bound these quantities in terms of the ratio d / d_min where d_min is the dimension of the smallest nontrivial representation of G. As an application, we bound the extent to which a function f : G -> H can be an approximate homomorphism where H is another finite group. We show that if H's representations are significantly smaller than G's, no such f can be much more homomorphic than a random function. We interpret these results as showing that if G is quasirandom, that is, if d_min is large, then G cannot be embedded in a small number of dimensi...
Lattice Boltzmann approaches to magnetohydrodynamics and electromagnetism
Dellar, Paul
2010-03-01
J u B E g We present a lattice Boltzmann approach for magnetohydrodynamics and electromagnetism that expresses the magnetic field using a discrete set of vector distribution functions i. The i were first postulated to evolve according to a vector Boltzmann equation of the form ti+ ξi.∇i= - 1τ ( i- i^(0) ), where the ξi are a discrete set of velocities. The right hand side relaxes the i towards some specified functions i^(0) of the fluid velocity , and of the macroscopic magnetic field given by = ∑ii. Slowly varying solutions obey the equations of resistive magnetohydrodynamics. This lattice Boltzmann formulation has been used in large-scale (up to 1800^3 resolution) simulations of magnetohydrodynamic turbulence. However, this is only the simplest form of Ohm's law. We may simulate more realistic extended forms of Ohm's law using more complex collision operators. A current-dependent relaxation time yields a current-dependent resistivity η(|∇x|), as used to model ``anomalous'' resistivity created by small-scale plasma processes. Using a hydrodynamic matrix collision operator that depends upon the magnetic field , we may simulate Braginskii's magnetohydrodynamics, in which the viscosity for strains parallel to the magnetic field lines is much larger than the viscosity for strains in perpendicular directions. Changing the collision operator again, from the above vector Boltzmann equation we may derive the full set of Maxwell's equations, including the displacement current, and Ohm's law, - 1c^2 tE+ ∇x= μo,= σ( E + x). The original lattice Boltzmann scheme was designed to reproduce resistive magnetohydrodynamics in the non-relativistic limit. However, the kinetic formulation requires a system of first order partial differential equations with collision terms. This system coincides with the full set of Maxwell's equations and Ohm's law, so we capture a much wider range of electromagnetic phenomena, including electromagnetic waves.
CERN. Geneva
2015-01-01
Most physics results at the LHC end in a likelihood ratio test. This includes discovery and exclusion for searches as well as mass, cross-section, and coupling measurements. The use of Machine Learning (multivariate) algorithms in HEP is mainly restricted to searches, which can be reduced to classification between two fixed distributions: signal vs. background. I will show how we can extend the use of ML classifiers to distributions parameterized by physical quantities like masses and couplings as well as nuisance parameters associated to systematic uncertainties. This allows for one to approximate the likelihood ratio while still using a high dimensional feature vector for the data. Both the MEM and ABC approaches mentioned above aim to provide inference on model parameters (like cross-sections, masses, couplings, etc.). ABC is fundamentally tied Bayesian inference and focuses on the “likelihood free” setting where only a simulator is available and one cannot directly compute the likelihood for the dat...
Schmidt, Wolfgang M
1980-01-01
"In 1970, at the U. of Colorado, the author delivered a course of lectures on his famous generalization, then just established, relating to Roth's theorem on rational approxi- mations to algebraic numbers. The present volume is an ex- panded and up-dated version of the original mimeographed notes on the course. As an introduction to the author's own remarkable achievements relating to the Thue-Siegel-Roth theory, the text can hardly be bettered and the tract can already be regarded as a classic in its field."(Bull.LMS) "Schmidt's work on approximations by algebraic numbers belongs to the deepest and most satisfactory parts of number theory. These notes give the best accessible way to learn the subject. ... this book is highly recommended." (Mededelingen van het Wiskundig Genootschap)
Nonaligned shocks for discrete velocity models of the Boltzmann equation
Directory of Open Access Journals (Sweden)
J. M. Greenberg
1991-05-01
Full Text Available At the conclusion of I. Bonzani's presentation on the existence of structured shock solutions to the six-velocity, planar, discrete Boltzmann equation (with binary and triple collisions, Greenberg asked whether such solutions were possible in directions e(α=(cosα ,sinα when α was not one of the particle flow directions. This question generated a spirited discussion but the question was still open at the conclusion of the conference. In this note the author will provide a partial resolution to the question raised above. Using formal perturbation arguments he will produce approximate solutions to the equation considered by Bonzani which represent traveling waves propagating in any direction e(α=(cosα ,sinα.
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
A bound for the convergence rate of parallel tempering for sampling restricted Boltzmann machines
DEFF Research Database (Denmark)
Fischer, Asja; Igel, Christian
2015-01-01
Abstract Sampling from restricted Boltzmann machines (RBMs) is done by Markov chain Monte Carlo (MCMC) methods. The faster the convergence of the Markov chain, the more efficiently can high quality samples be obtained. This is also important for robust training of RBMs, which usually relies...... for contrastive divergence learning, our bound on the mixing time implies an upper bound on the error of the gradient approximation when the method is used for RBM training....
Privacy-Preserving Restricted Boltzmann Machine
Yu Li; Yuan Zhang; Yue Ji
2014-01-01
With the arrival of the big data era, it is predicted that distributed data mining will lead to an information technology revolution. To motivate different institutes to collaborate with each other, the crucial issue is to eliminate their concerns regarding data privacy. In this paper, we propose a privacy-preserving method for training a restricted boltzmann machine (RBM). The RBM can be got without revealing their private data to each other when using our privacy-preserving method. We provi...
Geometry of the restricted Boltzmann machine
Cueto, Maria Angelica; Morton, Jason; Sturmfels, Bernd
2009-01-01
The restricted Boltzmann machine is a graphical model for binary random variables. Based on a complete bipartite graph separating hidden and observed variables, it is the binary analog to the factor analysis model. We study this graphical model from the perspectives of algebraic statistics and tropical geometry, starting with the observation that its Zariski closure is a Hadamard power of the first secant variety of the Segre variety of projective lines. We derive a dimension formula for the ...
Nuclear Flow in Consistent Boltzmann Algorithm Models
Kortemeyer, G.; Daffin, F.; Bauer, W.
1995-01-01
We investigate the stochastic Direct Simulation Monte Carlo method (DSMC) for numerically solving the collision-term in heavy-ion transport theories of the Boltzmann-Uehling-Uhlenbeck (BUU) type. The first major modification we consider is changes in the collision rates due to excluded volume and shadowing/screening effects (Enskog theory). The second effect studied by us is the inclusion of an additional advection term. These modifications ensure a non-vanishing second virial and change the ...
Lattice Boltzmann Model and Geophysical Hydrodynamic Equation
Institute of Scientific and Technical Information of China (English)
冯士德; 杨京龙; 郜宪林; 季仲贞
2002-01-01
A lattice Boltzmann equation model in a rotating system is developed by introducing the Coriolis force effect.The geophysical hydrodynamic equation can be derived from this model. Numerical computations are performed to simulate the cylindrical annulus experiment and Benard convection. The numerical results have shown the flow behaviour of large-scale geostrophic current and Benard convection cells, which verifies the applicability of this model to both theory and experiment.
An exact energy conservation property of the quantum lattice Boltzmann algorithm
International Nuclear Information System (INIS)
The quantum lattice Boltzmann algorithm offers a unitary and readily parallelisable discretisation of the Dirac equation that is free of the fermion-doubling problem. The expectation of the discrete time-advance operator is an exact invariant of the algorithm. Its imaginary part determines the expectation of the Hamiltonian operator, the energy of the solution, with an accuracy that is consistent with the overall accuracy of the algorithm. In the one-dimensional case, this accuracy may be increased from first to second order using a variable transformation. The three-dimensional quantum lattice Boltzmann algorithm uses operator splitting to approximate evolution under the three-dimensional Dirac equation by a sequence of solutions of one-dimensional Dirac equations. The three-dimensional algorithm thus inherits the energy conservation property of the one-dimensional algorithm, although the implementation shown remains only first-order accurate due to the splitting error. -- Highlights: ► The quantum lattice Boltzmann algorithm approximates the Dirac equation. ► It has an exact invariant: the expectation of the discrete time-advance operator. ► The invariant consistently approximates the energy of the continuous system. ► We achieve second-order accuracy through a variable transformation.
Molnár, Etele; Rischke, Dirk H
2016-01-01
In Moln\\'ar et al. [Phys. Rev. D 93, 114025 (2016)] the equations of anisotropic dissipative fluid dynamics were obtained from the moments of the Boltzmann equation based on an expansion around an arbitrary anisotropic single-particle distribution function. In this paper we make a particular choice for this distribution function and consider the boost-invariant expansion of a fluid in one dimension. In order to close the conservation equations, we need to choose an additional moment of the Boltzmann equation. We discuss the influence of the choice of this moment on the time evolution of fluid-dynamical variables and identify the moment that provides the best match of anisotropic fluid dynamics to the solution of the Boltzmann equation in the relaxation-time approximation.
Energy Technology Data Exchange (ETDEWEB)
Stoenescu, M.L.
1977-06-01
The terms in Boltzmann kinetic equation corresponding to elastic short range collisions, inelastic excitational collisions, coulomb interactions and electric field acceleration are evaluated numerically for a standard distribution function minimizing the computational volume by expressing the terms as linear combinations with recalculable coefficients, of the distribution function and its derivatives. The present forms are suitable for spatial distribution calculations.
International Nuclear Information System (INIS)
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 PN approximation, the SN method (discrete ordinates method) and their variants. In the discrete ordinates (SN) 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 SN 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 PN 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)
Privacy-Preserving Restricted Boltzmann Machine
Directory of Open Access Journals (Sweden)
Yu Li
2014-01-01
Full Text Available With the arrival of the big data era, it is predicted that distributed data mining will lead to an information technology revolution. To motivate different institutes to collaborate with each other, the crucial issue is to eliminate their concerns regarding data privacy. In this paper, we propose a privacy-preserving method for training a restricted boltzmann machine (RBM. The RBM can be got without revealing their private data to each other when using our privacy-preserving method. We provide a correctness and efficiency analysis of our algorithms. The comparative experiment shows that the accuracy is very close to the original RBM model.
Privacy-preserving restricted boltzmann machine.
Li, Yu; Zhang, Yuan; Ji, Yue
2014-01-01
With the arrival of the big data era, it is predicted that distributed data mining will lead to an information technology revolution. To motivate different institutes to collaborate with each other, the crucial issue is to eliminate their concerns regarding data privacy. In this paper, we propose a privacy-preserving method for training a restricted boltzmann machine (RBM). The RBM can be got without revealing their private data to each other when using our privacy-preserving method. We provide a correctness and efficiency analysis of our algorithms. The comparative experiment shows that the accuracy is very close to the original RBM model. PMID:25101139
Application of lattice Boltzmann scheme to nanofluids
Institute of Scientific and Technical Information of China (English)
XUAN Yimin; LI Qiang; YAO Zhengping
2004-01-01
A nanofluid is a particle suspension that consists of base liquids and nanoparticles. Nanofluid has greater potential for heat transfer enhancement than traditional solid-liquid mixture. By accounting for the external and internal forces acting on the suspended nanoparticles and interactions among the nanoparticles and fluid particles,a lattice Boltzmann model for simulating flow and energy transport processes inside the nanofluids is proposed. The irregular motion of the nanoparticles and inherent dynamic behavior of nanofluids are discussed. The distributions of suspended nanoparticles inside nanofluids are calculated.
Lattice-Boltzmann-based Simulations of Diffusiophoresis
Castigliego, Joshua; Kreft Pearce, Jennifer
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 by 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. This simulation, in particular, was intended to model an oceanic system where the particles of interest were zooplankton, phytoplankton and microplastics. The separation of plankton from the microplastics was achieved.
Celebrating Cercignani's conjecture for the Boltzmann equation
Desvillettes, Laurent; Villani, Cédric
2010-01-01
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.
Generalized Boltzmann equations for on-shell particle production in a hot plasma
Jakovác, A
2002-01-01
A novel refinement of the conventional treatment of Kadanoff--Baym equations is suggested. Besides the Boltzmann equation another differential equation is used for calculating the evolution of the non-equilibrium two-point function. Although it was usually interpreted as a constraint on the solution of the Boltzmann equation, we argue that its dynamics is relevant to the determination and resummation of the particle production cut contributions. The differential equation for this new contribution is illustrated in the example of the cubic scalar model. The analogue of the relaxation time approximation is suggested. It results in the shift of the threshold location and in smearing out of the non-analytic threshold behaviour of the spectral function. Possible consequences for the dilepton production are discussed.
Influence of asperities on fluid and thermal flow in a fracture: a coupled Lattice Boltzmann study
Neuville, Amélie; Toussaint, Renaud
2013-01-01
The characteristics of the hydro-thermal flow which occurs when a cold fluid is injected into a hot fractured bedrock depend on the morphology of the fracture. We consider a sharp triangular asperity, invariant in one direction, perturbing an otherwise flat fracture. We investigate its influence on the macroscopic hydraulic transmissivity and heat transfer efficiency, at fixed low Reynolds number. In this study, numerical simulations are done with a coupled lattice Boltzmann method that solves both the complete Navier-Stokes and advection-diffusion equations in three dimensions. The results are compared with those obtained under lubrication approximations which rely on many hypotheses and neglect the three-dimensional (3D) effects. The lubrication results are obtained by analytically solving the Stokes equation and a two-dimensional (integrated over the thickness) advection-diffusion equation. We use a lattice Boltzmann method with a double distribution (for mass and energy transport) on hypercubic and cubic ...
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 Poisson-Boltzmann: fluctuations and fluid structure in a self-consistent theory.
Buyukdagli, S; Blossey, R
2016-09-01
Poisson-Boltzmann (PB) theory is the classic approach to soft matter electrostatics and has been applied to numerous physical chemistry and biophysics problems. Its essential limitations are in its neglect of correlation effects and fluid structure. Recently, several theoretical insights have allowed the formulation of approaches that go beyond PB theory in a systematic way. In this topical review, we provide an update on the developments achieved in the self-consistent formulations of correlation-corrected Poisson-Boltzmann theory. We introduce a corresponding system of coupled non-linear equations for both continuum electrostatics with a uniform dielectric constant, and a structured solvent-a dipolar Coulomb fluid-including non-local effects. While the approach is only approximate and also limited to corrections in the so-called weak fluctuation regime, it allows us to include physically relevant effects, as we show for a range of applications of these equations. PMID:27357125
Can the Higgs Boson Save Us From the Menace of the Boltzmann Brains?
Boddy, Kimberly K
2013-01-01
The standard $\\Lambda$CDM model provides an excellent fit to current cosmological observations but suffers from a potentially serious Boltzmann Brain problem. If the universe enters a de Sitter vacuum phase that is truly eternal, there will be a finite temperature in empty space and corresponding thermal fluctuations. Among these fluctuations will be intelligent observers, as well as configurations that reproduce any local region of the current universe to arbitrary precision. We discuss the possibility that the escape from this unacceptable situation may be found in known physics: vacuum instability induced by the Higgs field. Avoiding Boltzmann Brains in a measure-independent way requires a decay timescale of order the current age of the universe, which can be achieved if the top quark pole mass is approximately 178 GeV. Otherwise we must invoke new physics or a particular cosmological measure before we can consider $\\Lambda$CDM to be an empirical success.
Adaptive Finite Element Modeling Techniques for the Poisson-Boltzmann Equation
Holst, Michael; Yu, Zeyun; Zhou, Yongcheng; Zhu, Yunrong
2010-01-01
We develop an efficient and reliable adaptive finite element method (AFEM) for the nonlinear Poisson-Boltzmann equation (PBE). We first examine the regularization technique of Chen, Holst, and Xu; this technique made possible the first a priori pointwise estimates and the first complete solution and approximation theory for the Poisson-Boltzmann equation. It also made possible the first provably convergent discretization of the PBE, and allowed for the development of a provably convergent AFEM for the PBE. However, in practice the regularization turns out to be numerically ill-conditioned. In this article, we examine a second regularization, and establish a number of basic results to ensure that the new approach produces the same mathematical advantages of the original regularization, without the ill-conditioning property. We then design an AFEM scheme based on the new regularized problem, and show that the resulting AFEM scheme is accurate and reliable, by proving a contraction result for the error. This res...
Formulating Weak Lensing from the Boltzmann Equation and Application to Lens-lens Couplings
Su, S -C
2014-01-01
The Planck mission has conclusively detected lensing of the Cosmic Microwave Background (CMB) radiation from foreground sources to an overall significance of greater than $25\\sigma$. The high precision of this measurement motivates the development of a more complete formulation of the calculation of this effect. While most effects on the CMB anisotropies are widely studied through direct solutions of the Boltzmann equation, the non-linear effect of CMB lensing is formulated through the solutions of the geodesic equation. In this paper, we present a new formalism to the calculation of the lensing effect by \\emph{directly solving the Boltzmann equation}, as we did in the calculation of the CMB anisotropies at recombination. In particular, we developed a diagrammatic approach to efficiently keep track of all the interaction terms and calculate all possible non-trivial correlations to arbitrary high orders. Using this formalism, we explicitly articulate the approximations required to recover the usual remapping a...
Beyond Poisson–Boltzmann: fluctuations and fluid structure in a self-consistent theory
Buyukdagli, S.; Blossey, R.
2016-09-01
Poisson–Boltzmann (PB) theory is the classic approach to soft matter electrostatics and has been applied to numerous physical chemistry and biophysics problems. Its essential limitations are in its neglect of correlation effects and fluid structure. Recently, several theoretical insights have allowed the formulation of approaches that go beyond PB theory in a systematic way. In this topical review, we provide an update on the developments achieved in the self-consistent formulations of correlation-corrected Poisson–Boltzmann theory. We introduce a corresponding system of coupled non-linear equations for both continuum electrostatics with a uniform dielectric constant, and a structured solvent—a dipolar Coulomb fluid—including non-local effects. While the approach is only approximate and also limited to corrections in the so-called weak fluctuation regime, it allows us to include physically relevant effects, as we show for a range of applications of these equations.
Convolution Inequalities for the Boltzmann Collision Operator
Alonso, Ricardo J.; Carneiro, Emanuel; Gamba, Irene M.
2010-09-01
We study integrability properties of a general version of the Boltzmann collision operator for hard and soft potentials in n-dimensions. A reformulation of the collisional integrals allows us to write the weak form of the collision operator as a weighted convolution, where the weight is given by an operator invariant under rotations. Using a symmetrization technique in L p we prove a Young’s inequality for hard potentials, which is sharp for Maxwell molecules in the L 2 case. Further, we find a new Hardy-Littlewood-Sobolev type of inequality for Boltzmann collision integrals with soft potentials. The same method extends to radially symmetric, non-increasing potentials that lie in some {Ls_{weak}} or L s . The method we use resembles a Brascamp, Lieb and Luttinger approach for multilinear weighted convolution inequalities and follows a weak formulation setting. Consequently, it is closely connected to the classical analysis of Young and Hardy-Littlewood-Sobolev inequalities. In all cases, the inequality constants are explicitly given by formulas depending on integrability conditions of the angular cross section (in the spirit of Grad cut-off). As an additional application of the technique we also obtain estimates with exponential weights for hard potentials in both conservative and dissipative interactions.
Thermal equation of state for lattice Boltzmann gases
Institute of Scientific and Technical Information of China (English)
Ran Zheng
2009-01-01
The Galilean invaxiance and the induced thermo-hydrodynamics of the lattice Boltzmann Bhatnagar-Gross-Krook model axe proposed together with their rigorous theoretical background. From the viewpoint of group invariance,recovering the Galilean invariance for the isothermal lattice Boltzmann Bhatnagar-Gross-Krook equation (LBGKE) induces a new natural thermal-dynamical system, which is compatible with the elementary statistical thermodynamics.
Thermal equation of state for lattice Boltzmann gases
Ran, Zheng
2009-06-01
The Galilean invariance and the induced thermo-hydrodynamics of the lattice Boltzmann Bhatnagar-Gross-Krook model are proposed together with their rigorous theoretical background. From the viewpoint of group invariance, recovering the Galilean invariance for the isothermal lattice Boltzmann Bhatnagar-Gross-Krook equation (LBGKE) induces a new natural thermal-dynamical system, which is compatible with the elementary statistical thermodynamics.
Droplet collision simulation by multi-speed lattice Boltzmann method
Lycett-Brown, D.; Karlin, I.V.; Luo, K. H.
2011-01-01
Realization of the Shan-Chen multiphase flow lattice Boltzmann model is considered in the framework of the higher-order Galilean invariant lattices. The present multiphase lattice Boltzmann model is used in two dimensional simulation of droplet collisions at high Weber numbers. Results are found to be in a good agreement with experimental findings.
Reciprocal Symmetric Boltzmann Function and Unified Boson-Fermion Statistics
Ahmad, Mushfiq; Talukder, Muhammad O. G.
2007-01-01
The differential equation for Boltzmann's function is replaced by the corresponding discrete finite difference equation. The difference equation is, then, symmetrized so that the equation remains invariant when step d is replaced by -d. The solutions of this equation come in Boson-Fermion pairs. Reciprocal symmetric Boltzmann's function, thus, unifies both Bosonic and Fermionic distributions.
A probabilistic view on the general relativistic Boltzmann equation
Bailleul, Ismael
2011-01-01
A new probalistic approach to general relativistic kinetic theory is proposed. The general relativistic Boltzmann equation is linked to a new Markov process in a completely intrinsic way. This treatment is then used to prove the causal character of the relativistic Boltzmann model.
Monte Carlo variance reduction approaches for non-Boltzmann tallies
International Nuclear Information System (INIS)
Quantities that depend on the collective effects of groups of particles cannot be obtained from the standard Boltzmann transport equation. Monte Carlo estimates of these quantities are called non-Boltzmann tallies and have become increasingly important recently. Standard Monte Carlo variance reduction techniques were designed for tallies based on individual particles rather than groups of particles. Experience with non-Boltzmann tallies and analog Monte Carlo has demonstrated the severe limitations of analog Monte Carlo for many non-Boltzmann tallies. In fact, many calculations absolutely require variance reduction methods to achieve practical computation times. Three different approaches to variance reduction for non-Boltzmann tallies are described and shown to be unbiased. The advantages and disadvantages of each of the approaches are discussed
Expected energy-based restricted Boltzmann machine for classification.
Elfwing, S; Uchibe, E; Doya, K
2015-04-01
In classification tasks, restricted Boltzmann machines (RBMs) have predominantly been used in the first stage, either as feature extractors or to provide initialization of neural networks. In this study, we propose a discriminative learning approach to provide a self-contained RBM method for classification, inspired by free-energy based function approximation (FE-RBM), originally proposed for reinforcement learning. For classification, the FE-RBM method computes the output for an input vector and a class vector by the negative free energy of an RBM. Learning is achieved by stochastic gradient-descent using a mean-squared error training objective. In an earlier study, we demonstrated that the performance and the robustness of FE-RBM function approximation can be improved by scaling the free energy by a constant that is related to the size of network. In this study, we propose that the learning performance of RBM function approximation can be further improved by computing the output by the negative expected energy (EE-RBM), instead of the negative free energy. To create a deep learning architecture, we stack several RBMs on top of each other. We also connect the class nodes to all hidden layers to try to improve the performance even further. We validate the classification performance of EE-RBM using the MNIST data set and the NORB data set, achieving competitive performance compared with other classifiers such as standard neural networks, deep belief networks, classification RBMs, and support vector machines. The purpose of using the NORB data set is to demonstrate that EE-RBM with binary input nodes can achieve high performance in the continuous input domain. PMID:25318375
Gas kinetic algorithm for flows in Poiseuille-like microchannels using Boltzmann model equation
Institute of Scientific and Technical Information of China (English)
LI; Zhihui; ZHANG; Hanxin; FU; Song
2005-01-01
The gas-kinetic unified algorithm using Boltzmann model equation have been extended and developed to solve the micro-scale gas flows in Poiseuille-like micro-channels from Micro-Electro-Mechanical Systems (MEMS). The numerical modeling of the gas kinetic boundary conditions suitable for micro-scale gas flows is presented. To test the present method, the classical Couette flows with various Knudsen numbers, the gas flows from short microchannels like plane Poiseuille and the pressure-driven gas flows in two-dimensional short microchannels have been simulated and compared with the approximate solutions of the Boltzmann equation, the related DSMC results, the modified N-S solutions with slip-flow boundary theory, the gas-kinetic BGK-Burnett solutions and the experimental data. The comparisons show that the present gas-kinetic numerical algorithm using the mesoscopic Boltzmann simplified velocity distribution function equation can effectively simulate and reveal the gas flows in microchannels. The numerical experience indicates that this method may be a powerful tool in the numerical simulation of micro-scale gas flows from MEMS.
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.
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. PMID:26986435
Thermal lattice Boltzmann method for complex microflows
Yasuoka, Haruka; Kaneda, Masayuki; Suga, Kazuhiko
2016-07-01
A methodology to simulate thermal fields in complex microflow geometries is proposed. For the flow fields, the regularized multiple-relaxation-time lattice Boltzmann method (LBM) is applied coupled with the diffusive-bounce-back boundary condition for wall boundaries. For the thermal fields, the regularized lattice Bhatnagar-Gross-Krook model is applied. For the thermal wall boundary condition, a newly developed boundary condition, which is a mixture of the diffuse scattering and constant temperature conditions, is applied. The proposed set of schemes is validated by reference data in the Fourier flows and square cylinder flows confined in a microchannel. The obtained results confirm that it is essential to apply the regularization to the thermal LBM for avoiding kinked temperature profiles in complex thermal flows. The proposed wall boundary condition is successful to obtain thermal jumps at the walls with good accuracy.
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.
Lattice Boltzmann model for resistive relativistic magnetohydrodynamics
Mohseni, F; Succi, S; Herrmann, H J
2015-01-01
In this paper, we develop a lattice Boltzmann model for relativistic magnetohydrodynamics (MHD). Even though the model is derived for resistive MHD, it is shown that it is numerically robust even in the high conductivity (ideal MHD) limit. In order to validate the numerical method, test simulations are carried out for both ideal and resistive limits, namely the propagation of Alfv\\'en waves in the ideal MHD and the evolution of current sheets in the resistive regime, where very good agreement is observed comparing to the analytical results. Additionally, two-dimensional magnetic reconnection driven by Kelvin-Helmholtz instability is studied and the effects of different parameters on the reconnection rate are investigated. It is shown that the density ratio has negligible effect on the magnetic reconnection rate, while an increase in shear velocity decreases the reconnection rate. Additionally, it is found that the reconnection rate is proportional to $\\sigma^{-\\frac{1}{2}}$, $\\sigma$ being the conductivity, w...
Autotagging music with conditional restricted Boltzmann machines
Mandel, Michael; Larochelle, Hugo; Bengio, Yoshua
2011-01-01
This paper describes two applications of conditional restricted Boltzmann machines (CRBMs) to the task of autotagging music. The first consists of training a CRBM to predict tags that a user would apply to a clip of a song based on tags already applied by other users. By learning the relationships between tags, this model is able to pre-process training data to significantly improve the performance of a support vector machine (SVM) autotagging. The second is the use of a discriminative RBM, a type of CRBM, to autotag music. By simultaneously exploiting the relationships among tags and between tags and audio-based features, this model is able to significantly outperform SVMs, logistic regression, and multi-layer perceptrons. In order to be applied to this problem, the discriminative RBM was generalized to the multi-label setting and four different learning algorithms for it were evaluated, the first such in-depth analysis of which we are aware.
Lattice Boltzmann modelling of intrinsic permeability
Li, Jun; Wu, Lei; Zhang, Yonghao
2016-01-01
Lattice Boltzmann method (LBM) has been applied to predict flow properties of porous media including intrinsic permeability, where it is implicitly assumed that the LBM is equivalent to the incompressible (or near incompressible) Navier-Stokes equation. However, in LBM simulations, high-order moments, which are completely neglected in the Navier-Stokes equation, are still available through particle distribution functions. To ensure that the LBM simulation is correctly working at the Navier-Stokes hydrodynamic level, the high-order moments have to be negligible. This requires that the Knudsen number (Kn) is small so that rarefaction effect can be ignored. In this technical note, we elaborate this issue in LBM modelling of porous media flows, which is particularly important for gas flows in ultra-tight media.
Flux Limiter Lattice Boltzmann for Compressible Flows
Institute of Scientific and Technical Information of China (English)
陈峰; 许爱国; 张广财; 李英骏
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.
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.
Accurate deterministic solutions for the classic Boltzmann shock profile
Yue, Yubei
The Boltzmann equation or Boltzmann transport equation is a classical kinetic equation devised by Ludwig Boltzmann in 1872. It is regarded as a fundamental law in rarefied gas dynamics. Rather than using macroscopic quantities such as density, temperature, and pressure to describe the underlying physics, the Boltzmann equation uses a distribution function in phase space to describe the physical system, and all the macroscopic quantities are weighted averages of the distribution function. The information contained in the Boltzmann equation is surprisingly rich, and the Euler and Navier-Stokes equations of fluid dynamics can be derived from it using series expansions. Moreover, the Boltzmann equation can reach regimes far from the capabilities of fluid dynamical equations, such as the realm of rarefied gases---the topic of this thesis. Although the Boltzmann equation is very powerful, it is extremely difficult to solve in most situations. Thus the only hope is to solve it numerically. But soon one finds that even a numerical simulation of the equation is extremely difficult, due to both the complex and high-dimensional integral in the collision operator, and the hyperbolic phase-space advection terms. For this reason, until few years ago most numerical simulations had to rely on Monte Carlo techniques. In this thesis I will present a new and robust numerical scheme to compute direct deterministic solutions of the Boltzmann equation, and I will use it to explore some classical gas-dynamical problems. In particular, I will study in detail one of the most famous and intrinsically nonlinear problems in rarefied gas dynamics, namely the accurate determination of the Boltzmann shock profile for a gas of hard spheres.
Lattice Boltzmann simulations of settling behaviors of irregularly shaped particles
Zhang, Pei; Galindo-Torres, S. A.; Tang, Hongwu; Jin, Guangqiu; Scheuermann, A.; Li, Ling
2016-06-01
We investigated the settling dynamics of irregularly shaped particles in a still fluid under a wide range of conditions with Reynolds numbers Re varying between 1 and 2000, sphericity ϕ and circularity c both greater than 0.5, and Corey shape factor (CSF) less than 1. To simulate the particle settling process, a modified lattice Boltzmann model combined with a turbulence module was adopted. This model was first validated using experimental data for particles of spherical and cubic shapes. For irregularly shaped particles, two different types of settling behaviors were observed prior to particles reaching a steady state: accelerating and accelerating-decelerating, which could be distinguished by a critical CSF value of approximately 0.7. The settling dynamics were analyzed with a focus on the projected areas and angular velocities of particles. It was found that a minor change in the starting projected area, an indicator of the initial particle orientation, would not strongly affect the settling velocity for low Re. Periodic oscillations developed for all simulated particles when Re>100 . The amplitude of these oscillations increased with Re. However, the periods were not sensitive to Re. The critical Re that defined the transition between the steady and periodically oscillating behaviors depended on the inertia tensor. In particular, the maximum eigenvalue of the inertia tensor played a major role in signaling this transition in comparison to the intermediate and minimum eigenvalues.
New Boundary Treatment Methods for Lattice Boltzmann Method
Institute of Scientific and Technical Information of China (English)
Cheng Yong-guang; Suo Li-sheng
2003-01-01
In practical fluid dynamic simulations, the boundary condition should be treated carefully because it always has crucial influence on the numerical accuracy, stability and efficiency. Two types of boundary treatment methods for lattice Boltzmann method (LBM) are proposed. One is for the treatment of boundaries situated at lattice nodes, and the other is for the approximation of boundaries that are not located at the regular lattice nodes. The first type of boundary treatment method can deal with various dynamic boundaries on complex geometries by using a general set of formulas, which can maintain second-order accuracy. Based on the fact that the fluid flows simulated by LBM are not far from equilibrium, the unknown distributions at a boundary node are expressed as the analogous forms of their corresponding equilibrium distributions. Therefore, the number of unknowns can be reduced and an always-closed set of equations can be obtained for the solutions to pressure, velocity and special boundary conditions on various geometries. The second type of boundary treatment is a complete interpolation scheme to treat curved boundaries. It comes from careful analysis of the relations between distribution functions at boundary nodes and their neighboring lattice nodes. It is stable for all situations and of second-order accuracy. Basic ideas, implementation procedures and verifications with typical examples for the both treatments are presented. Numerical simulations and analyses show that they are accurate, stable, general and efficient for practical simulations.
Lattice Boltzmann Simulation of Viscous Flow past Elliptical Cylinder
Directory of Open Access Journals (Sweden)
D.Arumuga Perumal
2012-09-01
Full Text Available This work is concerned with the vortex structures of two-dimensional elliptical cylinder by lattice Boltzmann method. It is known that, the nature of the flow past cylindrical obstacles is very complex. Therefore, in the present work a kinetic based approach, namely, lattice Boltzmann method is used to compute both for steady and unsteady flows. A two dimensional nine-velocity square lattice (D2Q9 model is used in the present simulation. Effects of blockage ratio, Reynolds number and channel length are studied in detail. Here we conclude that lattice Boltzmann method can be effectively used to capture vortex shedding and other features.
Diophantine approximation and badly approximable sets
DEFF Research Database (Denmark)
Kristensen, S.; Thorn, R.; Velani, S.
2006-01-01
. The classical set Bad of `badly approximable' numbers in the theory of Diophantine approximation falls within our framework as do the sets Bad(i,j) of simultaneously badly approximable numbers. Under various natural conditions we prove that the badly approximable subsets of Omega have full Hausdorff dimension....... Applications of our general framework include those from number theory (classical, complex, p-adic and formal power series) and dynamical systems (iterated function schemes, rational maps and Kleinian groups)....
The isotropic diffusion source approximation for supernova neutrino transport
Liebendoerfer, M.; Whitehouse, S. C.; Fischer, T
2007-01-01
Astrophysical observations originate from matter that interacts with radiation or transported particles. We develop a pragmatic approximation in order to enable multi-dimensional simulations with basic spectral radiative transfer when the computational resources are not sufficient to solve the complete Boltzmann transport equation. The distribution function of the transported particles is decomposed into trapped and streaming particle components. Their separate evolution equations are coupled...
On measure solutions of the Boltzmann equation, part I: Moment production and stability estimates
Lu, Xuguang; Mouhot, Clément
The spatially homogeneous Boltzmann equation with hard potentials is considered for measure valued initial data having finite mass and energy. We prove the existence of weak measure solutions, with and without angular cutoff on the collision kernel; the proof in particular makes use of an approximation argument based on the Mehler transform. Moment production estimates in the usual form and in the exponential form are obtained for these solutions. Finally for the Grad angular cutoff, we also establish uniqueness and strong stability estimate on these solutions.
Lattice Boltzmann method to study the contraction of a viscous ligament
Srivastava, Sudhir; Jeurissen, Roger; Wijshoff, Herman; Toschi, Federico
2013-01-01
We employ a recently formulated axisymmetric version of the multiphase Shan-Chen (SC) lattice Boltzmann method (LBM) [Srivastava et al, in preparation (2013)] to simulate the contraction of a liquid ligament. We compare the axisymmetric LBM simulation against the slender jet (SJ) approximation model [T. Driessen and R. Jeurissen, IJCFD {\\bf 25}, 333 (2011)]. We compare the retraction dynamics of the tail-end of the liquid ligament from the LBM simulation, the SJ model, Flow3D simulations and a simple model based on the force balance (FB). We find good agreement between the theoretical prediction (FB), the SJ model, and the LBM simulations.
High order numerical methods for the space non-homogeneous Boltzmann equation
International Nuclear Information System (INIS)
In this paper we present accurate methods for the numerical solution of the Boltzmann equation of rarefied gas. The methods are based on a time splitting technique. The transport is solved by a third order accurate (in space) positive and flux conservative (PFC) method. The collision step is treated by a Fourier approximation of the collision integral, which guarantees spectral accuracy in velocity, coupled with several high order integrators in time. Strang splitting is used to achieve second order accuracy in space and time. Several numerical tests illustrate the properties of the methods
Mohammadipoor, O R; Niazmand, H; Mirbozorgi, S A
2014-01-01
Since the lattice Boltzmann method originally carries out the simulations on the regular Cartesian lattices, curved boundaries are often approximated as a series of stair steps. The most commonly employed technique for resolving curved-boundary problems is extrapolating or interpolating macroscopic properties of boundary nodes. Previous investigations have indicated that using more than one equation for extrapolation or interpolation in boundary conditions potentially causes abrupt changes in particle distributions. Therefore, a curved-boundary treatment is introduced to improve computational accuracy of the conventional stair-shaped approximation used in lattice Boltzmann simulations by using a unified equation for extrapolation of macroscopic variables. This boundary condition is not limited to fluid flow and can be extended to potential fields. The proposed treatment is tested against several well-established problems and the solutions order of accuracy is evaluated. Numerical results show that the present treatment is of second-order accuracy and has reliable stability characteristics. PMID:24580362
Langevin theory of fluctuations in the discrete Boltzmann equation
Gross, M; Varnik, F; Adhikari, R
2010-01-01
The discrete Boltzmann equation for both the ideal and a non-ideal fluid is extended by adding Langevin noise terms in order to incorporate the effects of thermal fluctuations. After casting the fluctuating discrete Boltzmann equation in a form appropriate to the Onsager-Machlup theory of linear fluctuations, the statistical properties of the noise are determined by invoking a fluctuation-dissipation theorem at the kinetic level. By integrating the fluctuating discrete Boltzmann equation, the fluctuating lattice Boltzmann equation is obtained, which provides an efficient way to solve the equations of fluctuating hydrodynamics for ideal and non-ideal fluids. Application of the framework to a generic force-based non-ideal fluid model leads to ideal gas-type thermal noise. Simulation results indicate proper thermalization of all degrees of freedom.
Computational Aeroacoustics Using the Generalized Lattice Boltzmann Equation Project
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...
Permit Allocation in Emissions Trading using the Boltzmann Distribution
Park, Ji-Won; Isard, Walter
2011-01-01
In emissions trading, the initial permit allocation is an intractable issue because it needs to be essentially fair to the participating countries. There are many ways to distribute a given total amount of emissions permits among countries, but the existing distribution methods such as auctioning and grandfathering have been debated. Here we describe a new model for permit allocation in emissions trading using the Boltzmann distribution. The Boltzmann distribution is introduced to permit allocation by combining it with concepts in emissions trading. A price determination mechanism for emission permits is then developed in relation to the {\\beta} value in the Boltzmann distribution. Finally, it is demonstrated how emissions permits can be practically allocated among participating countries in empirical results. The new allocation model using the Boltzmann distribution describes a most probable, natural, and unbiased distribution of emissions permits among multiple countries. Based on its simplicity and versati...
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-conserving classical dynamics in quantum time-correlation functions: “Matsubara dynamics”
Energy Technology Data Exchange (ETDEWEB)
Hele, Timothy J. H.; Willatt, Michael J.; Muolo, Andrea; Althorpe, Stuart C., E-mail: sca10@cam.ac.uk [Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW (United Kingdom)
2015-04-07
We show that a single change in the derivation of the linearized semiclassical-initial value representation (LSC-IVR or “classical Wigner approximation”) results in a classical dynamics which conserves the quantum Boltzmann distribution. We rederive the (standard) LSC-IVR approach by writing the (exact) quantum time-correlation function in terms of the normal modes of a free ring-polymer (i.e., a discrete imaginary-time Feynman path), taking the limit that the number of polymer beads N → ∞, such that the lowest normal-mode frequencies take their “Matsubara” values. The change we propose is to truncate the quantum Liouvillian, not explicitly in powers of ħ{sup 2} at ħ{sup 0} (which gives back the standard LSC-IVR approximation), but in the normal-mode derivatives corresponding to the lowest Matsubara frequencies. The resulting “Matsubara” dynamics is inherently classical (since all terms O(ħ{sup 2}) disappear from the Matsubara Liouvillian in the limit N → ∞) and conserves the quantum Boltzmann distribution because the Matsubara Hamiltonian is symmetric with respect to imaginary-time translation. Numerical tests show that the Matsubara approximation to the quantum time-correlation function converges with respect to the number of modes and gives better agreement than LSC-IVR with the exact quantum result. Matsubara dynamics is too computationally expensive to be applied to complex systems, but its further approximation may lead to practical methods.
A new lattice Boltzmann model for incompressible magnetohydrodynamics
Institute of Scientific and Technical Information of China (English)
Chen Xing-Wang; Shi Bao-Chang
2005-01-01
Most of the existing lattice Boltzmann magnetohydrodynamics (MHD) models can be viewed as compressible schemes to simulate incompressible MHD flows. The compressible effect might lead to some undesired errors in numerical simulations. In this paper a new incompressible lattice Boltzmann MHD model without compressible effect is presented for simulating incompressible MHD flows. Numerical simulations of the Hartmann flow are performed. We do numerous tests and make comparison with Dellar's model in detail. The numerical results are in good agreement with the analytical error.
A Boltzmann model for rod alignment and schooling fish
Carlen, Eric A.; Carvalho, Maria C.; Degond, Pierre; Wennberg, Bernt
2014-01-01
We consider a Boltzmann model introduced by Bertin, Droz and Greegoire as a binary interaction model of the Vicsek alignment interaction. This model considers particles lying on the circle. Pairs of particles interact by trying to reach their mid-point (on the circle) up to some noise. We study the equilibria of this Boltzmann model and we rigorously show the existence of a pitchfork bifurcation when a parameter measuring the inverse of the noise intensity crosses a critical threshold. The an...
Analysis of Jeans instability from Boltzmann equation
Kremer, Gilberto M
2015-01-01
The dynamics of self-gravitating fluids is analyzed within the framework of a collisionless Boltzmann equation in the presence of gravitational fields and Poisson equation. The equilibrium distribution function takes into account the expansion of the Universe and a pressureless fluid in the matter dominated Universe. Without invoking Jeans "swindle" a dispersion relation is obtained by considering small perturbations of the equilibrium values of the distribution function and gravitational potential. The collapse criterion -- which happens in an unstable region where the solution grows exponentially with time -- is determined from the dispersion relation. The collapse criterion in a static Universe occurs when the wavenumber $k$ is smaller than the Jeans wavenumber $k_J$, which was the solution found by Jeans. For an expanding Universe it is shown that this criterion is $k\\leq\\sqrt{7/6}\\,k_J$. As a consequence the ratio of the mass contained in a sphere of diameter equal to the wavelength $\\lambda=2\\pi/k$ to t...
Modeling adsorption with lattice Boltzmann equation.
Guo, Long; Xiao, Lizhi; Shan, Xiaowen; Zhang, Xiaoling
2016-01-01
The research of adsorption theory has recently gained renewed attention due to its critical relevance to a number of trending industrial applications, hydrogen storage and shale gas exploration for instance. The existing theoretical foundation, laid mostly in the early twentieth century, was largely based on simple heuristic molecular interaction models and static interaction potential which, although being insightful in illuminating the fundamental mechanisms, are insufficient for computations with realistic adsorbent structure and adsorbate hydrodynamics, both critical for real-life applications. Here we present and validate a novel lattice Boltzmann model incorporating both adsorbate-adsorbate and adsorbate-adsorbent interactions with hydrodynamics which, for the first time, allows adsorption to be computed with real-life details. Connection with the classic Ono-Kondo lattice theory is established and various adsorption isotherms, both within and beyond the IUPAC classification are observed as a pseudo-potential is varied. This new approach not only enables an important physical to be simulated for real-life applications, but also provides an enabling theoretical framework within which the fundamentals of adsorption can be studied. PMID:27256325
Lattice Boltzmann model for resistive relativistic magnetohydrodynamics.
Mohseni, F; Mendoza, M; Succi, S; Herrmann, H J
2015-08-01
In this paper, we develop a lattice Boltzmann model for relativistic magnetohydrodynamics (MHD). Even though the model is derived for resistive MHD, it is shown that it is numerically robust even in the high conductivity (ideal MHD) limit. In order to validate the numerical method, test simulations are carried out for both ideal and resistive limits, namely the propagation of Alfvén waves in the ideal MHD and the evolution of current sheets in the resistive regime, where very good agreement is observed comparing to the analytical results. Additionally, two-dimensional magnetic reconnection driven by Kelvin-Helmholtz instability is studied and the effects of different parameters on the reconnection rate are investigated. It is shown that the density ratio has a negligible effect on the magnetic reconnection rate, while an increase in shear velocity decreases the reconnection rate. Additionally, it is found that the reconnection rate is proportional to σ-1/2, σ being the conductivity, which is in agreement with the scaling law of the Sweet-Parker model. Finally, the numerical model is used to study the magnetic reconnection in a stellar flare. Three-dimensional simulation suggests that the reconnection between the background and flux rope magnetic lines in a stellar flare can take place as a result of a shear velocity in the photosphere. PMID:26382548
Modeling adsorption with lattice Boltzmann equation
Guo, Long; Xiao, Lizhi; Shan, Xiaowen; Zhang, Xiaoling
2016-01-01
The research of adsorption theory has recently gained renewed attention due to its critical relevance to a number of trending industrial applications, hydrogen storage and shale gas exploration for instance. The existing theoretical foundation, laid mostly in the early twentieth century, was largely based on simple heuristic molecular interaction models and static interaction potential which, although being insightful in illuminating the fundamental mechanisms, are insufficient for computations with realistic adsorbent structure and adsorbate hydrodynamics, both critical for real-life applications. Here we present and validate a novel lattice Boltzmann model incorporating both adsorbate-adsorbate and adsorbate-adsorbent interactions with hydrodynamics which, for the first time, allows adsorption to be computed with real-life details. Connection with the classic Ono-Kondo lattice theory is established and various adsorption isotherms, both within and beyond the IUPAC classification are observed as a pseudo-potential is varied. This new approach not only enables an important physical to be simulated for real-life applications, but also provides an enabling theoretical framework within which the fundamentals of adsorption can be studied. PMID:27256325
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
Lattice Boltzmann method and its applications in engineering thermophysics
Institute of Scientific and Technical Information of China (English)
HE YaLing; LI Qing; WANG Yong; TANG GuiHua
2009-01-01
The lattice Boltzmann method (LBM),a mesoscopic method between the molecular dynamics method and the conventional numerical methods,has been developed into a very efficient numerical alternative in the past two decades.Unlike conventional numerical methods,the kinetic theory based LBM simulates fluid flows by tracking the evolution of the particle distribution function,and then accumulates the distribution to obtain macroscopic averaged properties.In this article we review some work on LBM applications in engineering thermophysics:(1) brief introduction to the development of the LBM; (2)fundamental theory of LBM including the Boltzmann equation,Maxwell distribution function,Boltzmann-BGK equation,and the lattice Boltzmann-BGK equation; (3) lattice Boltzmann models for compressible flows and non-equilibrium gas flows,bounce back-specular-reflection boundary scheme for microscale gaseous flows,the mass modified outlet boundary scheme for fully developed flows,and an implicit-explicit finite-difference-based LBM; and (4) applications of the LBM to oscillating flow,compressible flow,porous media flow,non-equilibrium flow,and gas resonant oscillating flow.
The intellectual quadrangle: Mach-Boltzmann-Planck-Einstein
International Nuclear Information System (INIS)
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.)
The Cosmic Linear Anisotropy Solving System (CLASS). Part II: Approximation schemes
Energy Technology Data Exchange (ETDEWEB)
Blas, Diego; Lesgourgues, Julien [Institut de Théorie des Phénomènes Physiques, École Polytechnique Fédérale de Lausanne, CH-1015, Lausanne (Switzerland); Tram, Thomas, E-mail: diego.blas@epfl.ch, E-mail: julien.lesgourgues@cern.ch, E-mail: tram@phys.au.dk [CERN, Theory Division, CH-1211 Geneva 23 (Switzerland)
2011-07-01
Boltzmann codes are used extensively by several groups for constraining cosmological parameters with Cosmic Microwave Background and Large Scale Structure data. This activity is computationally expensive, since a typical project requires from 10{sup 4} to 10{sup 5} Boltzmann code executions. The newly released code CLASS (Cosmic Linear Anisotropy Solving System) incorporates improved approximation schemes leading to a simultaneous gain in speed and precision. We describe here the three approximations used by CLASS for basic ΛCDM models, namely: a baryon-photon tight-coupling approximation which can be set to first order, second order or to a compromise between the two; an ultra-relativistic fluid approximation which had not been implemented in public distributions before; and finally a radiation streaming approximation taking reionisation into account.
Analytical estimation of effective charges at saturation in Poisson-Boltzmann cell models
Trizac, E; Bocquet, L
2003-01-01
We propose a simple approximation scheme for computing the effective charges of highly charged colloids (spherical or cylindrical with infinite length). Within non-linear Poisson-Boltzmann theory, we start from an expression for the effective charge in the infinite-dilution limit which is asymptotically valid for large salt concentrations; this result is then extended to finite colloidal concentration, approximating the salt partitioning effect which relates the salt content in the suspension to that of a dialysing reservoir. This leads to an analytical expression for the effective charge as a function of colloid volume fraction and salt concentration. These results compare favourably with the effective charges at saturation (i.e. in the limit of large bare charge) computed numerically following the standard prescription proposed by Alexander et al within the cell model.
Systematic Study of the Boundary Composition in Poisson Boltzmann Calculations
Kar, P; Hansmann, U H E; Hoefinger, S
2007-01-01
We describe a three-stage procedure to analyze the dependence of Poisson Boltzmann calculations on the shape, size and geometry of the boundary between solute and solvent. Our study is carried out within the boundary element formalism, but our results are also of interest to finite difference techniques of Poisson Boltzmann calculations. At first, we identify the critical size of the geometrical elements for discretizing the boundary, and thus the necessary resolution required to establish numerical convergence. In the following two steps we perform reference calculations on a set of dipeptides in different conformations using the Polarizable Continuum Model and a high-level Density Functional as well as a high-quality basis set. Afterwards, we propose a mechanism for defining appropriate boundary geometries. Finally, we compare the classic Poisson Boltzmann description with the Quantum Chemical description, and aim at finding appropriate fitting parameters to get a close match to the reference data. Surprisi...
Lattice Boltzmann Model for Compressible Fluid on a Square Lattice
Institute of Scientific and Technical Information of China (English)
SUN Cheng-Hai
2000-01-01
A two-level four-direction lattice Boltzmann model is formulated on a square lattice to simulate compressible flows with a high Mach number. The particle velocities are adaptive to the mean velocity and internal energy. Therefore, the mean flow can have a high Mach number. Due to the simple form of the equilibrium distribution, the 4th order velocity tensors are not involved in the calculations. Unlike the standard lattice Boltzmann model, o special treatment is need for the homogeneity of 4th order velocity tensors on square lattices. The Navier-Stokes equations were derived by the Chapman-Enskog method from the BGK Boltzmann equation. The model can be easily extended to three-dimensional cubic lattices. Two-dimensional shock-wave propagation was simulated
Accounting for adsorption and desorption in Lattice Boltzmann simulations
Levesque, Maximilien; Pagonabarraga, Ignacio; Frenkel, Daan; Rotenberg, Benjamin
2013-01-01
We report a Lattice-Boltzmann scheme that accounts for adsorption and desorption in the calculation of mesoscale dynamical properties of tracers in media of arbitrary complexity. Lattice Boltzmann simulations made it possible to solve numerically the coupled Navier-Stokes equations of fluid dynamics and Nernst-Planck equations of electrokinetics in complex, heterogeneous media. Associated to the moment propagation scheme, it became possible to extract the effective diffusion and dispersion coefficients of tracers, or solutes, of any charge, e.g. in porous media. Nevertheless, the dynamical properties of tracers depend on the tracer-surface affinity, which is not purely electrostatic, but also includes a species-specific contribution. In order to capture this important feature, we introduce specific adsorption and desorption processes in a Lattice-Boltzmann scheme through a modified moment propagation algorithm, in which tracers may adsorb and desorb from surfaces through kinetic reaction rates. The method is ...
Lattice Boltzmann Large Eddy Simulation Model of MHD
Flint, Christopher
2016-01-01
The work of Ansumali \\textit{et al.}\\cite{Ansumali} is extended to Two Dimensional Magnetohydrodynamic (MHD) turbulence in which energy is cascaded to small spatial scales and thus requires subgrid modeling. Applying large eddy simulation (LES) modeling of the macroscopic fluid equations results in the need to apply ad-hoc closure schemes. LES is applied to a suitable mesoscopic lattice Boltzmann representation from which one can recover the MHD equations in the long wavelength, long time scale Chapman-Enskog limit (i.e., the Knudsen limit). Thus on first performing filter width expansions on the lattice Boltzmann equations followed by the standard small Knudsen expansion on the filtered lattice Boltzmann system results in a closed set of MHD turbulence equations provided we enforce the physical constraint that the subgrid effects first enter the dynamics at the transport time scales. In particular, a multi-time relaxation collision operator is considered for the density distribution function and a single rel...
An integrable 3D lattice model with positive Boltzmann weights
Mangazeev, Vladimir V; Sergeev, Sergey M
2013-01-01
In this paper we construct a three-dimensional (3D) solvable lattice model with non-negative Boltzmann weights. The spin variables in the model are assigned to edges of the 3D cubic lattice and run over an infinite number of discrete states. The Boltzmann weights satisfy the tetrahedron equation, which is a 3D generalisation of the Yang-Baxter equation. The weights depend on a free parameter 0Boltzmann weights.
Leike, Reimar H
2016-01-01
In Bayesian statistics probability distributions express beliefs. However, for many problems the beliefs cannot be computed analytically and approximations of beliefs are needed. We seek a ranking function that quantifies how "embarrassing" it is to communicate a given approximation. We show that there is only one ranking under the requirements that (1) the best ranked approximation is the non-approximated belief and (2) that the ranking judges approximations only by their predictions for actual outcomes. We find that this ranking is equivalent to the Kullback-Leibler divergence that is frequently used in the literature. However, there seems to be confusion about the correct order in which its functional arguments, the approximated and non-approximated beliefs, should be used. We hope that our elementary derivation settles the apparent confusion. We show for example that when approximating beliefs with Gaussian distributions the optimal approximation is given by moment matching. This is in contrast to many su...
Lattice gas cellular automata and lattice Boltzmann models an introduction
Wolf-Gladrow, Dieter A
2000-01-01
Lattice-gas cellular automata (LGCA) and lattice Boltzmann models (LBM) are relatively new and promising methods for the numerical solution of nonlinear partial differential equations. The book provides an introduction for graduate students and researchers. Working knowledge of calculus is required and experience in PDEs and fluid dynamics is recommended. Some peculiarities of cellular automata are outlined in Chapter 2. The properties of various LGCA and special coding techniques are discussed in Chapter 3. Concepts from statistical mechanics (Chapter 4) provide the necessary theoretical background for LGCA and LBM. The properties of lattice Boltzmann models and a method for their construction are presented in Chapter 5.
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...
Spinor Boltzmann Equation with Two Momenta at the Fermi Level
Institute of Scientific and Technical Information of China (English)
王正川
2012-01-01
Based on the formalism of Keldysh's nonequilibrium Green function, we establish a two momenta spinor Boltzmann equation for longitudinal scalar distribution function and transverse vector distribution function. The lon- gitudinal charge currents, transverse spin currents and the continuity equations satisfied by them are then studied, it indicates that both the charge currents and spin currents decay oscillately along with position, which is due to the momenta integral over the Fermi surface. We also compare our charge currents and spin currents with the corresponding results of one momentum spinor Boltzmann equation, the differences are obvious.
Asymptotic-preserving Boltzmann model equations for binary gas mixture
Liu, Sha; Liang, Yihua
2016-02-01
An improved system of Boltzmann model equations is developed for binary gas mixture. This system of model equations has a complete asymptotic preserving property that can strictly recover the Navier-Stokes equations in the continuum limit with the correct constitutive relations and the correct viscosity, thermal conduction, diffusion, and thermal diffusion coefficients. In this equation system, the self- and cross-collision terms in Boltzmann equations are replaced by single relaxation terms. In monocomponent case, this system of equations can be reduced to the commonly used Shakhov equation. The conservation property and the H theorem which are important for model equations are also satisfied by this system of model equations.
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.
Multiphase lattice Boltzmann simulations for porous media applications -- a review
Liu, Haihu; Leonardi, Christopher R; Jones, Bruce D; Schmieschek, Sebastian; Narváez, Ariel; Williams, John R; Valocchi, Albert J; Harting, Jens
2014-01-01
Over the last two decades, lattice Boltzmann methods have become an increasingly popular tool to compute the flow in complex geometries such as porous media. In addition to single phase simulations allowing, for example, a precise quantification of the permeability of a porous sample, a number of extensions to the lattice Boltzmann method are available which allow to study multiphase and multicomponent flows on a pore scale level. In this article we give an extensive overview on a number of these diffuse interface models and discuss their advantages and disadvantages. Furthermore, we shortly report on multiphase flows containing solid particles, as well as implementation details and optimization issues.
Learning Feature Hierarchies with Centered Deep Boltzmann Machines
Montavon, Grégoire
2012-01-01
Deep Boltzmann machines are in principle powerful models for extracting the hierarchical structure of data. Unfortunately, attempts to train layers jointly (without greedy layer-wise pretraining) have been largely unsuccessful. We propose a modification of the learning algorithm that initially recenters the output of the activation functions to zero. This modification leads to a better conditioned Hessian and thus makes learning easier. We test the algorithm on real data and demonstrate that our suggestion, the centered deep Boltzmann machine, learns a hierarchy of increasingly abstract representations and a better generative model of data.
CORRECTIONS TO THE COLLISION TERM IN THE BGK BOLTZMANN EQUATION
Institute of Scientific and Technical Information of China (English)
FENG SHI-DE; REN RONG-CAI; CUI XIAO-PENG; JI ZHONG-ZHEN
2001-01-01
With the discrete method of the hexagonal cell and three different velocities of particle population in each cell,a two-dimensional lattice Boltzmann model is developed in this paper.[1,2] The collision operator in the Boltzmann equation is expanded to fourth order using the Taylor expansion.[3,4] With this model, good results have been obtained from the numerical simulation of the reflection phenomenon of the shock wave on the surface of an obstacle, and the numerical stability is also good. Thus the applicability of the D2Q19 model is verified.
Bin Qin
2014-01-01
Relationships between fuzzy relations and fuzzy topologies are deeply researched. The concept of fuzzy approximating spaces is introduced and decision conditions that a fuzzy topological space is a fuzzy approximating space are obtained.
Rasin, A
1994-01-01
We discuss the idea of approximate flavor symmetries. Relations between approximate flavor symmetries and natural flavor conservation and democracy models is explored. Implications for neutrino physics are also discussed.
Chebyshev collocation spectral lattice Boltzmann method for simulation of low-speed flows.
Hejranfar, Kazem; Hajihassanpour, Mahya
2015-01-01
In this study, the Chebyshev collocation spectral lattice Boltzmann method (CCSLBM) is developed and assessed for the computation of low-speed flows. Both steady and unsteady flows are considered here. The discrete Boltzmann equation with the Bhatnagar-Gross-Krook approximation based on the pressure distribution function is considered and the space discretization is performed by the Chebyshev collocation spectral method to achieve a highly accurate flow solver. To provide accurate unsteady solutions, the time integration of the temporal term in the lattice Boltzmann equation is made by the fourth-order Runge-Kutta scheme. To achieve numerical stability and accuracy, physical boundary conditions based on the spectral solution of the governing equations implemented on the boundaries are used. An iterative procedure is applied to provide consistent initial conditions for the distribution function and the pressure field for the simulation of unsteady flows. The main advantage of using the CCSLBM over other high-order accurate lattice Boltzmann method (LBM)-based flow solvers is the decay of the error at exponential rather than at polynomial rates. Note also that the CCSLBM applied does not need any numerical dissipation or filtering for the solution to be stable, leading to highly accurate solutions. Three two-dimensional (2D) test cases are simulated herein that are a regularized cavity, the Taylor vortex problem, and doubly periodic shear layers. The results obtained for these test cases are thoroughly compared with the analytical and available numerical results and show excellent agreement. The computational efficiency of the proposed solution methodology based on the CCSLBM is also examined by comparison with those of the standard streaming-collision (classical) LBM and two finite-difference LBM solvers. The study indicates that the CCSLBM provides more accurate and efficient solutions than these LBM solvers in terms of CPU and memory usage and an exponential
Approximate iterative algorithms
Almudevar, Anthony Louis
2014-01-01
Iterative algorithms often rely on approximate evaluation techniques, which may include statistical estimation, computer simulation or functional approximation. This volume presents methods for the study of approximate iterative algorithms, providing tools for the derivation of error bounds and convergence rates, and for the optimal design of such algorithms. Techniques of functional analysis are used to derive analytical relationships between approximation methods and convergence properties for general classes of algorithms. This work provides the necessary background in functional analysis a
Approximation of distributed delays
Lu, Hao; Eberard, Damien; Simon, Jean-Pierre
2010-01-01
We address in this paper the approximation problem of distributed delays. Such elements are convolution operators with kernel having bounded support, and appear in the control of time-delay systems. From the rich literature on this topic, we propose a general methodology to achieve such an approximation. For this, we enclose the approximation problem in the graph topology, and work with the norm defined over the convolution Banach algebra. The class of rational approximates is described, and a constructive approximation is proposed. Analysis in time and frequency domains is provided. This methodology is illustrated on the stabilization control problem, for which simulations results show the effectiveness of the proposed methodology.
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...
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.
Convection-diffusion lattice Boltzmann scheme for irregular lattices
Sman, van der R.G.M.; Ernst, M.H.
2000-01-01
In this paper, a lattice Boltzmann (LB) scheme for convection diffusion on irregular lattices is presented, which is free of any interpolation or coarse graining step. The scheme is derived using the axioma that the velocity moments of the equilibrium distribution equal those of the Maxwell-Boltzman
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 the...
Measuring Boltzmann's Constant with Carbon Dioxide
Ivanov, Dragia; Nikolov, Stefan
2013-01-01
In this paper we present two experiments to measure Boltzmann's constant--one of the fundamental constants of modern-day physics, which lies at the base of statistical mechanics and thermodynamics. The experiments use very basic theory, simple equipment and cheap and safe materials yet provide very precise results. They are very easy and…
Boundary conditions for surface reactions in lattice Boltzmann simulations
Gillissen, J.J.J.; Looije, N.
2014-01-01
A surface reaction boundary condition in multicomponent lattice Boltzmann simulations is developed. The method is applied to a test case with nonlinear reaction rates and nonlinear density profiles. The results are compared to the corresponding analytical solution, which shows that the error of the
Thermal creep problems by the discrete Boltzmann equation
Directory of Open Access Journals (Sweden)
L. Preziosi
1991-05-01
Full Text Available This paper deals with an initial-boundary value problem for the discrete Boltzmann equation confined between two moving walls at different temperature. A model suitable for the quantitative analysis of the initial boundary value problem and the relative existence theorem are given.
An exactly solvable non-linear Boltzmann equation
Ernst, M.H.; Hendriks, E.M.
1979-01-01
The initial value problem for a model Boltzmann equation of a two dimensional gas with a continuous or discrete energy distribution function and a transition probability δ(ε - ε') is solved exactly; ε and ε' are the total energies before and after collision.
Sparse approximation with bases
2015-01-01
This book systematically presents recent fundamental results on greedy approximation with respect to bases. Motivated by numerous applications, the last decade has seen great successes in studying nonlinear sparse approximation. Recent findings have established that greedy-type algorithms are suitable methods of nonlinear approximation in both sparse approximation with respect to bases and sparse approximation with respect to redundant systems. These insights, combined with some previous fundamental results, form the basis for constructing the theory of greedy approximation. Taking into account the theoretical and practical demand for this kind of theory, the book systematically elaborates a theoretical framework for greedy approximation and its applications. The book addresses the needs of researchers working in numerical mathematics, harmonic analysis, and functional analysis. It quickly takes the reader from classical results to the latest frontier, but is written at the level of a graduate course and do...
Fiorentini, Mattia; Bonini, Nicola
2016-08-01
We present a first-principles computational approach to calculate thermoelectric transport coefficients via the exact solution of the linearized Boltzmann transport equation, also including the effect of nonequilibrium phonon populations induced by a temperature gradient. We use density functional theory and density functional perturbation theory for an accurate description of the electronic and vibrational properties of a system, including electron-phonon interactions; carriers' scattering rates are computed using standard perturbation theory. We exploit Wannier interpolation (both for electronic bands and electron-phonon matrix elements) for an efficient sampling of the Brillouin zone, and the solution of the Boltzmann equation is achieved via a fast and stable conjugate gradient scheme. We discuss the application of this approach to n -doped silicon. In particular, we discuss a number of thermoelectric properties such as the thermal and electrical conductivities of electrons, the Lorenz number and the Seebeck coefficient, including the phonon drag effect, in a range of temperatures and carrier concentrations. This approach gives results in good agreement with experimental data and provides a detailed characterization of the nature and the relative importance of the individual scattering mechanisms. Moreover, the access to the exact solution of the Boltzmann equation for a realistic system provides a direct way to assess the accuracy of different flavors of relaxation time approximation, as well as of models that are popular in the thermoelectric community to estimate transport coefficients.
Wei, Linsheng; Xu, Min; Yuan, Dingkun; Zhang, Yafang; Hu, Zhaoji; Tan, Zhihong
2014-10-01
The electron drift velocity, electron energy distribution function (EEDF), density-normalized effective ionization coefficient and density-normalized longitudinal diffusion velocity are calculated in SF6-O2 and SF6-Air mixtures. The experimental results from a pulsed Townsend discharge are plotted for comparison with the numerical results. The reduced field strength varies from 40 Td to 500 Td (1 Townsend=10-17 V·cm2) and the SF6 concentration ranges from 10% to 100%. A Boltzmann equation associated with the two-term spherical harmonic expansion approximation is utilized to gain the swarm parameters in steady-state Townsend. Results show that the accuracy of the Boltzmann solution with a two-term expansion in calculating the electron drift velocity, electron energy distribution function, and density-normalized effective ionization coefficient is acceptable. The effective ionization coefficient presents a distinct relationship with the SF6 content in the mixtures. Moreover, the E/Ncr values in SF6-Air mixtures are higher than those in SF6-O2 mixtures and the calculated value E/Ncr in SF6-O2 and SF6-Air mixtures is lower than the measured value in SF6-N2. Parametric studies conducted on these parameters using the Boltzmann analysis offer substantial insight into the plasma physics, as well as a basis to explore the ozone generation process.
International Nuclear Information System (INIS)
The electron drift velocity, electron energy distribution function (EEDF), density-normalized effective ionization coefficient and density-normalized longitudinal diffusion velocity are calculated in SF6-O2 and SF6-Air mixtures. The experimental results from a pulsed Townsend discharge are plotted for comparison with the numerical results. The reduced field strength varies from 40 Td to 500 Td (1 Townsend=10−17 V·cm2) and the SF6 concentration ranges from 10% to 100%. A Boltzmann equation associated with the two-term spherical harmonic expansion approximation is utilized to gain the swarm parameters in steady-state Townsend. Results show that the accuracy of the Boltzmann solution with a two-term expansion in calculating the electron drift velocity, electron energy distribution function, and density-normalized effective ionization coefficient is acceptable. The effective ionization coefficient presents a distinct relationship with the SF6 content in the mixtures. Moreover, the E/Ncr values in SF6-Air mixtures are higher than those in SF6-O2 mixtures and the calculated value E/Ncr in SF6-O2 and SF6-Air mixtures is lower than the measured value in SF6-N2. Parametric studies conducted on these parameters using the Boltzmann analysis offer substantial insight into the plasma physics, as well as a basis to explore the ozone generation process. (low temperature plasma)
Institute of Scientific and Technical Information of China (English)
YueShihong; ZhangKecun
2002-01-01
In a dot product space with the reproducing kernel (r. k. S. ) ,a fuzzy system with the estimation approximation errors is proposed ,which overcomes the defect that the existing fuzzy control system is difficult to estimate the errors of approximation for a desired function,and keeps the characteristics of fuzzy system as an inference approach. The structure of the new fuzzy approximator benefits a course got by other means.
Approximation techniques for engineers
Komzsik, Louis
2006-01-01
Presenting numerous examples, algorithms, and industrial applications, Approximation Techniques for Engineers is your complete guide to the major techniques used in modern engineering practice. Whether you need approximations for discrete data of continuous functions, or you''re looking for approximate solutions to engineering problems, everything you need is nestled between the covers of this book. Now you can benefit from Louis Komzsik''s years of industrial experience to gain a working knowledge of a vast array of approximation techniques through this complete and self-contained resource.
Entropic Lattice Boltzmann study of hydrodynamics in a microcavity - Part 1
Energy Technology Data Exchange (ETDEWEB)
Karlin, I. V.; Ansumali, S.; Frouzakis, Ch. E.; Boulouchos, K. [Eidgenoessische Technische Hochschule (ETH), Labor fuer Aerothermochemie und Verbrennungssysteme ETHZ, ETH-Zentrum, Zuerich (Switzerland)
2005-07-01
This yearly report for 2004 presents a review of work being done on behalf of the Swiss Federal Office of Energy (SFOE) at the Laboratory for Aero-thermochemistry and Combustion Systems at the Federal Institute of Technology ETH in Zurich, Switzerland, on the development of a new approximation method for use in micrometer-scale flow calculations. The method, based on recently-developed so-called minimal entropy-kinetic models of the Boltzmann-kinetic equation, is discussed. Two detailed studies of micro-flows in specific geometries are discussed. The potential of the new method as a replacement for costly microscopic simulation methods is examined. The development and testing of a new thermal model - the so-called Thermal D2Q9 model - is discussed. A second study examined flows in a micro-cavity. A detailed parametric study of the quantitative and qualitative properties of the flows for a comprehensive range of dilution is mentioned.
Lattice Boltzmann technique for heat transport phenomena coupled with melting process
Ibrahem, A. M.; El-Amin, M. F.; Mohammadein, A. A.; Gorla, Rama Subba Reddy
2016-04-01
In this work, the heat transport phenomena coupled with melting process are studied by using the enthalpy-based lattice Boltzmann method (LBM). The proposed model is a modified version of thermal LB model, where could avoid iteration steps and ensures high accuracy. The Bhatnagar-Gross-Krook (BGK) approximation with a D1Q2 lattice was used to determine the temperature field for one-dimensional melting by conduction and multi-distribution functions (MDF) with D2Q9 lattice was used to determine the density, velocity and temperature fields for two-dimensional melting by natural convection. Different boundary conditions including Dirichlet, adiabatic and bounce-back boundary conditions were used. The influence of increasing Rayleigh number (from 103 to 105) on temperature distribution and melting process is studied. The obtained results show that a good agreement with the analytical solution for melting by conduction case and with the benchmark solution for melting by convection.
Lattice Boltzmann modeling of self-propelled Leidenfrost droplets on ratchet surfaces
Li, Q; Francois, M M; Hu, A J
2015-01-01
In this paper, the self-propelled motion of Leidenfrost droplets on ratchet surfaces is numerically investigated with a thermal multiphase lattice Boltzmann model with liquid-vapor phase change. The capability of the model for simulating evaporation is validated via the D2 law. Using the model, we first study the performances of Leidenfrost droplets on horizontal ratchet surfaces. It is numerically shown that the motion of self-propelled Leidenfrost droplets on ratchet surfaces is owing to the asymmetry of the ratchets and the vapor flows beneath the droplets. It is found that the Leidenfrost droplets move in the direction toward the slowly inclined side from the ratchet peaks, which agrees with the direction of droplet motion in experiments [Linke et al., Phys. Rev. Lett., 2006, 96, 154502]. Moreover, the influences of the ratchet aspect ratio are investigated. For the considered ratchet surfaces, a critical value of the ratchet aspect ratio is approximately found, which corresponds to the maximum droplet mo...
Fushiki, M; Svensson, B; Jönsson, B; Woodward, C E
1991-09-01
The accuracy of the Poisson-Boltzmann (PB) approximation and its linearized version is investigated by comparison to results obtained from Monte Carlo simulations. The dependence of the calcium binding constant of the protein calbindin as a function of salt concentration and mutation is used as a test case. The protein is modeled as a collection of charged and neutral spheres immersed in the electrolyte solution. The PB equation is solved using a finite difference technique on a grid in a spherical polar coordinate system, which is the preferred choice for a globular protein like calbindin. Both MC and PB give quantitative agreement with experimental results. The linearized PB equation is almost as accurate, but it becomes less reliable in systems with divalent ions. However, the linearized PB equation fails to describe the concentration profiles for cations and anions outside the protein even in a 1:1 salt solution. PMID:1790295
Tervo, J; Frank, M; Herty, M
2016-01-01
The paper considers a coupled system of linear Boltzmann transport equation (BTE), and its Continuous Slowing Down Approximation (CSDA). This system can be used to model the relevant transport of particles used e.g. in dose calculation in radiation therapy. The evolution of charged particles (e.g. electrons and positrons) are in practice often modelled using the CSDA version of BTE because of the so-called forward peakedness of scattering events contributing to the particle fluencies (or particle densities), which causes severe problems for numerical methods. First, we prove the existence and uniqueness of solutions, under sufficient criteria and in appropriate $L^2$-based spaces, of a single (particle) CSDA-equation by using two complementary techniques, the Lions-Lax-Milgram Theorem (variational approach), and the theory evolution operators (semigroup approach). The necessary a priori estimates are shown. In addition, we prove the corresponding results and estimates for the system of coupled transport equat...
Research of Micro-Rectangular-Channel Flow Based on Lattice Boltzmann Method
Directory of Open Access Journals (Sweden)
Xiao Hong
2013-08-01
Full Text Available The source codes of Lattice Boltzmann Method (LBM based on the D3Q15 model were developed in the current study. In the simulation process, the pressure boundary conditions were developed and the rectangular micro-channel flow was investigated. The Width-to-Height ratio (W/H is the main influencing parameter of the rectangular micro-channel flow in low Reynolds number condition. A smaller W/H of the rectangular micro-channel results in a greater difference of drag coefficient between LBM simulation data and empirical formula data. On the empirical data of drag coefficient approximating laminar flow, when the Reynolds number is more than 10, the drag coefficients of LBM simulation and empirical data of laminar flow are in substantial agreement. Thus, in more than 10 conditions of Reynolds number, the empirical data on laminar flow can be used in rectangular micro-channel flow.
Expectation Consistent Approximate Inference
DEFF Research Database (Denmark)
Opper, Manfred; Winther, Ole
2005-01-01
We propose a novel framework for approximations to intractable probabilistic models which is based on a free energy formulation. The approximation can be understood from replacing an average over the original intractable distribution with a tractable one. It requires two tractable probability dis...
Ordered cones and approximation
Keimel, Klaus
1992-01-01
This book presents a unified approach to Korovkin-type approximation theorems. It includes classical material on the approximation of real-valuedfunctions as well as recent and new results on set-valued functions and stochastic processes, and on weighted approximation. The results are notonly of qualitative nature, but include quantitative bounds on the order of approximation. The book is addressed to researchers in functional analysis and approximation theory as well as to those that want to applythese methods in other fields. It is largely self- contained, but the readershould have a solid background in abstract functional analysis. The unified approach is based on a new notion of locally convex ordered cones that are not embeddable in vector spaces but allow Hahn-Banach type separation and extension theorems. This concept seems to be of independent interest.
Approximate Modified Policy Iteration
Scherrer, Bruno; Ghavamzadeh, Mohammad; Geist, Matthieu
2012-01-01
Modified policy iteration (MPI) is a dynamic programming (DP) algorithm that contains the two celebrated policy and value iteration methods. Despite its generality, MPI has not been thoroughly studied, especially its approximation form which is used when the state and/or action spaces are large or infinite. In this paper, we propose three approximate MPI (AMPI) algorithms that are extensions of the well-known approximate DP algorithms: fitted-value iteration, fitted-Q iteration, and classification-based policy iteration. We provide an error propagation analysis for AMPI that unifies those for approximate policy and value iteration. We also provide a finite-sample analysis for the classification-based implementation of AMPI (CBMPI), which is more general (and somehow contains) than the analysis of the other presented AMPI algorithms. An interesting observation is that the MPI's parameter allows us to control the balance of errors (in value function approximation and in estimating the greedy policy) in the fina...
Big-Bang Nucleosynthesis verifies classical Maxwell-Boltzmann distribution
Hou, S Q; Parikh, A; Daid, K; Bertulani, C
2014-01-01
We provide the most stringent constraint to date on possible deviations from the usually-assumed Maxwell-Boltzmann (MB) velocity distribution for nuclei in the Big-Bang plasma. The impact of non-extensive Tsallis statistics on thermonuclear reaction rates involved in standard models of Big-Bang Nucleosynthesis (BBN) has been investigated. We find that the non-extensive parameter $q$ may deviate by, at most, $|\\delta q|$=6$\\times$10$^{-4}$ from unity for BBN predictions to be consistent with observed primordial abundances; $q$=1 represents the classical Boltzmann-Gibbs statistics. This constraint arises primarily from the {\\em super}sensitivity of endothermic rates on the value of $q$, which is found for the first time. As such, the implications of non-extensive statistics in other astrophysical environments should be explored. This may offer new insight into the nucleosynthesis of heavy elements.
Accelerate Monte Carlo Simulations with Restricted Boltzmann Machines
Huang, Li
2016-01-01
Despite their exceptional flexibility and popularity, the Monte Carlo methods often suffer from slow mixing times for challenging statistical physics problems. We present a general strategy to overcome this difficulty by adopting ideas and techniques from the machine learning community. We fit the unnormalized probability of the physical model to a feedforward neural network and reinterpret the architecture as a restricted Boltzmann machine. Then, exploiting its feature detection ability, we utilize the restricted Boltzmann machine for efficient Monte Carlo updates and to speed up the simulation of the original physical system. We implement these ideas for the Falicov-Kimball model and demonstrate improved acceptance ratio and autocorrelation time near the phase transition point.
Shock-wave structure using nonlinear model Boltzmann equations.
Segal, B. M.; Ferziger, J. H.
1972-01-01
The structure of strong plane shock waves in a perfect monatomic gas was studied using four nonlinear models of the Boltzmann equation. The models involved the use of a simplified collision operator with velocity-independent collision frequency, in place of the complicated Boltzmann collision operator. The models employed were the BGK and ellipsoidal models developed by earlier authors, and the polynomial and trimodal gain function models developed during the work. An exact set of moment equations was derived for the density, velocity, temperature, viscous stress, and heat flux within the shock. This set was reduced to a pair of coupled nonlinear integral equations and solved using specially adapted numerical techniques. A new and simple Gauss-Seidel iteration was developed during the work and found to be as efficient as the best earlier iteration methods.
Quadrature-based Lattice Boltzmann Model for Relativistic Flows
Blaga, Robert
2016-01-01
A quadrature-based finite-difference lattice Boltzmann model is developed that is suitable for simulating relativistic flows of massless particles. We briefly review the relativistc Boltzmann equation and present our model. The quadrature is constructed such that the stress-energy tensor is obtained as a second order moment of the distribution function. The results obtained with our model are presented for a particular instance of the Riemann problem (the Sod shock tube). We show that the model is able to accurately capture the behavior across the whole domain of relaxation times, from the hydrodynamic to the ballistic regime. The property of the model of being extendable to arbitrarily high orders is shown to be paramount for the recovery of the analytical result in the ballistic regime.
Contact angles in the pseudopotential lattice Boltzmann modeling of wetting
Li, Q; Kang, Q J; Chen, Q
2014-01-01
In this paper, we aim to investigate the implementation of contact angles in the pseudopotential lattice Boltzmann modeling of wetting at a large density ratio. The pseudopotential lattice Boltzmann model [X. Shan and H. Chen, Phys. Rev. E 49, 2941 (1994)] is a popular mesoscopic model for simulating multiphase flows and interfacial dynamics. In this model, the contact angle is usually realized by a fluid-solid interaction. Two widely used fluid-solid interactions: the density-based interaction and the pseudopotential-based interaction, as well as a modified pseudopotential-based interaction formulated in the present paper, are numerically investigated and compared in terms of the achievable contact angles, the maximum and the minimum densities, and the spurious currents. It is found that the pseudopotential-based interaction works well for simulating small static (liquid) contact angles, however, is unable to reproduce static contact angles close to 180 degrees. Meanwhile, it is found that the proposed modif...
Conjugate heat transfer with the entropic lattice Boltzmann method.
Pareschi, G; Frapolli, N; Chikatamarla, S S; Karlin, I V
2016-07-01
A conjugate heat-transfer model is presented based on the two-population entropic lattice Boltzmann method. The present approach relies on the extension of Grad's boundary conditions to the two-population model for thermal flows, as well as on the appropriate exact conjugate heat-transfer condition imposed at the fluid-solid interface. The simplicity and efficiency of the lattice Boltzmann method (LBM), and in particular of the entropic multirelaxation LBM, are retained in the present approach, thus enabling simulations of turbulent high Reynolds number flows and complex wall boundaries. The model is validated by means of two-dimensional parametric studies of various setups, including pure solid conduction, conjugate heat transfer with a backward-facing step flow, and conjugate heat transfer with the flow past a circular heated cylinder. Further validations are performed in three dimensions for the case of a turbulent flow around a heated mounted cube.
Quantitative and qualitative Kac's chaos on the Boltzmann's sphere
Carrapatoso, Kleber
2012-01-01
We investigate the construction of chaotic probability measures on the Boltzmann's sphere, which is the state space of the stochastic process of a many-particle system undergoing a dynamics preserving energy and momentum. Firstly, based on a version of the local Central Limit Theorem (or Berry-Essenn theorem), we construct a sequence of probabilities that is Kac chaotic and we prove a quantitative rate of convergence. Then, we investigate a stronger notion of chaos, namely entropic chaos introduced in \\cite{CCLLV}, and we prove, with quantitative rate, that this same sequence is also entropically chaotic. Furthermore, we investigate more general class of probability measures on the Boltzmann's sphere. Using the HWI inequality we prove that a Kac chaotic probability with bounded Fisher's information is entropically chaotic and we give a quantitative rate. We also link different notions of chaos, proving that Fisher's information chaos, introduced in \\cite{HaurayMischler}, is stronger than entropic chaos, which...
Learning Feature Hierarchies with Centered Deep Boltzmann Machines
Montavon, Grégoire; Müller, Klaus-Robert
2012-01-01
Deep Boltzmann machines are in principle powerful models for extracting the hierarchical structure of data. Unfortunately, attempts to train layers jointly (without greedy layer-wise pretraining) have been largely unsuccessful. We propose a modification of the learning algorithm that initially recenters the output of the activation functions to zero. This modification leads to a better conditioned Hessian and thus makes learning easier. We test the algorithm on real data and demonstrate that ...
Multi-reflection boundary conditions for lattice Boltzmann models
d´Humiéres, D.; Ginzburg, I
2002-01-01
We present a unified approach of several boundary conditions for lattice Boltzmann models. Its general framework is a generalization of previously introduced schemes such as the bounce-back rule, linear or quadratic interpolations, etc. The objectives are two fold: first to give theoretical tools to study the existing boundary conditions and their corresponding accuracy; secondly to design formally third- order accurate boundary conditions for general flows. Using these boundary conditions, C...
Volume-Based Fabric Tensors through Lattice-Boltzmann Simulations
Moreno, Rodrigo; Smedby, Örjan
2014-01-01
This paper introduces a new methodology to compute fabric tensors from computational fluid dynamics simulations performed through the lattice-Boltzmann method. Trabecular bone is modeled as a pipeline where a synthetic viscous fluid can flow from a single source located at the center of a spherical region of interest toward its boundaries. Two fabric tensors are computed from local velocities at the steady state estimated from the simulations, a tortuosity and a normalized tortuosity tensor.T...
Multi-component lattice-Boltzmann model with interparticle interaction
Shan, Xiaowen; Doolen, Gary
1995-01-01
A previously proposed [X. Shan and H. Chen, Phys. Rev. E {\\bf 47}, 1815, (1993)] lattice Boltzmann model for simulating fluids with multiple components and interparticle forces is described in detail. Macroscopic equations governing the motion of each component are derived by using Chapman-Enskog method. The mutual diffusivity in a binary mixture is calculated analytically and confirmed by numerical simulation. The diffusivity is generally a function of the concentrations of the two component...
A Lattice Boltzmann model for diffusion of binary gas mixtures
Bennett, Sam
2010-01-01
This thesis describes the development of a Lattice Boltzmann (LB) model for a binary gas mixture. Specifically, channel flow driven by a density gradient with diffusion slip occurring at the wall is studied in depth. The first part of this thesis sets the foundation for the multi-component model used in the subsequent chapters. Commonly used single component LB methods use a non-physical equation of state, in which the relationship between pressure and density varies according to the sca...
Lattice Boltzmann Method for mixtures at variable Schmidt number
Monteferrante, Michele; Melchionna, Simone; Marconi, Umberto Marini Bettolo
2015-01-01
When simulating multicomponent mixtures via the Lattice Boltzmann Method, it is desirable to control the mutual diffusivity between species while maintaining the viscosity of the solution fixed. This goal is herein achieved by a modification of the multicomponent Bhatnagar-Gross-Krook (BGK) evolution equations by introducing two different timescales for mass and momentum diffusion. Diffusivity is thus controlled by an effective drag force acting between species. Numerical simulations confirm ...
Derivation of anisotropic dissipative fluid dynamics from the Boltzmann equation
Molnar, E.; Niemi, H.; Rischke, D. H.
2016-01-01
Fluid-dynamical equations of motion can be derived from the Boltzmann equation in terms of an expansion around a single-particle distribution function which is in local thermodynamical equilibrium, i.e., isotropic in momentum space in the rest frame of a fluid element. However, in situations where the single-particle distribution function is highly anisotropic in momentum space, such as the initial stage of heavy-ion collisions at relativistic energies, such an expansion is bound to break dow...
Acoustic levitation and the Boltzmann-Ehrenfest principle
Putterman, S.; Rudnick, Joseph; Barmatz, M.
1989-01-01
The Boltzmann-Ehrenfest principle of adiabatic invariance relates the acoustic potential acting on a sample positioned in a single-mode cavity to the shift in resonant frequency caused by the presence of this sample. This general and simple relation applies to samples and cavities of arbitrary shape, dimension, and compressibility. Positioning forces and torques can, therefore, be determined from straightforward measurements of frequency shifts. Applications to the Rayleigh disk phenomenon and levitated cylinders are presented.
On the Spectral Problems for the Discrete Boltzmann Models
Institute of Scientific and Technical Information of China (English)
Aq Kwang-Hua Chu; J. FANG Jing
2000-01-01
The discrete Boltzmann models are used to study the spectral problems related to the one-dimensional plane wave propaogation in monatomic gases which are fundamental in the nonequilibrium tatistical thermodynamics. The results show that the 8-velocity model can only describe the propagation of the diffusion mode (entropy wave) in the intermediate Knudsen number regime. The 4- and 6-velocity models, instead, can describe the propagation of sound modes quite well, after comparison with the continuum-mechanical results.
Topological interactions in a Boltzmann-type framework
Blanchet, Adrien; Degond, Pierre
2015-01-01
We consider a finite number of particles characterised by their positions and velocities. At random times a randomly chosen particle, the follower, adopts the velocity of another particle, the leader. The follower chooses its leader according to the proximity rank of the latter with respect to the former. We study the limit of a system size going to infinity and, under the assumption of propagation of chaos, show that the limit equation is akin to the Boltzmann equation. However , it exhibits...
Average Contrastive Divergence for Training Restricted Boltzmann Machines
Xuesi Ma; Xiaojie Wang
2016-01-01
This paper studies contrastive divergence (CD) learning algorithm and proposes a new algorithm for training restricted Boltzmann machines (RBMs). We derive that CD is a biased estimator of the log-likelihood gradient method and make an analysis of the bias. Meanwhile, we propose a new learning algorithm called average contrastive divergence (ACD) for training RBMs. It is an improved CD algorithm, and it is different from the traditional CD algorithm. Finally, we obtain some experimental resul...
Discrete Boltzmann model of shallow water equations with polynomial equilibria
Meng, Jianping; Emerson, David R; Peng, Yong; Zhang, Jianmin
2016-01-01
A hierarchy of discrete Boltzmann model is proposed for simulating shallow water flows. By using the Hermite expansion and Gauss-Hermite quadrature, the conservation laws are automatically satisfied without extra effort. Moreover, the expansion order and quadrature can be chosen flexibly according to the problem for striking the balance of accuracy and efficiency. The models are then tested using the classical one-dimensional dam-breaking problem, and successes are found for both supercritical and subcritical flows.
The Karlqvist approximation revisited
Tannous, C.
2015-01-01
The Karlqvist approximation signaling the historical beginning of magnetic recording head theory is reviewed and compared to various approaches progressing from Green, Fourier, Conformal mapping that obeys the Sommerfeld edge condition at angular points and leads to exact results.
Approximations in Inspection Planning
DEFF Research Database (Denmark)
Engelund, S.; Sørensen, John Dalsgaard; Faber, M. H.;
2000-01-01
Planning of inspections of civil engineering structures may be performed within the framework of Bayesian decision analysis. The effort involved in a full Bayesian decision analysis is relatively large. Therefore, the actual inspection planning is usually performed using a number of approximations....... One of the more important of these approximations is the assumption that all inspections will reveal no defects. Using this approximation the optimal inspection plan may be determined on the basis of conditional probabilities, i.e. the probability of failure given no defects have been found...... by the inspection. In this paper the quality of this approximation is investigated. The inspection planning is formulated both as a full Bayesian decision problem and on the basis of the assumption that the inspection will reveal no defects....
Directory of Open Access Journals (Sweden)
Malvina Baica
1985-01-01
Full Text Available The author uses a new modification of Jacobi-Perron Algorithm which holds for complex fields of any degree (abbr. ACF, and defines it as Generalized Euclidean Algorithm (abbr. GEA to approximate irrationals.
Approximation Behooves Calibration
DEFF Research Database (Denmark)
da Silva Ribeiro, André Manuel; Poulsen, Rolf
2013-01-01
Calibration based on an expansion approximation for option prices in the Heston stochastic volatility model gives stable, accurate, and fast results for S&P500-index option data over the period 2005–2009.......Calibration based on an expansion approximation for option prices in the Heston stochastic volatility model gives stable, accurate, and fast results for S&P500-index option data over the period 2005–2009....
Gautschi, Walter; Rassias, Themistocles M
2011-01-01
Approximation theory and numerical analysis are central to the creation of accurate computer simulations and mathematical models. Research in these areas can influence the computational techniques used in a variety of mathematical and computational sciences. This collection of contributed chapters, dedicated to renowned mathematician Gradimir V. Milovanovia, represent the recent work of experts in the fields of approximation theory and numerical analysis. These invited contributions describe new trends in these important areas of research including theoretic developments, new computational alg
Diophantine approximations on fractals
Einsiedler, Manfred; Shapira, Uri
2009-01-01
We exploit dynamical properties of diagonal actions to derive results in Diophantine approximations. In particular, we prove that the continued fraction expansion of almost any point on the middle third Cantor set (with respect to the natural measure) contains all finite patterns (hence is well approximable). Similarly, we show that for a variety of fractals in [0,1]^2, possessing some symmetry, almost any point is not Dirichlet improvable (hence is well approximable) and has property C (after Cassels). We then settle by similar methods a conjecture of M. Boshernitzan saying that there are no irrational numbers x in the unit interval such that the continued fraction expansions of {nx mod1 : n is a natural number} are uniformly eventually bounded.
Dukkipati, Ambedkar; Murty, Narasimha M; Bhatnagar, Shalabh
2004-01-01
Boltzmann selection is an important selection mechanism in evolutionary algorithms as it has theoretical properties which help in theoretical analysis. However, Boltzmann selection is not used in practice because a good annealing schedule for the `inverse temperature' parameter is lacking. In this paper we propose a Cauchy annealing schedule for Boltzmann selection scheme based on a hypothesis that selection-strength should increase as evolutionary process goes on and distance between two sel...
Global Solutions of the Boltzmann Equation Over {{R}^D} Near Global Maxwellians with Small Mass
Bardos, Claude; Gamba, Irene M.; Golse, François; Levermore, C. David
2016-09-01
We study the dynamics defined by the Boltzmann equation set in the Euclidean space {{R}^D} in the vicinity of global Maxwellians with finite mass. A global Maxwellian is a special solution of the Boltzmann equation for which the collision integral vanishes identically. In this setting, the dispersion due to the advection operator quenches the dissipative effect of the Boltzmann collision integral. As a result, the large time limit of solutions of the Boltzmann equation in this regime is given by noninteracting, freely transported states and can be described with the tools of scattering theory.
Approximate and Incomplete Factorizations
Chan, T.F.; Vorst, H.A. van der
2001-01-01
In this chapter, we give a brief overview of a particular class of preconditioners known as incomplete factorizations. They can be thought of as approximating the exact LU factorization of a given matrix A (e.g. computed via Gaussian elimination) by disallowing certain ll-ins. As opposed to other PD
Prestack wavefield approximations
Alkhalifah, Tariq
2013-09-01
The double-square-root (DSR) relation offers a platform to perform prestack imaging using an extended single wavefield that honors the geometrical configuration between sources, receivers, and the image point, or in other words, prestack wavefields. Extrapolating such wavefields, nevertheless, suffers from limitations. Chief among them is the singularity associated with horizontally propagating waves. I have devised highly accurate approximations free of such singularities which are highly accurate. Specifically, I use Padé expansions with denominators given by a power series that is an order lower than that of the numerator, and thus, introduce a free variable to balance the series order and normalize the singularity. For the higher-order Padé approximation, the errors are negligible. Additional simplifications, like recasting the DSR formula as a function of scattering angle, allow for a singularity free form that is useful for constant-angle-gather imaging. A dynamic form of this DSR formula can be supported by kinematic evaluations of the scattering angle to provide efficient prestack wavefield construction. Applying a similar approximation to the dip angle yields an efficient 1D wave equation with the scattering and dip angles extracted from, for example, DSR ray tracing. Application to the complex Marmousi data set demonstrates that these approximations, although they may provide less than optimal results, allow for efficient and flexible implementations. © 2013 Society of Exploration Geophysicists.
White, Martin
2014-01-01
This year marks the 100th anniversary of the birth of Yakov Zel'dovich. Amongst his many legacies is the Zel'dovich approximation for the growth of large-scale structure, which remains one of the most successful and insightful analytic models of structure formation. We use the Zel'dovich approximation to compute the two-point function of the matter and biased tracers, and compare to the results of N-body simulations and other Lagrangian perturbation theories. We show that Lagrangian perturbation theories converge well and that the Zel'dovich approximation provides a good fit to the N-body results except for the quadrupole moment of the halo correlation function. We extend the calculation of halo bias to 3rd order and also consider non-local biasing schemes, none of which remove the discrepancy. We argue that a part of the discrepancy owes to an incorrect prediction of inter-halo velocity correlations. We use the Zel'dovich approximation to compute the ingredients of the Gaussian streaming model and show that ...
DEFF Research Database (Denmark)
Madsen, Rasmus Elsborg
2005-01-01
The Dirichlet compound multinomial (DCM), which has recently been shown to be well suited for modeling for word burstiness in documents, is here investigated. A number of conceptual explanations that account for these recent results, are provided. An exponential family approximation of the DCM that...
Discrete Ordinates Approximations to the First- and Second-Order Radiation Transport Equations
Fan, W C; Powell, J L
2002-01-01
The conventional discrete ordinates approximation to the Boltzmann transport equation can be described in a matrix form. Specifically, the within-group scattering integral can be represented by three components: a moment-to-discrete matrix, a scattering cross-section matrix and a discrete-to-moment matrix. Using and extending these entities, we derive and summarize the matrix representations of the second-order transport equations.
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.
Three-dimensional lattice Boltzmann model for compressible flows.
Sun, Chenghai; Hsu, Andrew T
2003-07-01
A three-dimensional compressible lattice Boltzmann model is formulated on a cubic lattice. A very large particle-velocity set is incorporated in order to enable a greater variation in the mean velocity. Meanwhile, the support set of the equilibrium distribution has only six directions. Therefore, this model can efficiently handle flows over a wide range of Mach numbers and capture shock waves. Due to the simple form of the equilibrium distribution, the fourth-order velocity tensors are not involved in the formulation. Unlike the standard lattice Boltzmann model, no special treatment is required for the homogeneity of fourth-order velocity tensors on square lattices. The Navier-Stokes equations were recovered, using the Chapman-Enskog method from the Bhatnagar-Gross-Krook (BGK) lattice Boltzmann equation. The second-order discretization error of the fluctuation velocity in the macroscopic conservation equation was eliminated by means of a modified collision invariant. The model is suitable for both viscous and inviscid compressible flows with or without shocks. Since the present scheme deals only with the equilibrium distribution that depends only on fluid density, velocity, and internal energy, boundary conditions on curved wall are easily implemented by an extrapolation of macroscopic variables. To verify the scheme for inviscid flows, we have successfully simulated a three-dimensional shock-wave propagation in a box and a normal shock of Mach number 10 over a wedge. As an application to viscous flows, we have simulated a flat plate boundary layer flow, flow over a cylinder, and a transonic flow over a NACA0012 airfoil cascade.
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.
Prestack traveltime approximations
Alkhalifah, Tariq Ali
2011-01-01
Most prestack traveltime relations we tend work with are based on homogeneous (or semi-homogenous, possibly effective) media approximations. This includes the multi-focusing or double square-root (DSR) and the common reflection stack (CRS) equations. Using the DSR equation, I analyze the associated eikonal form in the general source-receiver domain. Like its wave-equation counterpart, it suffers from a critical singularity for horizontally traveling waves. As a result, I derive expansion based solutions of this eikonal based on polynomial expansions in terms of the reflection and dip angles in a generally inhomogenous background medium. These approximate solutions are free of singularities and can be used to estimate travetimes for small to moderate offsets (or reflection angles) in a generally inhomogeneous medium. A Marmousi example demonstrates the usefulness of the approach. © 2011 Society of Exploration Geophysicists.
Appendix: Chapman-Enskog Expansion in the Lattice Boltzmann Method
Li, Jun
2015-01-01
The Chapman-Enskog expansion was used in the lattice Boltzmann method (LBM) to derive a Navier-Stokes-like equation and a formula was obtained to correlate the LBM model parameters to the kinematic viscosity implicitly implemented in LBM simulations. The obtained correlation formula usually works as long as the model parameters are carefully selected to make the Mach number and Knudsen number small although the validity of Chapman-Enskog expansion that has a formal definition of time derivative without tangible mathematical sense is not recognized by many mathematicians.
Lattice-Boltzmann Method for Geophysical Plastic Flows
Leonardi, Alessandro; Mendoza, Miller; Herrmann, Hans J
2015-01-01
We explore possible applications of the Lattice-Boltzmann Method for the simulation of geophysical flows. This fluid solver, while successful in other fields, is still rarely used for geotechnical applications. We show how the standard method can be modified to represent free-surface realization of mudflows, debris flows, and in general any plastic flow, through the implementation of a Bingham constitutive model. The chapter is completed by an example of a full-scale simulation of a plastic fluid flowing down an inclined channel and depositing on a flat surface. An application is given, where the fluid interacts with a vertical obstacle in the channel.
LATTICE BOLTZMANN EQUATION MODEL IN THE CORIOLIS FIELD
Institute of Scientific and Technical Information of China (English)
FENG SHI-DE; MAO JIANG-YU; ZHANG QIONG
2001-01-01
In a large-scale field of rotational fluid, various unintelligible and surprising dynamic phenomena are produced due to the effect of the Coriolis force. The lattice Boltzmann equation (LBE) model in the Coriolis field is developed based on previous works.[1-4] Geophysical fluid dynamics equations are derived from the model. Numerical simulations have been made on an ideal atmospheric circulation of the Northern Hemisphere by using the model and they reproduce the Rossby wave motion well. Hence the applicability of the model is verified in both theory and experiment.
Entropy inequality and hydrodynamic limits for the Boltzmann equation.
Saint-Raymond, Laure
2013-12-28
Boltzmann brought a fundamental contribution to the understanding of the notion of entropy, by giving a microscopic formulation of the second principle of thermodynamics. His ingenious idea, motivated by the works of his contemporaries on the atomic nature of matter, consists of describing gases as huge systems of identical and indistinguishable elementary particles. The state of a gas can therefore be described in a statistical way. The evolution, which introduces couplings, loses part of the information, which is expressed by the decay of the so-called mathematical entropy (the opposite of physical entropy!). PMID:24249776
On the Krook-Wu model of the Boltzmann equation
Cornille, H.
1980-08-01
The distribution function of the Krook-Wu model of the nonlinear Boltzmann equation (elastic differential cross sections inversely proportional to the relative speed of the colliding particles) is obtained as a generalized Laguerre polynomial expansion where the only time dependence is provided by the coefficients. In a recent paper M. Barnsley and the present author have shown that these coefficients are recursively determined from the resolution of a nonlinear differential system. Here we explicitly show how to construct the solutions of the Krook-Wu model and study the properties of the corresponding Krook-Wu distribution functions.
An alternative method for simulating particle suspensions using lattice Boltzmann
Santos, Luís Orlando Emerich dos
2011-01-01
In this study, we propose an alternative way to simulate particle suspensions using the lattice Boltzmann method. The main idea is to impose the non-slip boundary condition in the lattice sites located on the particle boundaries. The focus on the lattice sites, instead of the links between them, as done in the more used methods, represents a great simplification in the algorithm. A fully description of the method will be presented, in addition to simulations comparing the proposed method with other methods and, also, with experimental results.
Multi-component lattice-Boltzmann model with interparticle interaction
Shan, X; Shan, Xiaowen; Doolen, Gary
1995-01-01
Abstract: A previously proposed [X. Shan and H. Chen, Phys. Rev. E {\\bf 47}, 1815, (1993)] lattice Boltzmann model for simulating fluids with multiple components and interparticle forces is described in detail. Macroscopic equations governing the motion of each component are derived by using Chapman-Enskog method. The mutual diffusivity in a binary mixture is calculated analytically and confirmed by numerical simulation. The diffusivity is generally a function of the concentrations of the two components but independent of the fluid velocity so that the diffusion is Galilean invariant. The analytically calculated shear kinematic viscosity of this model is also confirmed numerically.
Boltzmann Machines and Denoising Autoencoders for Image Denoising
Cho, Kyunghyun
2013-01-01
Image denoising based on a probabilistic model of local image patches has been employed by various researchers, and recently a deep (denoising) autoencoder has been proposed by Burger et al. [2012] and Xie et al. [2012] as a good model for this. In this paper, we propose that another popular family of models in the field of deep learning, called Boltzmann machines, can perform image denoising as well as, or in certain cases of high level of noise, better than denoising autoencoders. We empiri...
A lattice Boltzmann method for dilute polymer solutions.
Singh, Shiwani; Subramanian, Ganesh; Ansumali, Santosh
2011-06-13
We present a lattice Boltzmann approach for the simulation of non-Newtonian fluids. The method is illustrated for the specific case of dilute polymer solutions. With the appropriate local equilibrium distribution, phase-space dynamics on a lattice, driven by a Bhatnagar-Gross-Krook (BGK) relaxation term, leads to a solution of the Fokker-Planck equation governing the probability density of polymer configurations. Results for the bulk rheological characteristics for steady and start-up shear flow are presented, and compare favourably with those obtained using Brownian dynamics simulations. The new method is less expensive than stochastic simulation techniques, particularly in the range of small to moderate Weissenberg numbers (Wi).
Relativistic Rotating Boltzmann Gas Using the Tetrad Formalism
Directory of Open Access Journals (Sweden)
Ambrus Victor E.
2015-12-01
Full Text Available We consider an application of the tetrad formalism introduced by Cardall et al. [Phys. Rev. D 88 (2013 023011] to the problem of a rigidly rotating relativistic gas in thermal equilibrium and discuss the possible applications of this formalism to rel- ativistic lattice Boltzmann simulations. We present in detail the transformation to the comoving frame, the choice of tetrad, as well as the explicit calculation and analysis of the components of the equilibrium particle ow four-vector and of the equilibrium stress-energy tensor.
Numerical Poisson-Boltzmann Model for Continuum Membrane Systems.
Botello-Smith, Wesley M; Liu, Xingping; Cai, Qin; Li, Zhilin; Zhao, Hongkai; Luo, Ray
2013-01-01
Membrane protein systems are important computational research topics due to their roles in rational drug design. In this study, we developed a continuum membrane model utilizing a level set formulation under the numerical Poisson-Boltzmann framework within the AMBER molecular mechanics suite for applications such as protein-ligand binding affinity and docking pose predictions. Two numerical solvers were adapted for periodic systems to alleviate possible edge effects. Validation on systems ranging from organic molecules to membrane proteins up to 200 residues, demonstrated good numerical properties. This lays foundations for sophisticated models with variable dielectric treatments and second-order accurate modeling of solvation interactions.
Lattice Boltzmann Simulation for the Spiral Wave Dynamics
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
We investigated the dynamics of the simple spiral waves of the Selkov reaction-diffusion system with the Lattice Boltzmann method. The results of computer simulation lead to the conclusion that the trajectory of the spiral tip is a small circle, the wavelength and the period decay exponentially when the value of parameter b increases; and the relation between the wavelength and the period is A oc T1 , which is qualitatively the same as that obtained by Ou-Yang Qi from Belousov-Zhabotinsky reaction system.
Boltzmann-conserving classical dynamics in quantum time-correlation functions: Matsubara dynamics
Hele, Timothy J H; Muolo, Andrea; Althorpe, Stuart C
2015-01-01
We show that a single change in the derivation of the linearized semiclassical-initial value representation (LSC-IVR or classical Wigner approximation) results in a classical dynamics which conserves the quantum Boltzmann distribution. We rederive the (standard) LSC-IVR approach by writing the (exact) quantum time-correlation function in terms of the normal modes of a free ring-polymer (i.e. a discrete imaginary-time Feynman path), taking the limit that the number of polymer beads $N \\to \\infty$, such that the lowest normal-mode frequencies take their Matsubara values. The change we propose is to truncate the quantum Liouvillian, not explicitly in powers of $\\hbar^2$ at $\\hbar^0$ (which gives back the standard LSC-IVR approximation), but in the normal-mode derivatives corresponding to the lowest Matsubara frequencies. The resulting Matsubara dynamics is inherently classical (since all terms $\\mathcal{O}\\left(\\hbar^{2}\\right)$ disappear from the Matsubara Liouvillian in the limit $N \\to \\infty$), and conserves...
Energy Technology Data Exchange (ETDEWEB)
Chalasani, P.; Saias, I. [Los Alamos National Lab., NM (United States); Jha, S. [Carnegie Mellon Univ., Pittsburgh, PA (United States)
1996-04-08
As increasingly large volumes of sophisticated options (called derivative securities) are traded in world financial markets, determining a fair price for these options has become an important and difficult computational problem. Many valuation codes use the binomial pricing model, in which the stock price is driven by a random walk. In this model, the value of an n-period option on a stock is the expected time-discounted value of the future cash flow on an n-period stock price path. Path-dependent options are particularly difficult to value since the future cash flow depends on the entire stock price path rather than on just the final stock price. Currently such options are approximately priced by Monte carlo methods with error bounds that hold only with high probability and which are reduced by increasing the number of simulation runs. In this paper the authors show that pricing an arbitrary path-dependent option is {number_sign}-P hard. They show that certain types f path-dependent options can be valued exactly in polynomial time. Asian options are path-dependent options that are particularly hard to price, and for these they design deterministic polynomial-time approximate algorithms. They show that the value of a perpetual American put option (which can be computed in constant time) is in many cases a good approximation to the value of an otherwise identical n-period American put option. In contrast to Monte Carlo methods, the algorithms have guaranteed error bounds that are polynormally small (and in some cases exponentially small) in the maturity n. For the error analysis they derive large-deviation results for random walks that may be of independent interest.
Approximations to Euler's constant
International Nuclear Information System (INIS)
We study a problem of finding good approximations to Euler's constant γ=lim→∞ Sn, where Sn = Σk=Ln (1)/k-log(n+1), by linear forms in logarithms and harmonic numbers. In 1995, C. Elsner showed that slow convergence of the sequence Sn can be significantly improved if Sn is replaced by linear combinations of Sn with integer coefficients. In this paper, considering more general linear transformations of the sequence Sn we establish new accelerating convergence formulae for γ. Our estimates sharpen and generalize recent Elsner's, Rivoal's and author's results. (author)
Finite elements and approximation
Zienkiewicz, O C
2006-01-01
A powerful tool for the approximate solution of differential equations, the finite element is extensively used in industry and research. This book offers students of engineering and physics a comprehensive view of the principles involved, with numerous illustrative examples and exercises.Starting with continuum boundary value problems and the need for numerical discretization, the text examines finite difference methods, weighted residual methods in the context of continuous trial functions, and piecewise defined trial functions and the finite element method. Additional topics include higher o
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.
Lattice Boltzmann Simulation for Complex Flow in a Solar Wall
Institute of Scientific and Technical Information of China (English)
CHEN Rou; Shao Jiu-Gu; ZHENG You-Qu; YU Hui-Dan; XU You-Sheng
2013-01-01
In this letter,we present a lattice Boltzmann simulation for complex flow in a solar wall system which includes porous media flow and heat transfer,specifically for solar energy utilization through an unglazed transpired solar air collector (UTC).Besides the lattice Boltzmann equation (LBE) for time evolution of particle distribution function for fluid field,we introduce an analogy,LBE for time evolution of distribution function for temperature.Both temperature fields of fluid (air) and solid (porous media) are modeled.We study the effects of fan velocity,solar radiation intensity,porosity,etc.on the thermal performance of the UTC.In general,our simulation results are in good agreement with what in literature.With the current system setting,both fan velocity and solar radiation intensity have significant effect on the thermal performance of the UTC.However,it is shown that the porosity has negligible effect on the heat collector indicating the current system setting might not be realistic.Further examinations of thermal performance in different UTC systems are ongoing.The results are expected to present in near future.
Avoiding Boltzmann Brain domination in holographic dark energy models
Directory of Open Access Journals (Sweden)
R. Horvat
2015-11-01
Full Text Available In a spatially infinite and eternal universe approaching ultimately a de Sitter (or quasi-de Sitter regime, structure can form by thermal fluctuations as such a space is thermal. The models of Dark Energy invoking holographic principle fit naturally into such a category, and spontaneous formation of isolated brains in otherwise empty space seems the most perplexing, creating the paradox of Boltzmann Brains (BB. It is thus appropriate to ask if such models can be made free from domination by Boltzmann Brains. Here we consider only the simplest model, but adopt both the local and the global viewpoint in the description of the Universe. In the former case, we find that if a dimensionless model parameter c, which modulates the Dark Energy density, lies outside the exponentially narrow strip around the most natural c=1 line, the theory is rendered BB-safe. In the latter case, the bound on c is exponentially stronger, and seemingly at odds with those bounds on c obtained from various observational tests.
Fault diagnosis via neural networks: The Boltzmann machine
International Nuclear Information System (INIS)
The Boltzmann machine is a general-purpose artificial neural network that can be used as an associative memory as well as a mapping tool. The usual information entropy is introduced, and a network energy function is suitably defined. The network's training procedure is based on the simulated annealing during which a combination of energy minimization and entropy maximization is achieved. An application in the nuclear reactor field is presented in which the Boltzmann input-output machine is used to detect and diagnose a pipe break in a simulated auxiliary feedwater system feeding two coupled steam generators. The break may occur on either the hot or the cold leg of any of the two steam generators. The binary input data to the network encode only the trends of the thermohydraulic signals so that the network is actually a polarity device. The results indicate that the trained neural network is actually capable of performing its task. The method appears to be robust enough so that it may also be applied with success in the presence of substantial amounts of noise that cause the network to be fed with wrong signals
Three-dimensional lattice Boltzmann model for electrodynamics.
Mendoza, M; Muñoz, J D
2010-11-01
In this paper we introduce a three-dimensional Lattice-Boltzmann model that recovers in the continuous limit the Maxwell equations in materials. In order to build conservation equations with antisymmetric tensors, like the Faraday law, the model assigns four auxiliary vectors to each velocity vector. These auxiliary vectors, when combined with the distribution functions, give the electromagnetic fields. The evolution is driven by the usual Bhatnager-Gross-Krook (BGK) collision rule, but with a different form for the equilibrium distribution functions. This lattice Bhatnager-Gross-Krook (LBGK) model allows us to consider for both dielectrics and conductors with realistic parameters, and therefore it is adequate to simulate the most diverse electromagnetic problems, like the propagation of electromagnetic waves (both in dielectric media and in waveguides), the skin effect, the radiation pattern of a small dipole antenna and the natural frequencies of a resonant cavity, all with 2% accuracy. Actually, it shows to be one order of magnitude faster than the original Finite-difference time-domain (FDTD) formulation by Yee to reach the same accuracy. It is, therefore, a valuable alternative to simulate electromagnetic fields and opens lattice Boltzmann for a broad spectrum of new applications in electrodynamics.
Wall Orientation and Shear Stress in the Lattice Boltzmann Model
Matyka, Maciej; Mirosław, Łukasz
2013-01-01
The wall shear stress is a quantity of profound importance for clinical diagnosis of artery diseases. The lattice Boltzmann is an easily parallelizable numerical method of solving the flow problems, but it suffers from errors of the velocity field near the boundaries which leads to errors in the wall shear stress and normal vectors computed from the velocity. In this work we present a simple formula to calculate the wall shear stress in the lattice Boltzmann model and propose to compute wall normals, which are necessary to compute the wall shear stress, by taking the weighted mean over boundary facets lying in a vicinity of a wall element. We carry out several tests and observe an increase of accuracy of computed normal vectors over other methods in two and three dimensions. Using the scheme we compute the wall shear stress in an inclined and bent channel fluid flow and show a minor influence of the normal on the numerical error, implying that that the main error arises due to a corrupted velocity field near ...
Avoiding Boltzmann Brain domination in holographic dark energy models
Horvat, R.
2015-11-01
In a spatially infinite and eternal universe approaching ultimately a de Sitter (or quasi-de Sitter) regime, structure can form by thermal fluctuations as such a space is thermal. The models of Dark Energy invoking holographic principle fit naturally into such a category, and spontaneous formation of isolated brains in otherwise empty space seems the most perplexing, creating the paradox of Boltzmann Brains (BB). It is thus appropriate to ask if such models can be made free from domination by Boltzmann Brains. Here we consider only the simplest model, but adopt both the local and the global viewpoint in the description of the Universe. In the former case, we find that if a dimensionless model parameter c, which modulates the Dark Energy density, lies outside the exponentially narrow strip around the most natural c = 1 line, the theory is rendered BB-safe. In the latter case, the bound on c is exponentially stronger, and seemingly at odds with those bounds on c obtained from various observational tests.
Avoiding Boltzmann Brain domination in holographic dark energy models
Horvat, R
2015-01-01
In a spatially infinite and eternal universe approaching ultimately a de Sitter (or quasi-de Sitter) regime, structure can form by thermal fluctuations as such a space is thermal. The models of Dark Energy invoking holographic principle fit naturally into such a category, and spontaneous formation of isolated brains in otherwise empty space seems the most perplexing, creating the paradox of Boltzmann Brains (BB). It is thus appropriate to ask if such models can be made free from domination by Boltzmann Brains. Here we consider only the simplest model, but adopt both the local and the global viewpoint in the description of the Universe. In the former case, we find that if a parameter $c$, which modulates the Dark Energy density, lies outside the exponentially narrow strip around the most natural $c = 1$ line, the theory is rendered BB-safe. In the later case, the bound on $c$ is exponentially stronger, and seemingly at odds with those bounds on $c$ obtained from various observational tests.
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...
L2-stability of the Vlasov-Maxwell-Boltzmann system near global Maxwellians
International Nuclear Information System (INIS)
We present a L2-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 L2-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 L2-stability theory of the Boltzmann equation near global Maxwellians,” (submitted) and Ha, S.-Y., Yang, X.-F., and Yun, S.-B., “L2 stability theory of the Boltzmann equation near a global Maxwellian,” Arch. Ration. Mech. Anal. 197, 657–688 (2010)] on the L2-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 L2-stability estimate. This is the first result on the L2-stability of the Boltzmann equation coupled with self-consistent field equations in three dimensions
Haubold, H J; Saxena, R K
2004-01-01
Classical statistical mechanics of macroscopic systems in equilibrium is based on Boltzmann's principle. Tsallis has proposed a generalization of Boltzmann-Gibbs statistics. Its relation to dynamics and nonextensivity of statistical systems are matters of intense investigation and debate. This essay review has been prepared at the occasion of awarding the 'Mexico Prize for Science and Technology 2003'to Professor Constantino Tsallis from the Brazilian Center for Research in Physics.
Boltzmann and Einstein: Statistics and dynamics –An unsolved problem
Indian Academy of Sciences (India)
E G D Cohen
2005-05-01
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, arguing that a statistical description of a system should be based on the dynamics of the system. This opened the way, especially for complex systems, for other than Boltzmann statistics. The first non-Boltzmann statistics, not based on dynamics though, was proposed by Tsallis. A generalization of Tsallis' statistics as a special case of a new class of superstatistics, based on Einstein's criticism of Boltzmann, is discussed. It seems that perhaps a combination of dynamics and statistics is necessary to describe systems with complicated dynamics.
The Compact Approximation Property does not imply the Approximation Property
Willis, George A.
1992-01-01
It is shown how to construct, given a Banach space which does not have the approximation property, another Banach space which does not have the approximation property but which does have the compact approximation property.
Energy Technology Data Exchange (ETDEWEB)
Toelke, J.
2001-07-01
The first part of this work is concerned with the development of methodological foundations for the computer simulation of two-phase flows like gas-liquid-mixtures in complex, three-dimensional structures. The basic numerical approach is the Lattice-Boltzmann scheme which is very suitable for this class of problems. After the approach is verified using standard test cases, the method is applied to complex engineering problems. The most important application is the simulation of the two-phase flow (air/water) in a laboratory-scale biofilm reactor for wastewater treatment. The second part of the work deals with the development of efficient numerical methods for the stationary discrete Boltzmann equations. They are discretized by finite differences on uniform and non-uniform grids and fast solvers are applied to the resulting algebraic system of equations. Also a multigrid approach is developed and examined. For typical problems like boundary-layer and driven cavity flow a considerable gain in computing time is achieved. (orig.)
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.
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. PMID:27627290
Santillan, Arturo O; Cutanda-Henríquez, Vicente
2008-11-01
An investigation on the resonance frequency shift for a plane-wave mode in a cylindrical cavity produced by a rigid sphere is reported in this paper. This change of the resonance frequency has been previously considered as a cause of oscillational instabilities in single-mode acoustic levitation devices. It is shown that the use of the Boltzmann-Ehrenfest principle of adiabatic invariance allows the derivation of an expression for the resonance frequency shift in a simpler and more direct way than a method based on a Green's function reported in literature. The position of the sphere can be any point along the axis of the cavity. Obtained predictions of the resonance frequency shift with the deduced equation agree quite well with numerical simulations based on the boundary element method. The results are also confirmed by experiments. The equation derived from the Boltzmann-Ehrenfest principle appears to be more general, and for large spheres, it gives a better approximation than the equation previously reported. PMID:19045761
Interacting boson approximation
International Nuclear Information System (INIS)
Lectures notes on the Interacting Boson Approximation are given. Topics include: angular momentum tensors; properties of T/sub i//sup (n)/ matrices; T/sub i//sup (n)/ matrices as Clebsch-Gordan coefficients; construction of higher rank tensors; normalization: trace of products of two s-rank tensors; completeness relation; algebra of U(N); eigenvalue of the quadratic Casimir operator for U(3); general result for U(N); angular momentum content of U(3) representation; p-Boson model; Hamiltonian; quadrupole transitions; S,P Boson model; expectation value of dipole operator; S-D model: U(6); quadratic Casimir operator; an O(5) subgroup; an O(6) subgroup; properties of O(5) representations; quadratic Casimir operator; quadratic Casimir operator for U(6); decomposition via SU(5) chain; a special O(3) decomposition of SU(3); useful identities; a useful property of D/sub αβγ/(α,β,γ = 4-8) as coupling coefficients; explicit construction of T/sub x//sup (2)/ and d/sub αβγ/; D-coefficients; eigenstates of T3; and summary of T = 2 states
Prestack traveltime approximations
Alkhalifah, Tariq Ali
2012-05-01
Many of the explicit prestack traveltime relations used in practice are based on homogeneous (or semi-homogenous, possibly effective) media approximations. This includes the multifocusing, based on the double square-root (DSR) equation, and the common reflection stack (CRS) approaches. Using the DSR equation, I constructed the associated eikonal form in the general source-receiver domain. Like its wave-equation counterpart, it suffers from a critical singularity for horizontally traveling waves. As a result, I recasted the eikonal in terms of the reflection angle, and thus, derived expansion based solutions of this eikonal in terms of the difference between the source and receiver velocities in a generally inhomogenous background medium. The zero-order term solution, corresponding to ignoring the lateral velocity variation in estimating the prestack part, is free of singularities and can be used to estimate traveltimes for small to moderate offsets (or reflection angles) in a generally inhomogeneous medium. The higher-order terms include limitations for horizontally traveling waves, however, we can readily enforce stability constraints to avoid such singularities. In fact, another expansion over reflection angle can help us avoid these singularities by requiring the source and receiver velocities to be different. On the other hand, expansions in terms of reflection angles result in singularity free equations. For a homogenous background medium, as a test, the solutions are reasonably accurate to large reflection and dip angles. A Marmousi example demonstrated the usefulness and versatility of the formulation. © 2012 Society of Exploration Geophysicists.
Determination of the Boltzmann Constant Using the Differential - Cylindrical Procedure
Feng, X J; Lin, H; Gillis, K A; Moldover, M R
2015-01-01
We report in this paper the progresses on the determination of the Boltzmann constant using the acoustic gas thermometer (AGT) of fixed-length cylindrical cavities. First, we present the comparison of the molar masses of pure argon gases through comparing speeds of sound of gases. The procedure is independent from the methodology by Gas Chromatography-Mass Spectrometry (GC-MS). The experimental results show good agreement between both methods. The comparison offers an independent inspection of the analytical results by GC-MS. Second, we present the principle of the novel differential-cylindrical procedure based on the AGT of two fixed-length cavities. The deletion mechanism for some major perturbations is analyzed for the new procedure. The experimental results of the differential-cylindrical procedure demonstrate some major improvements on the first, second acoustic and third virial coefficients, and the excess half-widths. The three acoustic virial coefficients agree well with the stated-of-the-art experime...
Spreading Dynamics of Nanodrops: A Lattice Boltzmann Study
Gross, Markus
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 non-ideal lattice Boltzmann method [Gross et al., J. Stat. Mech., P03030 (2011)] for the study of this issue. Confirming the predictions of Davidovich and coworkers [PRL 95, 244905 (2005)], we provide the first independent evidence for the existence of an asymptotic, self-similar noise-driven spreading regime in both two- and three-dimensional geometry. The cross over from the deterministic Tanner's law, where the drop's base radius $b$ grows (in 3D) with time as $b \\sim t^{1/10}$ and the noise dominated regime where $b \\sim t^{1/6}$ is also observed by tuning the strength of thermal noise.
Boltzmann Equation Solver Adapted to Emergent Chemical Non-equilibrium
Birrell, Jeremiah
2014-01-01
We present a novel method to solve the spatially homogeneous and isotropic relativistic Boltzmann equation. We employ a basis set of orthogonal polynomials dynamically adapted to allow emergence of chemical non-equilibrium. Two time dependent parameters characterize the set of orthogonal polynomials, the effective temperature $T(t)$ and phase space occupation factor $\\Upsilon(t)$. In this first paper we address (effectively) massless fermions and derive dynamical equations for $T(t)$ and $\\Upsilon(t)$ such that the zeroth order term of the basis alone captures the number density and energy density of each particle distribution. We validate our method and illustrate the reduced computational cost and the ability to represent final state chemical non-equilibrium by studying a model problem that is motivated by the physics of the neutrino freeze-out processes in the early Universe, where the essential physical characteristics include reheating from another disappearing particle component ($e^\\pm$-annihilation).
Simulation of a Microfluidic Gradient Generator using Lattice Boltzmann Methods
Simon, Tanaka
2013-01-01
Microfluidics provides a powerful and versatile technology to accurately control spatial and temporal conditions for cell culturing and can therefore be used to study cellular responses to gradients. Here we use Lattice Boltzmann methods (LBM) to solve both the Navier-Stokes equation (NSE) for the fluid and the coupled convection-diffusion equation (CDE) for the compounds that form the diffusion-based gradient. The design of a microfluidic chamber for diffusion-based gradients must avoid flow through the cell chamber. This can be achieved by alternately opening the source and the sink channels. The fast toggling of microfluidic valves requires switching between different boundary conditions. We demonstrate that the LBM is a powerful method for handling complex geometries, high Peclet number conditions, discontinuities in the boundary conditions, and multiphysics coupling.
Beyond Poisson-Boltzmann: Numerical Sampling of Charge Density Fluctuations.
Poitevin, Frédéric; Delarue, Marc; Orland, Henri
2016-07-01
We present a method aimed at sampling charge density fluctuations in Coulomb systems. The derivation follows from a functional integral representation of the partition function in terms of charge density fluctuations. Starting from the mean-field solution given by the Poisson-Boltzmann equation, an original approach is proposed to numerically sample fluctuations around it, through the propagation of a Langevin-like stochastic partial differential equation (SPDE). The diffusion tensor of the SPDE can be chosen so as to avoid the numerical complexity linked to long-range Coulomb interactions, effectively rendering the theory completely local. A finite-volume implementation of the SPDE is described, and the approach is illustrated with preliminary results on the study of a system made of two like-charge ions immersed in a bath of counterions. PMID:27075231
Exact results for the Boltzmann equation and Smoluchowski's coagulation equation
International Nuclear Information System (INIS)
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.)
Lattice Boltzmann model for melting with natural convection
Energy Technology Data Exchange (ETDEWEB)
Huber, Christian [Department of Earth and Planetary Science, University of California - Berkeley, 307 McCone Hall 4767, Berkeley, CA 94720-4767 (United States)], E-mail: chuber@seismo.berkeley.edu; Parmigiani, Andrea [Computer Science Department, University of Geneva, 24, Rue du General Dufour, 1211 Geneva 4 (Switzerland)], E-mail: andrea.parmigiani@terre.unige.ch; Chopard, Bastien [Computer Science Department, University of Geneva, 24, Rue du General Dufour, 1211 Geneva 4 (Switzerland)], E-mail: Bastien.Chopard@cui.unige.ch; Manga, Michael [Department of Earth and Planetary Science, University of California - Berkeley, 177 McCone Hall 4767, Berkeley, CA 94720-4767 (United States)], E-mail: manga@seismo.berkeley.edu; Bachmann, Olivier [Department of Earth and Space Science, University of Washington, Johnson Hall 070, Seattle WA 98195-1310 (United States)], E-mail: bachmano@u.washington.edu
2008-10-15
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.
The peeling process of infinite Boltzmann planar maps
Budd, Timothy
2015-01-01
We start by studying a peeling process on finite random planar maps with faces of arbitrary degrees determined by a general weight sequence, which satisfies an admissibility criterion. The corresponding perimeter process is identified as a biased random walk, in terms of which the admissibility criterion has a very simple interpretation. The finite random planar maps under consideration were recently proved to possess a well-defined local limit known as the infinite Boltzmann planar map (IBPM). Inspired by recent work of Curien and Le Gall, we show that the peeling process on the IBPM can be obtained from the peeling process of finite random maps by conditioning the perimeter process to stay positive. The simplicity of the resulting description of the peeling process allows us to obtain the scaling limit of the associated perimeter and volume process for arbitrary regular critical weight sequences.
Lattice Boltzmann Simulation of Multiple Bubbles Motion under Gravity
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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.
Lattice Boltzmann method for shape optimization of fluid distributor
Wang, Limin; Luo, Lingai
2013-01-01
This paper presents the shape optimization of a flat-type arborescent fluid distributor for the purpose of process intensification. A shape optimization algorithm based on the lattice Boltzmann method (LBM) is proposed with the objective of decreasing the flow resistance of such distributor at the constraint of constant fluid volume. Prototypes of the initial distributor as well as the optimized one are designed. Fluid distribution and hydraulic characteristics of these distributors are investigated numerically. Results show that the pressure drop of the optimized distributor is between 15.9% and 25.1% lower than that of the initial reference while keeping a uniform flow distribution, demonstrating the process intensification in fluid distributor, and suggesting the interests of the proposed optimization algorithm in engineering optimal design.
Comparison of different Propagation Steps for the Lattice Boltzmann Method
Wittmann, Markus; Hager, Georg; Wellein, Gerhard
2011-01-01
Several possibilities exist to implement the propagation step of the lattice Boltzmann method. This paper describes common implementations which are compared according to the number of memory transfer operations they require per lattice node update. A memory bandwidth based performance model is then used to obtain an estimation of the maximal reachable performance on different machines. A subset of the discussed implementations of the propagation step were benchmarked on different Intel and AMD-based compute nodes using the framework of an existing flow solver which is specially adapted to simulate flow in porous media. Finally the estimated performance is compared to the measured one. As expected, the number of memory transfers has a significant impact on performance. Advanced approaches for the propagation step like "AA pattern" or "Esoteric Twist" require more implementation effort but sustain significantly better performance than non-naive straight forward implementations.
Two Dimensional Lattice Boltzmann Method for Cavity Flow Simulation
Directory of Open Access Journals (Sweden)
Panjit MUSIK
2004-01-01
Full Text Available This paper presents a simulation of incompressible viscous flow within a two-dimensional square cavity. The objective is to develop a method originated from Lattice Gas (cellular Automata (LGA, which utilises discrete lattice as well as discrete time and can be parallelised easily. Lattice Boltzmann Method (LBM, known as discrete Lattice kinetics which provide an alternative for solving the Navier–Stokes equations and are generally used for fluid simulation, is chosen for the study. A specific two-dimensional nine-velocity square Lattice model (D2Q9 Model is used in the simulation with the velocity at the top of the cavity kept fixed. LBM is an efficient method for reproducing the dynamics of cavity flow and the results which are comparable to those of previous work.
Lattice Boltzmann modeling of water-like fluids
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Sauro eSucci
2014-04-01
Full Text Available We review recent advances on the mesoscopic modeling of water-like fluids,based on the lattice Boltzmann (LB methodology.The main idea is to enrich the basic LB (hydro-dynamics with angular degrees of freedom responding to suitable directional potentials between water-like molecules.The model is shown to reproduce some microscopic features of liquid water, such as an average number of hydrogen bonds per molecules (HBs between $3$ and $4$, as well as a qualitatively correctstatistics of the hydrogen bond angle as a function of the temperature.Future developments, based on the coupling the present water-like LB model with the dynamics of suspended bodies,such as biopolymers, may open new angles of attack to the simulation of complex biofluidic problems, such as protein folding and aggregation, and the motion of large biomolecules in complex cellular environments.
Modeling Image Structure with Factorized Phase-Coupled Boltzmann Machines
Cadieu, Charles F
2010-01-01
We describe a model for capturing the statistical structure of local amplitude and local spatial phase in natural images. The model is based on a recently developed, factorized third-order Boltzmann machine that was shown to be effective at capturing higher-order structure in images by modeling dependencies among squared filter outputs (Ranzato and Hinton, 2010). Here, we extend this model to $L_p$-spherically symmetric subspaces. In order to model local amplitude and phase structure in images, we focus on the case of two dimensional subspaces, and the $L_2$-norm. When trained on natural images the model learns subspaces resembling quadrature-pair Gabor filters. We then introduce an additional set of hidden units that model the dependencies among subspace phases. These hidden units form a combinatorial mixture of phase coupling distributions, concentrated in the sum and difference of phase pairs. When adapted to natural images, these distributions capture local spatial phase structure in natural images.
Chemical-potential-based Lattice Boltzmann Method for Nonideal Fluids
Wen, Binghai; He, Bing; Zhang, Chaoying; Fang, Haiping
2016-01-01
Chemical potential is an effective way to drive phase transition or express wettability. In this letter, we present a chemical-potential-based lattice Boltzmann model to simulate multiphase flows. The nonideal force is directly evaluated by a chemical potential. The model theoretically satisfies thermodynamics and Galilean invariance. The computational efficiency is improved owing to avoiding the calculation of pressure tensor. We have derived several chemical potentials of the popular equations of state from the free-energy density function. An effective chemical-potential boundary condition is implemented to investigate the wettability of a solid surface. Remarkably, the numerical results show that the contact angle can be linearly tuned by the surface chemical potential.
Lattice Boltzmann method for mixtures at variable Schmidt number
Monteferrante, Michele; Melchionna, Simone; Marconi, Umberto Marini Bettolo
2014-07-01
When simulating multicomponent mixtures via the Lattice Boltzmann Method, it is desirable to control the mutual diffusivity between species while maintaining the viscosity of the solution fixed. This goal is herein achieved by a modification of the multicomponent Bhatnagar-Gross-Krook evolution equations by introducing two different timescales for mass and momentum diffusion. Diffusivity is thus controlled by an effective drag force acting between species. Numerical simulations confirm the accuracy of the method for neutral binary and charged ternary mixtures in bulk conditions. The simulation of a charged mixture in a charged slit channel show that the conductivity and electro-osmotic mobility exhibit a departure from the Helmholtz-Smoluchowski prediction at high diffusivity.
Lattice-Boltzmann hydrodynamics of anisotropic active matter.
de Graaf, Joost; Menke, Henri; Mathijssen, Arnold J T M; Fabritius, Marc; Holm, Christian; Shendruk, Tyler N
2016-04-01
A plethora of active matter models exist that describe the behavior of self-propelled particles (or swimmers), both with and without hydrodynamics. However, there are few studies that consider shape-anisotropic swimmers and include hydrodynamic interactions. Here, we introduce a simple method to simulate self-propelled colloids interacting hydrodynamically in a viscous medium using the lattice-Boltzmann technique. Our model is based on raspberry-type viscous coupling and a force/counter-force formalism, which ensures that the system is force free. We consider several anisotropic shapes and characterize their hydrodynamic multipolar flow field. We demonstrate that shape-anisotropy can lead to the presence of a strong quadrupole and octupole moments, in addition to the principle dipole moment. The ability to simulate and characterize these higher-order moments will prove crucial for understanding the behavior of model swimmers in confining geometries. PMID:27059561
Sedimentation analysis of small ice crystals by Lattice Boltzmann Method
Giovacchini, Juan P
2016-01-01
Lattice Boltzmann Method (LBM) is used to simulate and analyze the sedimentation of small ($16-80 \\,\\mu m$) ice particles in the atmosphere. We are specially interested in evaluating the terminal falling velocity for two ice particle shapes: columnar ice crystals and six bullet-rosettes ice policrystal. The main objective in this paper is to investigate the LBM suitability to solve ice crystal sedimentation problems, as well as to evaluate these numerical methods as a powerful numerical tool to solve these problems for arbitrary ice crystal shapes and sizes. LBM results are presented in comparison with laboratory experimental results and theoretical proposals well known in the literature. The numerical results show good agreement with experimental and theoretical results for both geometrical configurations.
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...
Full Eulerian lattice Boltzmann model for conjugate heat transfer.
Hu, Yang; Li, Decai; Shu, Shi; Niu, Xiaodong
2015-12-01
In this paper a full Eulerian lattice Boltzmann model is proposed for conjugate heat transfer. A unified governing equation with a source term for the temperature field is derived. By introducing the source term, we prove that the continuity of temperature and its normal flux at the interface is satisfied automatically. The curved interface is assumed to be zigzag lines. All physical quantities are recorded and updated on a Cartesian grid. As a result, any complicated treatment near the interface is avoided, which makes the proposed model suitable to simulate the conjugate heat transfer with complex interfaces efficiently. The present conjugate interface treatment is validated by several steady and unsteady numerical tests, including pure heat conduction, forced convection, and natural convection problems. Both flat and curved interfaces are also involved. The obtained results show good agreement with the analytical and/or finite volume results. PMID:26764851
Moving Charged Particles in Lattice Boltzmann-Based Electrokinetics
Kuron, Michael; Schornbaum, Florian; Bauer, Martin; Godenschwager, Christian; Holm, Christian; de Graaf, Joost
2016-01-01
The motion of ionic solutes and charged particles under the influence of an electric field and the ensuing hydrodynamic flow of the underlying solvent is ubiquitous in aqueous colloidal suspensions. The physics of such systems is described by a coupled set of differential equations, along with boundary conditions, collectively referred to as the electrokinetic equations. Capuani et al. [J. Chem. Phys. 121, 973 (2004)] introduced a lattice-based method for solving this system of equations, which builds upon the lattice Boltzmann (LB) algorithm for the simulation of hydrodynamic flow and exploits computational locality. However, thus far, a description of how to incorporate moving boundary conditions, which are needed to simulate moving colloids, into the Capuani scheme has been lacking. In this paper, we detail how to introduce such moving boundaries, based on an analogue to the moving boundary method for the pure LB solver. The key ingredients in our method are mass and charge conservation for the solute spec...
Lattice Boltzmann implementation for Fluids Flow Simulation in Porous Media
Directory of Open Access Journals (Sweden)
Xinming Zhang
2011-06-01
Full Text Available In this paper, the lattice-Boltzmann method is developed to investigate the behavior of isothermal two-phase fluid flow in porous media. The method is based on the Shan–Chen multiphase model of nonideal fluids that allow coexistence of two phases of a single substance. We reproduce some different idealized situations (phase separation, surface tension, contact angle, pipe flow, and fluid droplet motion, et al in which the results are already known from theory or laboratory measurements and show the validity of the implementation for the physical two-phase flow in porous media. Application of the method to fluid intrusion in porous media is discussed and shows the effect of wettability on the fluid flow. The capability of reproducing critical flooding phenomena under strong wettability conditions is also proved.
Lattice Boltzmann simulation of turbulent natural convection in tall enclosures
Directory of Open Access Journals (Sweden)
Sajjadi Hasan
2015-01-01
Full Text Available In this paper Lattice Boltzmann simulation of turbulent natural convection with large-eddy simulations (LES in tall enclosures which is filled by air with Pr=0.71 has been studied. Calculations were performed for high Rayleigh numbers (Ra=107-109 and aspect ratios change between 0.5 to 2 (0.5
Heat conduction in multifunctional nanotrusses studied using Boltzmann transport equation
Energy Technology Data Exchange (ETDEWEB)
Dou, Nicholas G.; Minnich, Austin J. [Division of Engineering and Applied Science, California Institute of Technology, Pasadena, California 91125 (United States)
2016-01-04
Materials that possess low density, low thermal conductivity, and high stiffness are desirable for engineering applications, but most materials cannot realize these properties simultaneously due to the coupling between them. Nanotrusses, which consist of hollow nanoscale beams architected into a periodic truss structure, can potentially break these couplings due to their lattice architecture and nanoscale features. In this work, we study heat conduction in the exact nanotruss geometry by solving the frequency-dependent Boltzmann transport equation using a variance-reduced Monte Carlo algorithm. We show that their thermal conductivity can be described with only two parameters, solid fraction and wall thickness. Our simulations predict that nanotrusses can realize unique combinations of mechanical and thermal properties that are challenging to achieve in typical materials.
Modeling of urban traffic networks with lattice Boltzmann model
Meng, Jian-ping; Qian, Yue-hong; Dai, Shi-qiang
2008-02-01
It is of great importance to uncover the characteristics of traffic networks. However, there have been few researches concerning kinetics models for urban traffic networks. In this work, a lattice Boltzmann model (LBM) for urban traffic networks is proposed by incorporating the ideas of the Biham-Middleton-Levine (BML) model into the LBM for road traffic. In the present model, situations at intersections with the red and green traffic signals are treated as a kind of boundary conditions varying with time. Thus, the urban traffic network could be described in the mesoscopic level. By performing numerical simulations under the periodic boundary conditions, the behavior of average velocity is investigated in detail. The numerical results agree quite well with those given by the Chowdhury-Schadschneider (ChSch) model (Chowdhury D. and Schadschneider A., Phys. Rev. E, 59 (1999) R1311). Furthermore, the statistical noise is reduced in this discrete kinetics model, thus, the present model has considerably high computational efficiency.
Free Surface Lattice Boltzmann with Enhanced Bubble Model
Anderl, Daniela; Rauh, Cornelia; Rüde, Ulrich; Delgado, Antonio
2016-01-01
This paper presents an enhancement to the free surface lattice Boltzmann method (FSLBM) for the simulation of bubbly flows including rupture and breakup of bubbles. The FSLBM uses a volume of fluid approach to reduce the problem of a liquid-gas two-phase flow to a single-phase free surface simulation. In bubbly flows compression effects leading to an increase or decrease of pressure in the suspended bubbles cannot be neglected. Therefore, the free surface simulation is augmented by a bubble model that supplies the missing information by tracking the topological changes of the free surface in the flow. The new model presented here is capable of handling the effects of bubble breakup and coalesce without causing a significant computational overhead. Thus, the enhanced bubble model extends the applicability of the FSLBM to a new range of practically relevant problems, like bubble formation and development in chemical reactors or foaming processes.
A Lattice Boltzmann Model for Oscillating Reaction-Diffusion
Rodríguez-Romo, Suemi; Ibañez-Orozco, Oscar; Sosa-Herrera, Antonio
2016-07-01
A computational algorithm based on the lattice Boltzmann method (LBM) is proposed to model reaction-diffusion systems. In this paper, we focus on how nonlinear chemical oscillators like Belousov-Zhabotinsky (BZ) and the chlorite-iodide-malonic acid (CIMA) reactions can be modeled by LBM and provide with new insight into the nature and applications of oscillating reactions. We use Gaussian pulse initial concentrations of sulfuric acid in different places of a bidimensional reactor and nondiffusive boundary walls. We clearly show how these systems evolve to a chaotic attractor and produce specific pattern images that are portrayed in the reactions trajectory to the corresponding chaotic attractor and can be used in robotic control.
Boltzmann electron PIC simulation of the E-sail effect
Janhunen, P.
2015-12-01
The solar wind electric sail (E-sail) is a planned in-space propulsion device that uses the natural solar wind momentum flux for spacecraft propulsion with the help of long, charged, centrifugally stretched tethers. The problem of accurately predicting the E-sail thrust is still somewhat open, however, due to a possible electron population trapped by the tether. Here we develop a new type of particle-in-cell (PIC) simulation for predicting E-sail thrust. In the new simulation, electrons are modelled as a fluid, hence resembling hybrid simulation, but in contrast to normal hybrid simulation, the Poisson equation is used as in normal PIC to calculate the self-consistent electrostatic field. For electron-repulsive parts of the potential, the Boltzmann relation is used. For electron-attractive parts of the potential we employ a power law which contains a parameter that can be used to control the number of trapped electrons. We perform a set of runs varying the parameter and select the one with the smallest number of trapped electrons which still behaves in a physically meaningful way in the sense of producing not more than one solar wind ion deflection shock upstream of the tether. By this prescription we obtain thrust per tether length values that are in line with earlier estimates, although somewhat smaller. We conclude that the Boltzmann PIC simulation is a new tool for simulating the E-sail thrust. This tool enables us to calculate solutions rapidly and allows to easily study different scenarios for trapped electrons.
A Boltzmann machine for the organization of intelligent machines
Moed, Michael C.; Saridis, George N.
1989-01-01
In the present technological society, there is a major need to build machines that would execute intelligent tasks operating in uncertain environments with minimum interaction with a human operator. Although some designers have built smart robots, utilizing heuristic ideas, there is no systematic approach to design such machines in an engineering manner. Recently, cross-disciplinary research from the fields of computers, systems AI and information theory has served to set the foundations of the emerging area of the design of intelligent machines. Since 1977 Saridis has been developing an approach, defined as Hierarchical Intelligent Control, designed to organize, coordinate and execute anthropomorphic tasks by a machine with minimum interaction with a human operator. This approach utilizes analytical (probabilistic) models to describe and control the various functions of the intelligent machine structured by the intuitively defined principle of Increasing Precision with Decreasing Intelligence (IPDI) (Saridis 1979). This principle, even though resembles the managerial structure of organizational systems (Levis 1988), has been derived on an analytic basis by Saridis (1988). The purpose is to derive analytically a Boltzmann machine suitable for optimal connection of nodes in a neural net (Fahlman, Hinton, Sejnowski, 1985). Then this machine will serve to search for the optimal design of the organization level of an intelligent machine. In order to accomplish this, some mathematical theory of the intelligent machines will be first outlined. Then some definitions of the variables associated with the principle, like machine intelligence, machine knowledge, and precision will be made (Saridis, Valavanis 1988). Then a procedure to establish the Boltzmann machine on an analytic basis will be presented and illustrated by an example in designing the organization level of an Intelligent Machine. A new search technique, the Modified Genetic Algorithm, is presented and proved
Generalizing the Boltzmann equation in complex phase space.
Zadehgol, Abed
2016-08-01
In this work, a generalized form of the BGK-Boltzmann equation is proposed, where the velocity, position, and time can be represented by real or complex variables. The real representation leads to the conventional BGK-Boltzmann equation, which can recover the continuity and Navier-Stokes equations. We show that the complex representation yields a different set of equations, and it can also recover the conservation and Navier-Stokes equations, at low Mach numbers, provided that the imaginary component of the macroscopic mass can be neglected. We briefly review the Constant Speed Kinetic Model (CSKM), which was introduced in Zadehgol and Ashrafizaadeh [J. Comp. Phys. 274, 803 (2014)JCTPAH0021-999110.1016/j.jcp.2014.06.053] and Zadehgol [Phys. Rev. E 91, 063311 (2015)PLEEE81539-375510.1103/PhysRevE.91.063311]. The CSKM is then used as a basis to show that the complex-valued equilibrium distribution function of the present model can be identified with a simple singularity in the complex phase space. The virtual particles, in the present work, are concentrated on virtual "branes" which surround the computational nodes. Employing the Cauchy integral formula, it is shown that certain variations of the "branes," in the complex phase space, do not affect the local kinetic states. This property of the new model, which is referred to as the "apparent jumps" in the present work, is used to construct new models. The theoretical findings have been tested by simulating three benchmark flows. The results of the present simulations are in excellent agreement with the previous results reported by others. PMID:27627421
A simple lattice Boltzmann scheme for low Mach number reactive flows
Institute of Scientific and Technical Information of China (English)
CHEN; Sheng; LIU; Zhaohui; ZHANG; Chao; HE; Zhu; TIAN; Zhiwei; SHI; Baochang
2006-01-01
For simulating low Mach number reactive flows, a simple and coupled lattice Boltzmann (CLB) scheme is proposed, by which the fluid density can bear significant changes. Different from the existing hybrid lattice Boltzmann (HLB) scheme and non-coupled lattice Boltzmann (NCLB) scheme, this scheme is strictly lattice Boltzmann style and the fluid density couples directly with the temperature. Because it has got rid of the constraint of traditional thought in lattice Boltzmann scheme，on the basis of the equality among the particle speed c, the time step △t and the lattice grid spacing △x held, both c and △t can be adjusted in this scheme according to a "characteristic temperature" instead of the local temperature. The whole algorithm becomes more stable and efficient besides inheriting the intrinsically outstanding strong points of conventional lattice Boltzmann scheme. In this scheme, we also take into account different molecular weights of species, so it is more suitable for simulating actual low Mach number reactive flows than previous work. In this paper, we simulated a so-called "counter-flow" premixed propane-air flame, and the results got by our scheme are much better than that obtained by NCLB. And the more important thing is that the exploration in this work has offered a kind of brand-new train of thought for building other novel lattice Boltzmann scheme in the future.
The isotropic diffusion source approximation for supernova neutrino transport
Liebendörfer, M; Fischer, T
2007-01-01
Most astrophysical observations originate from matter that interacts with radiation or transported particles. We develop a pragmatic approximation in order to enable multi-dimensional simulations with basic radiative transfer when the computational resources are not sufficient to solve the complete Boltzmann transport equation. The distribution function of the transported particles is divided into trapped and streaming particle components. Their separate evolution equations are coupled by a source term that converts trapped particles into streaming particles. We determine this source term by requiring the correct diffusion limit. For a smooth transition to the free streaming regime, this 'diffusion source' is limited by the matter emissivity. The resulting streaming particle emission rates are integrated over space to obtain the streaming particle flux. A geometric estimate of the flux factor is used to convert the particle flux to the streaming particle density. The efficiency of the scheme results from the ...
Karakida, Ryo; Okada, Masato; Amari, Shun-Ichi
2016-07-01
The restricted Boltzmann machine (RBM) is an essential constituent of deep learning, but it is hard to train by using maximum likelihood (ML) learning, which minimizes the Kullback-Leibler (KL) divergence. Instead, contrastive divergence (CD) learning has been developed as an approximation of ML learning and widely used in practice. To clarify the performance of CD learning, in this paper, we analytically derive the fixed points where ML and CDn learning rules converge in two types of RBMs: one with Gaussian visible and Gaussian hidden units and the other with Gaussian visible and Bernoulli hidden units. In addition, we analyze the stability of the fixed points. As a result, we find that the stable points of CDn learning rule coincide with those of ML learning rule in a Gaussian-Gaussian RBM. We also reveal that larger principal components of the input data are extracted at the stable points. Moreover, in a Gaussian-Bernoulli RBM, we find that both ML and CDn learning can extract independent components at one of stable points. Our analysis demonstrates that the same feature components as those extracted by ML learning are extracted simply by performing CD1 learning. Expanding this study should elucidate the specific solutions obtained by CD learning in other types of RBMs or in deep networks. PMID:27131468
Accuracy analysis of high-order lattice Boltzmann models for rarefied gas flows
Meng, Jianping
2009-01-01
In this work, we have theoretically analyzed and numerically evaluated the accuracy of high-order lattice Boltzmann (LB) models for capturing non-equilibrium effects in rarefied gas flows. In the incompressible limit, the LB equation is proved to be equivalent to the linearized Bhatnagar-Gross-Krook (BGK) equation. Therefore, when the same Gauss-Hermite quadrature is used, LB method closely assembles the discrete velocity method (DVM). In addition, the order of Hermite expansion for the equilibrium distribution function is found not to be correlated with the approximation order in terms of the Knudsen number to the BGK equation, which was previously suggested by \\cite{2006JFM...550..413S}. Furthermore, we have numerically evaluated the LB models for a standing-shear-wave problem, which is designed specifically for assessing model accuracy by excluding the influence of gas molecule/surface interactions at wall boundaries. The numerical simulation results confirm that the high-order terms in the discrete equili...
Fluid flow and mass transfer over circular strands using the lattice Boltzmann method
Hossain, Md. Shakhawath; Chen, X. B.; Bergstrom, D. J.
2015-10-01
Knowledge of the fluid flow and mass transfer over circular strands is fundamental to the cell culture of tissue scaffolds in bio-reactors. This paper presents a study on the simulation of fluid flow and mass transfer over the circular strands of a tissue scaffold by using the multiple relaxation time lattice Boltzmann method for the low Reynolds number regime, with Re D = 0.01 and 0.1, respectively. The mass transfer problem approximates the transport of a scalar nutrient from the bulk fluid to the strand surface, such as is encountered in the flow through tissue scaffolds placed in bio-reactors. The circular geometry of the scaffold strand is treated and implemented by means of the interpolated bounce-back boundary condition formulation. Our simulation illustrates that the flow accelerates around the strand, resulting in the maximum shear stress at the shoulder of the strand and that diffusion mass transfer plays the dominant role in the scalar transport. The local Sherwood number varies significantly over the surface of the strand, with a peak value located on the upstream surface. Increasing the Schmidt number of the scalar and decreasing the blockage ratio results in higher mass transfer rates on the surface of the stand. Overall, the simulation results provide one with the insight into the fluid flow and mass transfer over the circular strands of a tissue scaffold in a bio-reactor, which would be impractical to obtain by experiments.
Investigation of an entropic stabilizer for the lattice-Boltzmann method.
Mattila, Keijo K; Hegele, Luiz A; Philippi, Paulo C
2015-06-01
The lattice-Boltzmann (LB) method is commonly used for the simulation of fluid flows at the hydrodynamic level of description. Due to its kinetic theory origins, the standard LB schemes carry more degrees of freedom than strictly needed, e.g., for the approximation of solutions to the Navier-stokes equation. In particular, there is freedom in the details of the so-called collision operator. This aspect was recently utilized when an entropic stabilizer, based on the principle of maximizing local entropy, was proposed for the LB method [I. V. Karlin, F. Bösch, and S. S. Chikatamarla, Phys. Rev. E 90, 031302(R) (2014)]. The proposed stabilizer can be considered as an add-on or extension to basic LB schemes. Here the entropic stabilizer is investigated numerically using the perturbed double periodic shear layer flow as a benchmark case. The investigation is carried out by comparing numerical results obtained with six distinct LB schemes. The main observation is that the unbounded, and not explicitly controllable, relaxation time for the higher-order moments will directly influence the leading-order error terms. As a consequence, the order of accuracy and, in general, the numerical behavior of LB schemes are substantially altered. Hence, in addition to systematic numerical validation, more detailed theoretical analysis of the entropic stabilizer is still required in order to properly understand its properties. PMID:26172795
Progress towards an acoustic determination of the Boltzmann constant at CEM-UVa
Pérez-Sanz, Fernando J.; Segovia, José J.; Martín, M. Carmen; Villamañán, Miguel A.; del Campo, Dolores; García, Carmen
2015-10-01
An acoustic gas thermometer was used to achieve a determination of the Boltzmann constant, kB, using a misaligned stainless steel (316L) spherical cavity with an internal volume of approximately 268 cm3. Measurements of the speed of sound while the cavity is filled with argon at the temperature of the triple point of water, 273.16 K, and at different pressures between 78.2 kPa and 0.9 MPa, were used to extrapolate the value of the speed of sound in argon at zero pressure. The internal volume of the resonator was accurately determined by measuring microwave resonance frequencies at the same temperature and pressure conditions as for the acoustic measurements. The measurements were taken at pressures from 78.2 kPa up to 901.3 kPa, and at 273.16 K. As the results of the measurements, we determined kB = (1.380 644 1 ± 0.000 022 1) × 10-23 J K-1 which means a relative standard uncertainty of 16 parts in 106.
Simulation of Thermal Flow Problems via a Hybrid Immersed Boundary-Lattice Boltzmann Method
Directory of Open Access Journals (Sweden)
J. Wu
2012-01-01
Full Text Available A hybrid immersed boundary-lattice Boltzmann method (IB-LBM is presented in this work to simulate the thermal flow problems. In current approach, the flow field is resolved by using our recently developed boundary condition-enforced IB-LBM (Wu and Shu, (2009. The nonslip boundary condition on the solid boundary is enforced in simulation. At the same time, to capture the temperature development, the conventional energy equation is resolved. To model the effect of immersed boundary on temperature field, the heat source term is introduced. Different from previous studies, the heat source term is set as unknown rather than predetermined. Inspired by the idea in (Wu and Shu, (2009, the unknown is calculated in such a way that the temperature at the boundary interpolated from the corrected temperature field accurately satisfies the thermal boundary condition. In addition, based on the resolved temperature correction, an efficient way to compute the local and average Nusselt numbers is also proposed in this work. As compared with traditional implementation, no approximation for temperature gradients is required. To validate the present method, the numerical simulations of forced convection are carried out. The obtained results show good agreement with data in the literature.
Qin, Feng; Zhao, Hua; Cai, Wei; Zhang, Zhiguo; Cao, Wenwu
2016-06-01
Noncontact monitoring temperature is very important in modern medicine, science, and technologies. The fluorescence intensity ratio (FIR) technique based on the Boltzmann distribution law exhibits excellent application potential, but the observed FIR deviates from the Boltzmann distribution law in the low temperature range. We propose a fluorescence intensity ratio relation FIR* = ηFIR by introducing a quantity η representing thermal population degree, which can be obtained from measured fluorescence decay curves of the upper emitting level. Using Eu3+ as an example, the method is confirmed that the deviated FIR is able to be corrected and return to follow the Boltzmann law.
THE NON-CUTOFF BOLTZMANN EQUATION WITH POTENTIAL FORCE IN THE WHOLE SPACE
Institute of Scientific and Technical Information of China (English)
Yuanjie LEI
2014-01-01
This paper is concerned with the non-cutoff Boltzmann equation for full-range interactions with potential force in the whole space. We establish the global existence and optimal temporal convergence rates of classical solutions to the Cauchy problem when initial data is a small perturbation of the stationary solution. The analysis is based on the time-weighted energy method building also upon the recent studies of the non-cutoff Boltzmann equation in [1-3, 15] and the non-cutoff Vlasov-Poisson-Boltzmann system [6].
Operators of Approximations and Approximate Power Set Spaces
Institute of Scientific and Technical Information of China (English)
ZHANG Xian-yong; MO Zhi-wen; SHU Lan
2004-01-01
Boundary inner and outer operators are introduced; and union, intersection, complement operators of approximations are redefined. The approximation operators have a good property of maintaining union, intersection, complement operators, so the rough set theory has been enriched from the operator-oriented and set-oriented views. Approximate power set spaces are defined, and it is proved that the approximation operators are epimorphisms from power set space to approximate power set spaces. Some basic properties of approximate power set space are got by epimorphisms in contrast to power set space.
Dyatko, Nikolay; Donkó, Zoltán
2015-08-01
At low reduced electric fields the electron energy distribution function in heavy noble gases can take two distinct shapes. This ‘bistability effect’—in which electron-electron (Coulomb) collisions play an essential role—is analyzed here for Xe with a Boltzmann equation approach and with a first principles particle simulation method. The solution of the Boltzmann equation adopts the usual approximations of (i) searching for the distribution function in the form of two terms (‘two-term approximation’), (ii) neglecting the Coulomb part of the collision integral for the anisotropic part of the distribution function, (iii) treating Coulomb collisions as binary events, and (iv) truncating the range of the electron-electron interaction beyond a characteristic distance. The particle-based simulation method avoids these approximations: the many-body interactions within the electron gas with a true (un-truncated) Coulomb potential are described by a molecular dynamics algorithm, while the collisions between electrons and the background gas atoms are treated with Monte Carlo simulation. We find a good general agreement between the results of the two techniques, which confirms, to a certain extent, the approximations used in the solution of the Boltzmann equation. The differences observed between the results are believed to originate from these approximations and from the presence of statistical noise in the particle simulations.
Ausloos, M.
2000-09-01
Recent observations have indicated that the traditional equilibrium market hypothesis (EMH; also known as Efficient Market Hypothesis) is unrealistic. It is shown here that it is the analog of a Boltzmann equation in physics, thus having some bad properties of mean-field approximations like a Gaussian distribution of price fluctuations. A kinetic theory for prices can be simply derived, considering in a first approach that market actors have all identical relaxation times, and solved within a Chapman-Enskog like formalism. In closing the set of equations, (i) an equation of state with a pressure and (ii) the equilibrium (isothermal) equation for the price (taken as the order parameter) of a stock as a function of the volume of money available are obtained.
Energy Technology Data Exchange (ETDEWEB)
Borges, P. D., E-mail: pdborges@gmail.com, E-mail: lscolfaro@txstate.edu; Scolfaro, L., E-mail: pdborges@gmail.com, E-mail: lscolfaro@txstate.edu [Department of Physics, Texas State University, San Marcos, Texas 78666 (United States)
2014-12-14
The thermoelectric properties of indium nitride in the most stable wurtzite phase (w-InN) as a function of electron and hole concentrations and temperature were studied by solving the semiclassical Boltzmann transport equations in conjunction with ab initio electronic structure calculations, within Density Functional Theory. Based on maximally localized Wannier function basis set and the ab initio band energies, results for the Seebeck coefficient are presented and compared with available experimental data for n-type as well as p-type systems. Also, theoretical results for electric conductivity and power factor are presented. Most cases showed good agreement between the calculated properties and experimental data for w-InN unintentionally and p-type doped with magnesium. Our predictions for temperature and concentration dependences of electrical conductivity and power factor revealed a promising use of InN for intermediate and high temperature thermoelectric applications. The rigid band approach and constant scattering time approximation were utilized in the calculations.
Takane, Yositake; Hayashi, Masahiko; Ebisawa, Hiromichi
2016-08-01
The time-dependent Ginzburg-Landau equation and the Boltzmann transport equation for charge-density-wave (CDW) conductors are derived from a microscopic one-dimensional model by applying the Keldysh Green's function approach under a quasiclassical approximation. The effects of an external electric field and impurity pinning of the CDW are fully taken into account without relying on a phenomenological argument. These equations simultaneously describe the spatiotemporal dynamics of both the CDW and quasiparticles; thus, they serve as a starting point to develop a general framework to analyze various nonequilibrium phenomena, such as current conversion between the CDW condensate and quasiparticles, in realistic CDW conductors. It is shown that, in typical situations, the equations correctly describe the nonlinear behavior of electric conductivity in a simpler manner.
Two-Dimensional Lattice Boltzmann for Reactive Rayleigh–Bénard and Bénard–Poiseuille Regimes
Directory of Open Access Journals (Sweden)
Suemi Rodríguez-Romo
2015-09-01
Full Text Available We perform a computer simulation of the reaction-diffusion and convection that takes place in Rayleigh–Bénard and Bénard–Poiseuille regimes. The lattice Boltzmann equation (LBE is used along with the Boussinesq approximation to solve the non-linear coupled differential equations that govern the systems’ thermo-hydrodynamics. Another LBE, is introduced to calculate the evolution concentration of the chemical species involved in the chemical reactions. The simulations are conducted at low Reynolds numbers and in terms of steady state between the first and second thermo-hydrodynamics instability. The results presented here (with no chemical reactions are in good agreement with those reported in the scientific literature which gives us high expectations about the reliability of the chemical kinetics simulation. Some examples are provided.
A multi-term solution of the space-time Boltzmann equation for electrons in gaseous and liquid Argon
Boyle, G J; Tattersall, W J; McEachran, R P; White, R D
2015-01-01
In a recent paper [1] the scattering and transport of excess electrons in liquid argon in the hydrodynamic regime was investigated, generalizing the seminal works of Lekner and Cohen [2,3] with modern scattering theory techniques and kinetic theory. In this paper, the discussion is extended to the non-hydrodynamic regime through the development of a full multi-term space-time solution of Boltzmann's equation for electron transport in gases and liquids using a novel operator-splitting method. A Green's function formalism is considered that enables flexible adaptation to various experimental systems. The spatio-temporal evolution of electrons in liquids in the hydrodynamic regime is studied for a benchmark model Percus-Yevick liquid as well as for liquid argon. The temporal evolution of Franck-Hertz oscillations are observed for liquids, with striking differences in the spatio-temporal development of the velocity distribution function components between the uncorrelated gas and true liquid approximations in arg...
Directory of Open Access Journals (Sweden)
Artin Laleian
2015-11-01
Full Text Available Two-dimensional (2D pore-scale models have successfully simulated microfluidic experiments of aqueous-phase flow with mixing-controlled reactions in devices with small aperture. A standard 2D model is not generally appropriate when the presence of mineral precipitate or biomass creates complex and irregular three-dimensional (3D pore geometries. We modify the 2D lattice Boltzmann method (LBM to incorporate viscous drag from the top and bottom microfluidic device (micromodel surfaces, typically excluded in a 2D model. Viscous drag from these surfaces can be approximated by uniformly scaling a steady-state 2D velocity field at low Reynolds number. We demonstrate increased accuracy by approximating the viscous drag with an analytically-derived body force which assumes a local parabolic velocity profile across the micromodel depth. Accuracy of the generated 2D velocity field and simulation permeability have not been evaluated in geometries with variable aperture. We obtain permeabilities within approximately 10% error and accurate streamlines from the proposed 2D method relative to results obtained from 3D simulations. In addition, the proposed method requires a CPU run time approximately 40 times less than a standard 3D method, representing a significant computational benefit for permeability calculations.
Approximation algorithms and hardness of approximation for knapsack problems
Buhrman, H.; Loff, B.; Torenvliet, L.
2012-01-01
We show various hardness of approximation algorithms for knapsack and related problems; in particular we will show that unless the Exponential-Time Hypothesis is false, then subset-sum cannot be approximated any better than with an FPTAS. We also give a simple new algorithm for approximating knapsac
Simulations of Bingham plastic flows with the multiple-relaxation-time lattice Boltzmann model
Chen, SongGui; Sun, QiCheng; Jin, Feng; Liu, JianGuo
2014-03-01
Fresh cement mortar is a type of workable paste, which can be well approximated as a Bingham plastic and whose flow behavior is of major concern in engineering. In this paper, Papanastasiou's model for Bingham fluids is solved by using the multiplerelaxation-time lattice Boltzmann model (MRT-LB). Analysis of the stress growth exponent m in Bingham fluid flow simulations shows that Papanastasiou's model provides a good approximation of realistic Bingham plastics for values of m > 108. For lower values of m, Papanastasiou's model is valid for fluids between Bingham and Newtonian fluids. The MRT-LB model is validated by two benchmark problems: 2D steady Poiseuille flows and lid-driven cavity flows. Comparing the numerical results of the velocity distributions with corresponding analytical solutions shows that the MRT-LB model is appropriate for studying Bingham fluids while also providing better numerical stability. We further apply the MRT-LB model to simulate flow through a sudden expansion channel and the flow surrounding a round particle. Besides the rich flow structures obtained in this work, the dynamics fluid force on the round particle is calculated. Results show that both the Reynolds number Re and the Bingham number Bn affect the drag coefficients C D , and a drag coefficient with Re and Bn being taken into account is proposed. The relationship of Bn and the ratio of unyielded zone thickness to particle diameter is also analyzed. Finally, the Bingham fluid flowing around a set of randomly dispersed particles is simulated to obtain the apparent viscosity and velocity fields. These results help simulation of fresh concrete flowing in porous media.
On the transparent conducting oxide Al doped ZnO: First Principles and Boltzmann equations study
Energy Technology Data Exchange (ETDEWEB)
Slassi, A. [Institute of Nanomaterials and Nanotechnology, MAScIR, Rabat (Morocco); LMPHE (URAC 12), Faculté des Sciences, Université Mohammed V-Agdal, Rabat (Morocco); Naji, S. [LMPHE (URAC 12), Faculté des Sciences, Université Mohammed V-Agdal, Rabat (Morocco); Department of Physics, Faculty of Science, Ibb University, Ibb (Yemen); Benyoussef, A. [Institute of Nanomaterials and Nanotechnology, MAScIR, Rabat (Morocco); LMPHE (URAC 12), Faculté des Sciences, Université Mohammed V-Agdal, Rabat (Morocco); Hamedoun, M., E-mail: hamedoun@hotmail.com [Institute of Nanomaterials and Nanotechnology, MAScIR, Rabat (Morocco); El Kenz, A. [LMPHE (URAC 12), Faculté des Sciences, Université Mohammed V-Agdal, Rabat (Morocco)
2014-08-25
Highlights: • The incorporation of Al in ZnO increases the optical band edge absorption. • Incorporated Al creates shallow donor states of Al-3s around Fermi level. • Transmittance decreases in the visible and IR regions, while it increases in the UV region. • Electrical conductivity increases and reaches almost the saturation for high concentration of Al. - Abstract: We report, in this work, a theoretical study on the electronic, optical and electrical properties of pure and Al doped ZnO with different concentrations. In fact, we investigate these properties using both First Principles calculations within TB-mBJ approximation and Boltzmann equations under the constant relaxation time approximation for charge carriers. It is found out that, the calculated lattice parameters and the optical band gap of pure ZnO are close to the experimental values and in a good agreement with the other theoretical studies. It is also observed that, the incorporations of Al in ZnO increase the optical band edge absorption which leads to a blue shift and no deep impurities levels are induced in the band gap as well. More precisely, these incorporations create shallow donor states around Fermi level in the conduction band minimum from mainly Al-3s orbital. Beside this, it is found that, the transmittance is decreased in the visible and IR regions, while it is significantly improved in UV region. Finally, our calculations show that the electrical conductivity is enhanced as a result of Al doping and it reaches almost the saturation for high concentration of Al. These features make Al doped ZnO a transparent conducting electrode for optoelectronic device applications.
Lattice Boltzmann flow simulations with applications of reduced order modeling techniques
Brown, Donald
2014-01-01
With the recent interest in shale gas, an understanding of the flow mechanisms at the pore scale and beyond is necessary, which has attracted a lot of interest from both industry and academia. One of the suggested algorithms to help understand flow in such reservoirs is the Lattice Boltzmann Method (LBM). The primary advantage of LBM is its ability to approximate complicated geometries with simple algorithmic modificatoins. In this work, we use LBM to simulate the flow in a porous medium. More specifically, we use LBM to simulate a Brinkman type flow. The Brinkman law allows us to integrate fast free-flow and slow-flow porous regions. However, due to the many scales involved and complex heterogeneities of the rock microstructure, the simulation times can be long, even with the speed advantage of using an explicit time stepping method. The problem is two-fold, the computational grid must be able to resolve all scales and the calculation requires a steady state solution implying a large number of timesteps. To help reduce the computational complexity and total simulation times, we use model reduction techniques to reduce the dimension of the system. In this approach, we are able to describe the dynamics of the flow by using a lower dimensional subspace. In this work, we utilize the Proper Orthogonal Decomposition (POD) technique, to compute the dominant modes of the flow and project the solution onto them (a lower dimensional subspace) to arrive at an approximation of the full system at a lowered computational cost. We present a few proof-of-concept examples of the flow field and the corresponding reduced model flow field.
Equivalence Between Forward and Backward Boltzmann Equations in Multi-Component Medium
Institute of Scientific and Technical Information of China (English)
张竹林
2002-01-01
The author generalized the propagator function theory introduced first by Sigmund, and gave a explicitly proof of a equivalence between forward and backward Boltzmann equations in a multi-component medium by using the generalized propagator function theory.
Gevrey Regularity for the Noncutoff Nonlinear Homogeneous Boltzmann Equation with Strong Singularity
Shi-you Lin
2014-01-01
The Cauchy problem of the nonlinear spatially homogeneous Boltzmann equation without angular cutoff is studied. By using analytic techniques, one proves the Gevrey regularity of the ${C}^{\\infty }$ solutions in non-Maxwellian and strong singularity cases.
Lattice Boltzmann simulation of transverse wave travelling in Maxwell viscoelastic fluid
Institute of Scientific and Technical Information of China (English)
Li Hua-Bing; Fang Hai-Ping
2004-01-01
A nine-velocity lattice Boltzmann method for Maxwell viscoelastic fluid is proposed. Travelling of transverse wave in Maxwell viscoelastic fluid is simulated. The instantaneous oscillating velocity, transverse shear speed and decay rate agree with theoretical results very well.
Corrected Stefan—Boltzmann Law and Lifespan of Schwarzschild-de-sitter Black Hole
Shi, Yan; Tang-Mei, He; Jing-Yi, Zhang
2016-06-01
In this paper, we correct the Stefan—Boltzmann law by considering the generalized uncertainty principle, and with this corrected Stefan—Boltzmann law, the lifespan of the Schwarzschild-de-sitter black holes is calculated. We find that the corrected Stefan—Boltzmann law contains two terms, the T4 term and the T6 term. Due to the modifications, at the end of the black hole radiation, it will arise a limited highest temperature and leave a residue. It is interesting to note that the mass of the residue and the Planck mass is in the same order of magnitude. The modified Stefan—Boltzmann law also gives a correction to the lifespan of the black hole, although it is very small. Supported by the National Natural Science Foundation of China under Grant Nos. 11273009 and 11303006
Immiscible multicomponent lattice Boltzmann model for fluids with high relaxation time ratio
Indian Academy of Sciences (India)
Tao Jiang; Qiwei Gong; Ruofan Qiu; Anlin Wang
2014-10-01
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 equilibrium velocity. Compared to the SC model, external forces in our model are discretized directly into the discrete lattice Boltzmann equation, as proposed by Guo et al. We develop it into a new multicomponent lattice Boltzmann (LB) model which has the ability to simulate immiscible multicomponent fluids with relaxation time ratio as large as 29.0 and to reduce `spurious velocity’. In this work, the improved model is validated and studied using the central bubble case and the rising bubble case. It finds good applications in both static and dynamic cases for multicomponent simulations with different relaxation time ratios.
Lattice Boltzmann model for the perfect gas flows with near-vacuum region
Institute of Scientific and Technical Information of China (English)
GuangwuYAN; LiYUAN
2000-01-01
It is known that the standard lattice Boltzmann method has near-vacuum limit,i. e., when the density is near zero, this method is invalid. In this letter, we propose a simple lattice Boltzmann model for one-dimensional flows. It possesses the ability of simulating nearvacuum area by setting a limitation of the relaxation factor. Thus, the model overcomes the disadvantage of non-physical pressure and the density. The numerical examples show these results are satisfactory.
A Lattice Boltzmann Model and Simulation of KdV-Burgers Equation
Institute of Scientific and Technical Information of China (English)
ZHANGChao-Ying; TANHui-Li; LIUMu-Ren; KONGLing-Jiang
2004-01-01
A lattice Boltzmann model of KdV-Burgers equation is derived by using the single-relaxation form of the lattice Boltzmann equation. With the present model, we simulate the traveling-wave solutions, the solitary-wave solutions, and the sock-wave solutions of KdV-Burgers equation, and calculate the decay factor and the wavelength of the sock-wave solution, which has exponential decay. The numerical results agree with the analytical solutions quite well.
Punshon-Smith, Samuel; Smith, Scott
2016-01-01
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 kin...
Note on Invariance of One-Dimensional Lattice-Boltzmann Equation
Institute of Scientific and Technical Information of China (English)
RAN Zheng
2007-01-01
Invariance of the one-dimensional lattice Boltzmann model is proposed together with its rigorous theoretical background.It is demonstrated that the symmetry inherent in Navier-Stokes equations is not really recovered in the one-dimensional lattice Boltzmann equation (LBE),especially for shock calculation.Symmetry breaking may be the inherent cause for the non-physical oscillations in the vicinity of the shock for LBE calculation.
A generalized linear Boltzmann equation for non-classical particle transport
International Nuclear Information System (INIS)
This paper presents a derivation and initial study of a new generalized linear Boltzmann equation (GLBE), which describes particle transport for random statistically homogeneous systems in which the distribution function for chord lengths between scattering centers is non-exponential. Such problems have recently been proposed for the description of photon transport in atmospheric clouds; this paper is a first attempt to develop a Boltzmann-like equation for these and other related applications.
Approximation by planar elastic curves
DEFF Research Database (Denmark)
Brander, David; Gravesen, Jens; Nørbjerg, Toke Bjerge
2015-01-01
We give an algorithm for approximating a given plane curve segment by a planar elastic curve. The method depends on an analytic representation of the space of elastic curve segments, together with a geometric method for obtaining a good initial guess for the approximating curve. A gradient......-driven optimization is then used to find the approximating elastic curve....
Approximate sine-Gordon solitons
Energy Technology Data Exchange (ETDEWEB)
Stratopoulos, G.N. (Dept. of Mathematical Sciences, Durham Univ. (United Kingdom)); Zakrzewski, W.J. (Dept. of Mathematical Sciences, Durham Univ. (United Kingdom))
1993-08-01
We look at the recently proposed scheme of approximating a sine-Gordon soliton by an expression derived from two dimensional instantons. We point out that the scheme of Sutcliffe in which he uses two dimensional instantons can be generalised to higher dimensions and that these generalisations produce even better approximations than the original approximation. We also comment on generalisations to other models. (orig.)
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.
Simulation of a Natural Convection by the Hybrid Thermal Lattice Boltzmann Equation
Energy Technology Data Exchange (ETDEWEB)
Ryu, Seungyeob; Kang, Hanok; Seo, Jaekwang; Yun, Juhyeon; Zee, Sung-Quun [Korea Atomic Energy Research Institute, Taejon (Korea, Republic of)
2006-07-01
Recently, the lattice Boltzmann method(LBM) has gained much attention for its ability to simulate fluid flows, and for its potential advantages over 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. In spite of its success in solving various challenging problems involving athermal fluids, the LBM has not been able to handle realistic thermal fluids with a satisfaction. The difficulty encountered in the thermal LBM seems to be the numerical instabilities. The existing thermal lattice Boltzmann models may be classified into three categories based on their approach in solving the Boltzmann equation, namely, the multispeed, the passive scalar and the thermal energy distribution approach. For more details see Ref. In the present work, the hybrid thermal lattice Boltzmann scheme proposed by Lallemand and Luo is used for simulating a natural convection in a square cavity. They proposed a hybrid thermal lattice Boltzmann equation(HTLBE) in which the mass and momentum conservation equations are solved by using the multiple-relaxation-time(MRT) model, whereas the diffusion-advection equations for the temperature are solved separately by using finite-difference technique. The main objective of the present work is to establish the lattice Boltzmann method as a viable tool for the simulation of temperature fields at high Rayleigh numbers.
Energy Technology Data Exchange (ETDEWEB)
Kawakami, H.; Urabe, J.; Yukimura, K. (Doshisha Univ., Kyoto (Japan))
1991-03-20
In a discharge excitation rare gas halide excima laser, uniform generation and stable maintenance of the excited discharge determines the laser characteristics. In this report, an approximate solution was obtained on the Boltzmann equation (frequently used for the theoretical analysis of this laser) to examine the nature of the solution. By optimizing the conversion of the variables, calculation of an electron swarm parameter in the hitherto uncertain range of the low conversion electric field was made possible, giving a generation mechanism of the uncertainty of the excited dischareg. The results are summarized as below. (1) The Boltzmann equation gives a linear solution for a logarithmic value of an electron energy in the range of low conversion electric field. (2) Time-wise responce ability between the measured voltage, current characteristics of the excitation discharge was clarified and the attachment and ionization coefficients calculated by Boltzmann equation. (3) Dependency of the attachment coefficient on the partial pressure of fluorine and kripton was examined, and the attachment coefficient was found to increase with the increase of the partial pressure for the both cases. 20 refs., 9 figs., 2 tabs.
Exact constants in approximation theory
Korneichuk, N
1991-01-01
This book is intended as a self-contained introduction for non-specialists, or as a reference work for experts, to the particular area of approximation theory that is concerned with exact constants. The results apply mainly to extremal problems in approximation theory, which in turn are closely related to numerical analysis and optimization. The book encompasses a wide range of questions and problems: best approximation by polynomials and splines; linear approximation methods, such as spline-approximation; optimal reconstruction of functions and linear functionals. Many of the results are base
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.
Multiple anisotropic collisions for advection-diffusion Lattice Boltzmann schemes
Ginzburg, Irina
2013-01-01
This paper develops a symmetrized framework for the analysis of the anisotropic advection-diffusion Lattice Boltzmann schemes. Two main approaches build the anisotropic diffusion coefficients either from the anisotropic anti-symmetric collision matrix or from the anisotropic symmetric equilibrium distribution. We combine and extend existing approaches for all commonly used velocity sets, prescribe most general equilibrium and build the diffusion and numerical-diffusion forms, then derive and compare solvability conditions, examine available anisotropy and stable velocity magnitudes in the presence of advection. Besides the deterioration of accuracy, the numerical diffusion dictates the stable velocity range. Three techniques are proposed for its elimination: (i) velocity-dependent relaxation entries; (ii) their combination with the coordinate-link equilibrium correction; and (iii) equilibrium correction for all links. Two first techniques are also available for the minimal (coordinate) velocity sets. Even then, the two-relaxation-times model with the isotropic rates often gains in effective stability and accuracy. The key point is that the symmetric collision mode does not modify the modeled diffusion tensor but it controls the effective accuracy and stability, via eigenvalue combinations of the opposite parity eigenmodes. We propose to reduce the eigenvalue spectrum by properly combining different anisotropic collision elements. The stability role of the symmetric, multiple-relaxation-times component, is further investigated with the exact von Neumann stability analysis developed in diffusion-dominant limit.
A Lattice-Boltzmann Method for Partially Saturated Computational Cells
Noble, D. R.; Torczynski, J. R.
The lattice-Boltzmann (LB) method is applied to complex, moving geometries in which computational cells are partially filled with fluid. The LB algorithm is modified to include a term that depends on the percentage of the cell saturated with fluid. The method is useful for modeling suspended obstacles that do not conform to the grid. Another application is to simulations of flow through reconstructed media that are not easily segmented into solid and liquid regions. A detailed comparison is made with FIDAP simulation results for the flow about a periodic line of cylinders in a channel at a non-zero Reynolds number. Two cases are examined. In the first simulation, the cylinders are given a constant velocity along the axis of the channel, and the steady solution is acquired. The transient behavior of the system is then studied by giving the cylinders an oscillatory velocity. For both steady and oscillatory flows, the method provides excellent agreement with FIDAP simulation results, even at locations close to the surface of a cylinder. In contrast to step-like solutions produced using the "bounce-back" condition, the proposed condition gives close agreement with the smooth FIDAP predictions. Computed drag forces with the proposed condition exhibit apparent quadratic convergence with grid refinement rather than the linear convergence exhibited by other LB boundary conditions.
Treatment of moving boundaries in lattice-Boltzmann simulations.
Indireshkumar, K.; Pal, A.; Brasseur, J. G.
2000-11-01
We consider the treatment of moving boundaries with the lattice-Boltzmann (LB) technique, where the treatment of the boundary often does not precisely conserve mass and spurious fluctuations in density/pressure result from boundary motion through fixed grids. First, we applied the extrapolation method proposed by Chen et. al.(S. Y. Chen, D. Martinez, and R Mei, Phys. Fluids) 8, 2527 (1996) to incompressible flow induced by the movement of a piston in a 2D ``cylinder'' with mass flow out of or into the cylinder. In these simulations, the velocity of the boundary nodes is set equal to the (known) velocity of the boundary (piston) in the equilibrium distribution function (Method I). In a second set of simulations, the boundary node velocities are obtained by interpolating between interior nodes and the boundary, thus including the effect of boundary position more precisely (Method II). Comparison of LB predictions with simulations using FIDAP show pressure agreement to witnin 2 %. The total mass is conserved to within 0.1% with Method I and improves to within 0.02 % using method II. Spurious fluctuations in density/pressure due to boundary movement is about 0.9% with Method I, which improves significantly to about 0.3% with Method II. The application of these simple techniques to more complex geometries and wall (and fluid) motions in a stomach during gastric emptying will be presented.
One-dimensional transient radiative transfer by lattice Boltzmann method.
Zhang, Yong; Yi, Hongliang; Tan, Heping
2013-10-21
The lattice Boltzmann method (LBM) is extended to solve transient radiative transfer in one-dimensional slab containing scattering media subjected to a collimated short laser irradiation. By using a fully implicit backward differencing scheme to discretize the transient term in the radiative transfer equation, a new type of lattice structure is devised. The accuracy and computational efficiency of this algorithm are examined firstly. Afterwards, effects of the medium properties such as the extinction coefficient, the scattering albedo and the anisotropy factor, and the shapes of laser pulse on time-resolved signals of transmittance and reflectance are investigated. Results of the present method are found to compare very well with the data from the literature. For an oblique incidence, the LBM results in this paper are compared with those by Monte Carlo method generated by ourselves. In addition, transient radiative transfer in a two-Layer inhomogeneous media subjected to a short square pulse irradiation is investigated. At last, the LBM is further extended to study the transient radiative transfer in homogeneous medium with a refractive index discontinuity irradiated by the short pulse laser. Several trends on the time-resolved signals different from those for refractive index of 1 (i.e. refractive-index-matched boundary) are observed and analysed. PMID:24150298
Lattice Boltzmann modeling of three-phase incompressible flows.
Liang, H; Shi, B C; Chai, Z H
2016-01-01
In this paper, based on multicomponent phase-field theory we intend to develop an efficient lattice Boltzmann (LB) model for simulating three-phase incompressible flows. In this model, two LB equations are used to capture the interfaces among three different fluids, and another LB equation is adopted to solve the flow field, where a new distribution function for the forcing term is delicately designed. Different from previous multiphase LB models, the interfacial force is not used in the computation of fluid velocity, which is more reasonable from the perspective of the multiscale analysis. As a result, the computation of fluid velocity can be much simpler. Through the Chapman-Enskog analysis, it is shown that the present model can recover exactly the physical formulations for the three-phase system. Numerical simulations of extensive examples including two circular interfaces, ternary spinodal decomposition, spreading of a liquid lens, and Kelvin-Helmholtz instability are conducted to test the model. It is found that the present model can capture accurate interfaces among three different fluids, which is attributed to its algebraical and dynamical consistency properties with the two-component model. Furthermore, the numerical results of three-phase flows agree well with the theoretical results or some available data, which demonstrates that the present LB model is a reliable and efficient method for simulating three-phase flow problems. PMID:26871191
Consistent lattice Boltzmann methods for incompressible axisymmetric flows
Zhang, Liangqi; Yang, Shiliang; Zeng, Zhong; Yin, Linmao; Zhao, Ya; Chew, Jia Wei
2016-08-01
In this work, consistent lattice Boltzmann (LB) methods for incompressible axisymmetric flows are developed based on two efficient axisymmetric LB models available in the literature. In accord with their respective original models, the proposed axisymmetric models evolve within the framework of the standard LB method and the source terms contain no gradient calculations. Moreover, the incompressibility conditions are realized with the Hermite expansion, thus the compressibility errors arising in the existing models are expected to be reduced by the proposed incompressible models. In addition, an extra relaxation parameter is added to the Bhatnagar-Gross-Krook collision operator to suppress the effect of the ghost variable and thus the numerical stability of the present models is significantly improved. Theoretical analyses, based on the Chapman-Enskog expansion and the equivalent moment system, are performed to derive the macroscopic equations from the LB models and the resulting truncation terms (i.e., the compressibility errors) are investigated. In addition, numerical validations are carried out based on four well-acknowledged benchmark tests and the accuracy and applicability of the proposed incompressible axisymmetric LB models are verified.
Boltzmann electron PIC simulation of the E-sail effect
Janhunen, Pekka
2015-01-01
The solar wind electric sail (E-sail) is a planned in-space propulsion device that uses the natural solar wind momentum flux for spacecraft propulsion with the help of long, charged, centrifugally stretched tethers. The problem of accurately predicting the E-sail thrust is still somewhat open, however, due to a possible electron population trapped by the tether. Here we develop a new type of particle-in-cell (PIC) simulation for predicting E-sail thrust. In the new simulation, electrons are modelled as a fluid, hence resembling hydrid simulation, but in contrast to normal hybrid simulation, the Poisson equation is used as in normal PIC to calculate the self-consistent electrostatic field. For electron-repulsive parts of the potential, the Boltzmann relation is used. For electron-attractive parts of the potential we employ a power law which contains a parameter that can be used to control the number of trapped electrons. We perform a set of runs varying the parameter and select the one with the smallest number...
Transition flow ion transport via integral Boltzmann equation
International Nuclear Information System (INIS)
A new approach is developed to solve the Integral Boltzmann Equation for the evolving velocity distribution of a source of ions, undergoing electrostatic acceleration through a neutral gas target. The theory is applicable to arbitrarily strong electric fields, any ion/neutral mass ratio greater than unity, and is not limited to spatially isotropic gas targets. A hard sphere collision model is used, with a provision for inelasticity. Both axial and radial velocity distributions are calculated for applications where precollision radial velocities are negligible, as is the case for ion beam extractions from high pressure sources. Theoretical predictions are tested through an experiment in which an atmospheric pressure ion source is coupled to a high vacuum energy analyser. Excellent agreement results for configurations in which the radial velocity remains small. Velocity distributions are applied to predicting the efficiency of coupling an atmospheric pressure ion source to a quadrupole mass spectrometer and results clearly indicate the most desirable extracting configuration. A method is devised to calculate ion-molecule hard sphere collision cross sections for easily fragmented organic ions
Derivation of anisotropic dissipative fluid dynamics from the Boltzmann equation
Molnar, E; Rischke, D H
2016-01-01
Fluid-dynamical equations of motion can be derived from the Boltzmann equation in terms of an expansion around a single-particle distribution function which is in local thermodynamical equilibrium, i.e., isotropic in momentum space in the rest frame of a fluid element. To zeroth order this expansion yields ideal fluid dynamics, to first order Navier-Stokes theory, and to second order transient theories of dissipative fluid dynamics. However, in situations where the single-particle distribution function is highly anisotropic in momentum space, such as the initial stage of heavy-ion collisions at relativistic energies, such an expansion is bound to break down. Nevertheless, one can still derive a fluid-dynamical theory, so-called anisotropic fluid dynamics, in terms of an expansion around a single-particle distribution function which incorporates (at least parts of) the momentum anisotropy via a suitable parametrization. In this paper we derive, up to terms of second order in this expansion, the equations of mo...
Lattice-Boltzmann Simulations of Microswimmer-Tracer Interactions
de Graaf, Joost
2016-01-01
Hydrodynamic interactions in systems comprised of self-propelled particles, such as swimming microorganisms, and passive tracers have a significant impact on the tracer dynamics compared to the equivalent "dry" sample. However, such interactions are often difficult to take into account in simulations due to their computational cost. Here, we perform a systematic investigation of swimmer-tracer interaction using an efficient force/counter-force based lattice-Boltzmann (LB) algorithm [J. de Graaf~\\textit{et al.}, J. Chem. Phys.~\\textbf{144}, 134106 (2016)], in order to validate its applicability to study large-scale microswimmer suspensions. We show that the LB algorithm reproduces far-field theoretical results well, both in a system with periodic boundary conditions and in a spherical cavity with no-slip walls, for which we derive expressions here. The LB algorithm has an inherent near-field renormalization of the flow field, due to the force interpolation between the swimmers and the lattice. This strongly pe...
Multiblock approach for the passive scalar thermal lattice Boltzmann method
Huang, Rongzong; Wu, Huiying
2014-04-01
A multiblock approach for the passive scalar thermal lattice Boltzmann method (TLBM) with multiple-relaxation-time collision scheme is proposed based on the Chapman-Enskog analysis. The interaction between blocks is executed in the moment space directly and an external force term is considered. Theoretical analysis shows that all the nonequilibrium parts of the nonconserved moments should be rescaled, while the nonequilibrium parts of the conserved moments can be calculated directly. Moreover, a local scheme based on the pseudoparticles for computing heat flux is proposed with no need to calculate temperature gradient based on the finite-difference scheme. In order to validate the multiblock approach and local scheme for computing heat flux, thermal Couette flow with wall injection is simulated and good results are obtained, which show that the adoption of the multiblock approach does not deteriorate the convergence rate of TLBM and the local scheme for computing heat flux has second-order convergence rate. Further application of the present approach is the simulation of natural convection in a square cavity with the Rayleigh number up to 109.
Macroscopic model and truncation error of discrete Boltzmann method
Hwang, Yao-Hsin
2016-10-01
A derivation procedure to secure the macroscopically equivalent equation and its truncation error for discrete Boltzmann method is proffered in this paper. Essential presumptions of two time scales and a small parameter in the Chapman-Enskog expansion are disposed of in the present formulation. Equilibrium particle distribution function instead of its original non-equilibrium form is chosen as key variable in the derivation route. Taylor series expansion encompassing fundamental algebraic manipulations is adequate to realize the macroscopically differential counterpart. A self-contained and comprehensive practice for the linear one-dimensional convection-diffusion equation is illustrated in details. Numerical validations on the incurred truncation error in one- and two-dimensional cases with various distribution functions are conducted to verify present formulation. As shown in the computational results, excellent agreement between numerical result and theoretical prediction are found in the test problems. Straightforward extensions to more complicated systems including convection-diffusion-reaction, multi-relaxation times in collision operator as well as multi-dimensional Navier-Stokes equations are also exposed in the Appendix to point out its expediency in solving complicated flow problems.
Application of Lattice Boltzmann Method to Flows in Microgeometries
Directory of Open Access Journals (Sweden)
Anoop K. Dass
2010-07-01
Full Text Available In the present investigation, Lattice Boltzmann Method (LBM is used to simulate rarefied gaseous microflows in three microgeometries. These are micro-couette, micro lid-driven cavity and micro-poiseuille flows. The Knudsen number is used to measure the degree of rarefaction in the microflows. First, micro-couette flow is computed with the effects of varying Knudsen number in the slip and threshold of the transition regime and the results compare well with existing results. After having thus established the credibility of the code and the method including boundary conditions, LBM is then used to investigate the micro lid-driven cavity flow with various aspect ratios. Simulation of microflow not only requires an appropriate method, it also requires suitable boundary conditions to provide a well-posed problem and unique solution. In this work, LBM and three slip boundary conditions, namely, diffuse scattering boundary condition, specular reflection and a combination of bounce-back and specular reflection is used to predict the micro lid-driven cavity flow fields. Then the LBM simulation is extended to micro-poiseuille flow. The results are substantiated through comparison with existing results and it is felt that the present methodology is reasonable to be employed in analyzing the flow in micro-systems.
Lattice Boltzmann models for the grain growth in polycrystalline systems
Directory of Open Access Journals (Sweden)
Yonggang Zheng
2016-08-01
Full Text Available In the present work, lattice Boltzmann models are proposed for the computer simulation of normal grain growth in two-dimensional systems with/without immobile dispersed second-phase particles and involving the temperature gradient effect. These models are demonstrated theoretically to be equivalent to the phase field models based on the multiscale expansion. Simulation results of several representative examples show that the proposed models can effectively and accurately simulate the grain growth in various single- and two-phase systems. It is found that the grain growth in single-phase polycrystalline materials follows the power-law kinetics and the immobile second-phase particles can inhibit the grain growth in two-phase systems. It is further demonstrated that the grain growth can be tuned by the second-phase particles and the introduction of temperature gradient is also an effective way for the fabrication of polycrystalline materials with grained gradient microstructures. The proposed models are useful for the numerical design of the microstructure of materials and provide effective tools to guide the experiments. Moreover, these models can be easily extended to simulate two- and three-dimensional grain growth with considering the mobile second-phase particles, transient heat transfer, melt convection, etc.
Lattice Boltzmann method for one-dimensional vector radiative transfer.
Zhang, Yong; Yi, Hongliang; Tan, Heping
2016-02-01
A one-dimensional vector radiative transfer (VRT) model based on lattice Boltzmann method (LBM) that considers polarization using four Stokes parameters is developed. The angular space is discretized by the discrete-ordinates approach, and the spatial discretization is conducted by LBM. LBM has such attractive properties as simple calculation procedure, straightforward and efficient handing of boundary conditions, and capability of stable and accurate simulation. To validate the performance of LBM for vector radiative transfer, four various test problems are examined. The first case investigates the non-scattering thermal-emitting atmosphere with no external collimated solar. For the other three cases, the external collimated solar and three different scattering types are considered. Particularly, the LBM is extended to solve VRT in the atmospheric aerosol system where the scattering function contains singularities and the hemisphere space distributions for the Stokes vector are presented and discussed. The accuracy and computational efficiency of this algorithm are discussed. Numerical results show that the LBM is accurate, flexible and effective to solve one-dimensional polarized radiative transfer problems. PMID:26906779
Lattice Boltzmann modeling of three-phase incompressible flows
Liang, H.; Shi, B. C.; Chai, Z. H.
2016-01-01
In this paper, based on multicomponent phase-field theory we intend to develop an efficient lattice Boltzmann (LB) model for simulating three-phase incompressible flows. In this model, two LB equations are used to capture the interfaces among three different fluids, and another LB equation is adopted to solve the flow field, where a new distribution function for the forcing term is delicately designed. Different from previous multiphase LB models, the interfacial force is not used in the computation of fluid velocity, which is more reasonable from the perspective of the multiscale analysis. As a result, the computation of fluid velocity can be much simpler. Through the Chapman-Enskog analysis, it is shown that the present model can recover exactly the physical formulations for the three-phase system. Numerical simulations of extensive examples including two circular interfaces, ternary spinodal decomposition, spreading of a liquid lens, and Kelvin-Helmholtz instability are conducted to test the model. It is found that the present model can capture accurate interfaces among three different fluids, which is attributed to its algebraical and dynamical consistency properties with the two-component model. Furthermore, the numerical results of three-phase flows agree well with the theoretical results or some available data, which demonstrates that the present LB model is a reliable and efficient method for simulating three-phase flow problems.
Lattice Boltzmann models for the grain growth in polycrystalline systems
Zheng, Yonggang; Chen, Cen; Ye, Hongfei; Zhang, Hongwu
2016-08-01
In the present work, lattice Boltzmann models are proposed for the computer simulation of normal grain growth in two-dimensional systems with/without immobile dispersed second-phase particles and involving the temperature gradient effect. These models are demonstrated theoretically to be equivalent to the phase field models based on the multiscale expansion. Simulation results of several representative examples show that the proposed models can effectively and accurately simulate the grain growth in various single- and two-phase systems. It is found that the grain growth in single-phase polycrystalline materials follows the power-law kinetics and the immobile second-phase particles can inhibit the grain growth in two-phase systems. It is further demonstrated that the grain growth can be tuned by the second-phase particles and the introduction of temperature gradient is also an effective way for the fabrication of polycrystalline materials with grained gradient microstructures. The proposed models are useful for the numerical design of the microstructure of materials and provide effective tools to guide the experiments. Moreover, these models can be easily extended to simulate two- and three-dimensional grain growth with considering the mobile second-phase particles, transient heat transfer, melt convection, etc.
Approximate solutions for the skyrmion
Ponciano, J A; Fanchiotti, H; Canal-Garcia, C A
2001-01-01
We reconsider the Euler-Lagrange equation for the Skyrme model in the hedgehog ansatz and study the analytical properties of the solitonic solution. In view of the lack of a closed form solution to the problem, we work on approximate analytical solutions. We show that Pade approximants are well suited to continue analytically the asymptotic representation obtained in terms of a power series expansion near the origin, obtaining explicit approximate solutions for the Skyrme equations. We improve the approximations by applying the 2-point Pade approximant procedure whereby the exact behaviour at spatial infinity is incorporated. An even better convergence to the exact solution is obtained by introducing a modified form for the approximants. The new representations share the same analytical properties with the exact solution at both small and large values of the radial variable r.
Hu, Kainan; Zhang, Hongwu; Geng, Shaojuan
2016-01-01
A new lattice Boltzmann scheme associated with flexible specific heat ratio is proposed. The new free degree is introduced via the internal energy associated with the internal structure. The evolution equation of the distribution function is reduced to two evolution equations. One is connected to the density and velocity, the other is of the energy. A two-dimensional lattice Boltzmann model and a three-dimensional lattice Boltzmann model are derived via the Hermite expansion. The two lattice ...
BDD Minimization for Approximate Computing
Soeken, Mathias; Grosse, Daniel; Chandrasekharan, Arun; Drechsler, Rolf
2016-01-01
We present Approximate BDD Minimization (ABM) as a problem that has application in approximate computing. Given a BDD representation of a multi-output Boolean function, ABM asks whether there exists another function that has a smaller BDD representation but meets a threshold w.r.t. an error metric. We present operators to derive approximated functions and present algorithms to exactly compute the error metrics directly on the BDD representation. An experimental evaluation demonstrates the app...
Directory of Open Access Journals (Sweden)
Anaïs Khuong
Full Text Available The goal of this study is to describe accurately how the directional information given by support inclinations affects the ant Lasius niger motion in terms of a behavioral decision. To this end, we have tracked the spontaneous motion of 345 ants walking on a 0.5×0.5 m plane canvas, which was tilted with 5 various inclinations by [Formula: see text] rad ([Formula: see text] data points. At the population scale, support inclination favors dispersal along uphill and downhill directions. An ant's decision making process is modeled using a version of the Boltzmann Walker model, which describes an ant's random walk as a series of straight segments separated by reorientation events, and was extended to take directional influence into account. From the data segmented accordingly ([Formula: see text] segments, this extension allows us to test separately how average speed, segments lengths and reorientation decisions are affected by support inclination and current walking direction of the ant. We found that support inclination had a major effect on average speed, which appeared approximately three times slower on the [Formula: see text] incline. However, we found no effect of the walking direction on speed. Contrastingly, we found that ants tend to walk longer in the same direction when they move uphill or downhill, and also that they preferentially adopt new uphill or downhill headings at turning points. We conclude that ants continuously adapt their decision making about where to go, and how long to persist in the same direction, depending on how they are aligned with the line of maximum declivity gradient. Hence, their behavioral decision process appears to combine klinokinesis with geomenotaxis. The extended Boltzmann Walker model parameterized by these effects gives a fair account of the directional dispersal of ants on inclines.
Lattice Boltzmann simulation of three-dimensional Rayleigh-Taylor instability.
Liang, H; Li, Q X; Shi, B C; Chai, Z H
2016-03-01
In this paper, the three-dimensional (3D) Rayleigh-Taylor instability (RTI) with low Atwood number (A(t)=0.15) in a long square duct (12W × W × W) is studied by using a multiple-relaxation-time lattice Boltzmann (LB) multiphase model. The effect of the Reynolds number on the interfacial dynamics and bubble and spike amplitudes at late time is investigated in detail. The numerical results show that at sufficiently large Reynolds numbers, a sequence of stages in the 3D immiscible RTI can be observed, which includes the linear growth, terminal velocity growth, reacceleration, and chaotic development stages. At late stage, the RTI induces a very complicated topology structure of the interface, and an abundance of dissociative drops are also observed in the system. The bubble and spike velocities at late stage are unstable and their values have exceeded the predictions of the potential flow theory [V. N. Goncharov, Phys. Rev. Lett. 88, 134502 (2002)]. The acceleration of the bubble front is also measured and it is found that the normalized acceleration at late time fluctuates around a constant value of 0.16. When the Reynolds number is reduced to small values, some later stages cannot be reached sequentially. The interface becomes relatively smoothed and the bubble velocity at late time is approximate to a constant value, which coincides with the results of the extended Layzer model [S.-I. Sohn, Phys. Rev. E 80, 055302(R) (2009)] and the modified potential theory [R. Banerjee, L. Mandal, S. Roy, M. Khan, and M. R. Guptae, Phys. Plasmas 18, 022109 (2011)]. In our simulations, the Graphics Processing Unit (GPU) parallel computing is also used to relieve the massive computational cost. PMID:27078453
Ion strength limit of computed excess functions based on the linearized Poisson-Boltzmann equation.
Fraenkel, Dan
2015-12-01
The linearized Poisson-Boltzmann (L-PB) equation is examined for its κ-range of validity (κ, Debye reciprocal length). This is done for the Debye-Hückel (DH) theory, i.e., using a single ion size, and for the SiS treatment (D. Fraenkel, Mol. Phys. 2010, 108, 1435), which extends the DH theory to the case of ion-size dissimilarity (therefore dubbed DH-SiS). The linearization of the PB equation has been claimed responsible for the DH theory's failure to fit with experiment at > 0.1 m; but DH-SiS fits with data of the mean ionic activity coefficient, γ± (molal), against m, even at m > 1 (κ > 0.33 Å(-1) ). The SiS expressions combine the overall extra-electrostatic potential energy of the smaller ion, as central ion-Ψa>b (κ), with that of the larger ion, as central ion-Ψb>a (κ); a and b are, respectively, the counterion and co-ion distances of closest approach. Ψa>b and Ψb>a are derived from the L-PB equation, which appears to conflict with their being effective up to moderate electrolyte concentrations (≈1 m). However, the L-PB equation can be valid up to κ ≥ 1.3 Å(-1) if one abandons the 1/κ criterion for its effectiveness and, instead, use, as criterion, the mean-field electrostatic interaction potential of the central ion with its ion cloud, at a radial distance dividing the cloud charge into two equal parts. The DH theory's failure is, thus, not because of using the L-PB equation; the lethal approximation is assigning a single size to the positive and negative ions.
Hu, Kainan; Geng, Shaojuan
2016-01-01
A new lattice Boltzmann scheme associated with flexible specific heat ratio is proposed. The new free degree is introduced via the internal energy associated with the internal structure. The evolution equation of the distribution function is reduced to two evolution equations. One is connected to the density and velocity, the other is of the energy. A two-dimensional lattice Boltzmann model and a three-dimensional lattice Boltzmann model are derived via the Hermite expansion. The two lattice Boltzmann models are applied to simulating the shock tube of one dimension. Good agreement between the numerical results and the analytical solutions are obtained.
Thermal conductivity for a bidimensional dilute gas within the Chapman-Enskog approximation
Mendez, A R; Escobar, Eric
2015-01-01
In this work we explicitly calculate the thermal conductivity for a bidimensional dilute gas of neutral molecules by solving Boltzmann's equation. Chapman-Enskog's method is used in order to analytically obtain the transport coefficient to first approximation. The result is expressed in terms of a collision integral for a unspecified molecular interaction model. The particular case of a hard disks model is addressed yielding a T 1/2 dependence with the temperature which is consistent with the one obtained by J. V. Sengers [1] and widely used in the literature as the low density limit in the Enskog expansion. The corresponding value for bidimensional Maxwellian molecules is also obtained.
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
Kaufman, Arie; Fan, Zhe; Petkov, Kaloian
2009-06-01
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
Implementing the lattice Boltzmann model on commodity graphics hardware
International Nuclear Information System (INIS)
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
Podolsky electromagnetism and a modification in Stefan-Boltzmann law
Energy Technology Data Exchange (ETDEWEB)
Bonin, Carlos Alberto; Bufalo, Rodrigo Santos; Escobar, Bruto Max Pimentel; Zambrano, German Enrique Ramos [Instituto de Fisica Teorica (IFT/UNESP), Sao Paulo, SP (Brazil)
2009-07-01
Full text. As it is well-known, gauge fields that emerge from the gauge principle are massless vector fields. Considering the photon as a Proca particle, experience sets an upper limit on its mass. This limit is m{sub Proca} < 6X10{sup -17}eV (PDG 2006). However, a mass term, regardless how small, breaks the gauge symmetry. Nevertheless, there exists a theory in which is possible to introduce a mass term preserving all symmetries of Maxwell electromagnetism, including the gauge one: such theory is known as Podolsky Electromagnetism. Podolsky theory is a second- order-derivative theory and has some remarkable properties, despite those already mentioned: the theory has two sectors, a massive one and massless one, it depends on a free parameter (which happens to be the mass of the massive sector) that, like all other elementary particles's masses of the Standard Model, must be fixed through experiences, and the fact that the electrostatic potential is finite everywhere, including over a punctual charge. Just like Maxwell electromagnetism, Podolsky's is a constrained theory and, since it is of second order in the derivatives, it consists in a much richer theoretical structure. Therefore, from both, theoretical and experimental points of view, Podolsky electromagnetism is a very attractive theory. In this work we study a gas of Podolsky photons at finite temperature through path integration. We show that the massless sector leads to the famous Planck's law for black-body radiation and, therefore, to the Stefan-Boltzmann law. We also show that the massive sector of the Podolsky theory induces a modification in both these laws. It is possible to set limits on the Podolsky parameter through comparison of our results with data from cosmic microwave background radiation. (author)
Podolsky electromagnetism and a modification in Stefan-Boltzmann law
International Nuclear Information System (INIS)
Full text. As it is well-known, gauge fields that emerge from the gauge principle are massless vector fields. Considering the photon as a Proca particle, experience sets an upper limit on its mass. This limit is mProca -17eV (PDG 2006). However, a mass term, regardless how small, breaks the gauge symmetry. Nevertheless, there exists a theory in which is possible to introduce a mass term preserving all symmetries of Maxwell electromagnetism, including the gauge one: such theory is known as Podolsky Electromagnetism. Podolsky theory is a second- order-derivative theory and has some remarkable properties, despite those already mentioned: the theory has two sectors, a massive one and massless one, it depends on a free parameter (which happens to be the mass of the massive sector) that, like all other elementary particles's masses of the Standard Model, must be fixed through experiences, and the fact that the electrostatic potential is finite everywhere, including over a punctual charge. Just like Maxwell electromagnetism, Podolsky's is a constrained theory and, since it is of second order in the derivatives, it consists in a much richer theoretical structure. Therefore, from both, theoretical and experimental points of view, Podolsky electromagnetism is a very attractive theory. In this work we study a gas of Podolsky photons at finite temperature through path integration. We show that the massless sector leads to the famous Planck's law for black-body radiation and, therefore, to the Stefan-Boltzmann law. We also show that the massive sector of the Podolsky theory induces a modification in both these laws. It is possible to set limits on the Podolsky parameter through comparison of our results with data from cosmic microwave background radiation. (author)
Nonlinear approximation with redundant dictionaries
DEFF Research Database (Denmark)
Borup, Lasse; Nielsen, M.; Gribonval, R.
2005-01-01
In this paper we study nonlinear approximation and data representation with redundant function dictionaries. In particular, approximation with redundant wavelet bi-frame systems is studied in detail. Several results for orthonormal wavelets are generalized to the redundant case. In general...
Approximate Inference in Probabilistic Models
DEFF Research Database (Denmark)
Opper, Manfred; Winther, Ole
2004-01-01
We present a framework for approximate inference in probabilistic data models which is based on free energies. The free energy is constructed from two approximating distributions which encode different aspects of the intractable model. Consistency between distributions is required on a chosen set...
The Logic of Approximate Dependence
Väänänen, Jouko
2014-01-01
We extend the treatment of functional dependence, the basic concept of dependence logic, to include the possibility of dependence with a limited number of exceptions. We call this approximate dependence. The main result of the paper is a Completeness Theorem for approximate dependence atoms. We point out some problematic features of this which suggests that we should consider multi-teams, not just teams.
Beyond the random phase approximation
DEFF Research Database (Denmark)
Olsen, Thomas; Thygesen, Kristian S.
2013-01-01
We assess the performance of a recently proposed renormalized adiabatic local density approximation (rALDA) for ab initio calculations of electronic correlation energies in solids and molecules. The method is an extension of the random phase approximation (RPA) derived from time-dependent density...
Approximate circuits for increased reliability
Energy Technology Data Exchange (ETDEWEB)
Hamlet, Jason R.; Mayo, Jackson R.
2015-12-22
Embodiments of the invention describe a Boolean circuit having a voter circuit and a plurality of approximate circuits each based, at least in part, on a reference circuit. The approximate circuits are each to generate one or more output signals based on values of received input signals. The voter circuit is to receive the one or more output signals generated by each of the approximate circuits, and is to output one or more signals corresponding to a majority value of the received signals. At least some of the approximate circuits are to generate an output value different than the reference circuit for one or more input signal values; however, for each possible input signal value, the majority values of the one or more output signals generated by the approximate circuits and received by the voter circuit correspond to output signal result values of the reference circuit.
Approximate circuits for increased reliability
Energy Technology Data Exchange (ETDEWEB)
Hamlet, Jason R.; Mayo, Jackson R.
2015-08-18
Embodiments of the invention describe a Boolean circuit having a voter circuit and a plurality of approximate circuits each based, at least in part, on a reference circuit. The approximate circuits are each to generate one or more output signals based on values of received input signals. The voter circuit is to receive the one or more output signals generated by each of the approximate circuits, and is to output one or more signals corresponding to a majority value of the received signals. At least some of the approximate circuits are to generate an output value different than the reference circuit for one or more input signal values; however, for each possible input signal value, the majority values of the one or more output signals generated by the approximate circuits and received by the voter circuit correspond to output signal result values of the reference circuit.
N-variable rational approximants
International Nuclear Information System (INIS)
''Desirable properties'' of a two-variable generalization of Pade approximants are laid down. The ''Chisholm approximants'' are defined and are shown to obey nearly all of these properties; the alternative ways of completing a unique definition are discussed, and the ''prong structure'' of the defining equations is elucidated. Several generalizations and variants of Chisholm approximants are described: N-variable diagonal, 2-variable simple off-diagonal, N-variable simple and general off-diagonal, and rotationally covariant 2-variable approximants. All of the 2-variable approximants are capable of representing singularities of functions of two variables, and of analytically continuing beyond the polycylinder of convergence of the double series. 8 figures
Coupling of replica exchange simulations to a non-Boltzmann structure reservoir.
Roitberg, Adrian E; Okur, Asim; Simmerling, Carlos
2007-03-15
Computing converged ensemble properties remains challenging for large biomolecules. Replica exchange molecular dynamics (REMD) can significantly increase the efficiency of conformational sampling by using high temperatures to escape kinetic traps. Several groups, including ours, introduced the idea of coupling replica exchange to a pre-converged, Boltzmann-populated reservoir, usually at a temperature higher than that of the highest temperature replica. This procedure reduces computational cost because the long simulation times needed for extensive sampling are only carried out for a single temperature. However, a weakness of the approach is that the Boltzmann-weighted reservoir can still be difficult to generate. We now present the idea of employing a non-Boltzmann reservoir, whose structures can be generated through more efficient conformational sampling methods. We demonstrate that the approach is rigorous and derive a correct statistical mechanical exchange criterion between the reservoir and the replicas that drives Boltzmann-weighted probabilities for the replicas. We test this approach on the trpzip2 peptide and demonstrate that the resulting thermal stability profile is essentially indistinguishable from that obtained using very long (>100 ns) standard REMD simulations. The convergence of this reservoir-aided REMD is significantly faster than for regular REMD. Furthermore, we demonstrate that modification of the exchange criterion is essential; REMD simulations using a standard exchange function with the non-Boltzmann reservoir produced incorrect results.
Parvan, A S
2016-01-01
The Tsallis statistics was applied to describe the experimental data on the transverse momentum distributions of hadrons for the first time. This result was achieved only for the massless Maxwell-Boltzmann particles. We considered the energy dependence of the parameters for both the Tsallis statistics and its zeroth term approximation for the pions produced in $pp$ collisions at high energies and found that the results of the zeroth term approximation deviate from the results of the Tsallis statistics only at low NA61/SHINE energies when the value of the parameter $q$ is close to unity. At higher energies, when the value of the parameter $q$ deviates essentially from the unity, the zeroth term approximation satisfactorily recovers the results of the Tsallis statistics.
André, Raíla
2014-01-01
In this work we analyze the dynamics of collisionless self-gravitating systems described by the f(R)-gravity and Boltzmann equation in the weak field approximation, focusing on the Jeans instability for theses systems. The field equations in this approximation were obtained within the Palatini formalism. Through the solution of coupled equations we achieved the collapse criterion for infinite homogeneous fluid and stellar systems, which is given by a dispersion relation. This result is compared with the results of the standard case and the case for f(R)-gravity in metric formalism, in order to see the difference among them. The limit of instability varies according to which theory of gravity is adopted.
The efficiency of Flory approximation
International Nuclear Information System (INIS)
The Flory approximation for the self-avoiding chain problem is compared with a conventional perturbation theory expansion. While in perturbation theory each term is averaged over the unperturbed set of configurations, the Flory approximation is equivalent to the perturbation theory with the averaging over the stretched set of configurations. This imposes restrictions on the integration domain in higher order terms and they can be treated self-consistently. The accuracy δν/ν of Flory approximation for self-avoiding chain problems is estimated to be 2-5% for 1 < d < 4. (orig.)
Goffin, Mark A.; Baker, Christopher M. J.; Buchan, Andrew G.; Pain, Christopher C.; Eaton, Matthew D.; Smith, Paul N.
2013-06-01
This article presents a method for goal-based anisotropic adaptive methods for the finite element method applied to the Boltzmann transport equation. The neutron multiplication factor, k, is used as the goal of the adaptive procedure. The anisotropic adaptive algorithm requires error measures for k with directional dependence. General error estimators are derived for any given functional of the flux and applied to k to acquire the driving force for the adaptive procedure. The error estimators require the solution of an appropriately formed dual equation. Forward and dual error indicators are calculated by weighting the Hessian of each solution with the dual and forward residual respectively. The Hessian is used as an approximation of the interpolation error in the solution which gives rise to the directional dependence. The two indicators are combined to form a single error metric that is used to adapt the finite element mesh. The residual is approximated using a novel technique arising from the sub-grid scale finite element discretisation. Two adaptive routes are demonstrated: (i) a single mesh is used to solve all energy groups, and (ii) a different mesh is used to solve each energy group. The second method aims to capture the benefit from representing the flux from each energy group on a specifically optimised mesh. The k goal-based adaptive method was applied to three examples which illustrate the superior accuracy in criticality problems that can be obtained.
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
Development of a coarse-grained water forcefield via multistate iterative Boltzmann inversion
Moore, Timothy C; McCabe, Clare
2015-01-01
A coarse-grained water model is developed using multistate iterative Boltzmann inversion. Following previous work, the k-means algorithm is used to dynamically map multiple water molecules to a single coarse-grained bead, allowing the use of structure-based coarse-graining methods. The model is derived to match the bulk and interfacial properties of liquid water and improves upon previous work that used single state iterative Boltzmann inversion. The model accurately reproduces the density and structural correlations of water at 305 K and 1.0 atm, stability of a liquid droplet at 305 K, and shows little tendency to crystallize at physiological conditions. This work also illustrates several advantages of using multistate iterative Boltzmann inversion for deriving generally applicable coarse-grained forcefields.
Comment on ‘A low-uncertainty measurement of the Boltzmann constant’
Macnaughton, Donald B.
2016-02-01
The International Committee for Weights and Measures has projected a major revision of the International System of Units in which all the base units will be defined by fixing the values of certain fundamental constants of nature. To assist, de Podesta et al recently experimentally obtained a precise new estimate of the Boltzmann constant. This estimate is proposed as a basis for the redefinition of the unit of temperature, the kelvin. The present paper reports a reanalysis of de Podesta et al’s data that reveals systematic non-random patterns in the residuals of the key fitted model equation. These patterns violate the assumptions underlying the analysis and thus they raise questions about the validity of de Podesta et al’s estimate of the Boltzmann constant. An approach is discussed to address these issues, which should lead to an accurate estimate of the Boltzmann constant with a lower uncertainty.
The standard map: From Boltzmann-Gibbs statistics to Tsallis statistics.
Tirnakli, Ugur; Borges, Ernesto P
2016-03-23
As well known, Boltzmann-Gibbs statistics is the correct way of thermostatistically approaching ergodic systems. On the other hand, nontrivial ergodicity breakdown and strong correlations typically drag the system into out-of-equilibrium states where Boltzmann-Gibbs statistics fails. For a wide class of such systems, it has been shown in recent years that the correct approach is to use Tsallis statistics instead. Here we show how the dynamics of the paradigmatic conservative (area-preserving) stan-dard map exhibits, in an exceptionally clear manner, the crossing from one statistics to the other. Our results unambiguously illustrate the domains of validity of both Boltzmann-Gibbs and Tsallis statistical distributions. Since various important physical systems from particle confinement in magnetic traps to autoionization of molecular Rydberg states, through particle dynamics in accelerators and comet dynamics, can be reduced to the standard map, our results are expected to enlighten and enable an improved interpretation of diverse experimental and observational results.
Boltzmann-Fokker-Planck calculations using standard discrete-ordinates codes
International Nuclear Information System (INIS)
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
Entropic Lattice Boltzmann Method for Moving and Deforming Geometries in Three Dimensions
Dorschner, B; Karlin, I V
2016-01-01
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 Dorschner et al. [11] as well as for three dimensional one-way coupled simulations of engine-type geometries in Dorschner et al. [12] 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, turbulence and deformable meshes. 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 at a Reynolds number of Re = 40000 an...
Measuring the usefulness of hidden units in Boltzmann machines with mutual information.
Berglund, Mathias; Raiko, Tapani; Cho, Kyunghyun
2015-04-01
Restricted Boltzmann machines (RBMs) and deep Boltzmann machines (DBMs) are important models in deep learning, but it is often difficult to measure their performance in general, or measure the importance of individual hidden units in specific. We propose to use mutual information to measure the usefulness of individual hidden units in Boltzmann machines. The measure is fast to compute, and serves as an upper bound for the information the neuron can pass on, enabling detection of a particular kind of poor training results. We confirm experimentally that the proposed measure indicates how much the performance of the model drops when some of the units of an RBM are pruned away. We demonstrate the usefulness of the measure for early detection of poor training in DBMs.
Simulation of Magnetorheological Fluids Based on Lattice Boltzmann Method with Double Meshes
Directory of Open Access Journals (Sweden)
Xinhua Liu
2012-01-01
Full Text Available In order to study the rheological characteristics of magnetorheological fluids, a novel approach based on the two-component Lattice Boltzmann method with double meshes was proposed, and the micro-scale structures of magnetorheological fluids in different strength magnetic fields were simulated. The framework composed of three steps for the simulation of magnetorheological fluids was addressed, and the double meshes method was elaborated. Moreover, the various internal and external forces acting on the magnetic particles were analyzed and calculated. The two-component Lattice Boltzmann model was set up, and the flowchart for the simulation of magnetorheological fluids based on the two-component Lattice Boltzmann method with double meshes was designed. Finally, a physics experiment was carried out, and the simulation examples were provided. The comparison results indicated that the proposed approach was feasible, efficient, and outperforming others.
Michael Sukop,; Cunningham, Kevin J.
2014-01-01
Digital optical borehole images at approximately 2 mm vertical resolution and borehole caliper data were used to create three-dimensional renderings of the distribution of (1) matrix porosity and (2) vuggy megaporosity for the karst carbonate Biscayne aquifer in southeastern Florida. The renderings based on the borehole data were used as input into Lattice Boltzmann methods to obtain intrinsic permeability estimates for this extremely transmissive aquifer, where traditional aquifer test methods may fail due to very small drawdowns and non-Darcian flow that can reduce apparent hydraulic conductivity. Variogram analysis of the borehole data suggests a nearly isotropic rock structure at lag lengths up to the nominal borehole diameter. A strong correlation between the diameter of the borehole and the presence of vuggy megaporosity in the data set led to a bias in the variogram where the computed horizontal spatial autocorrelation is strong at lag distances greater than the nominal borehole size. Lattice Boltzmann simulation of flow across a 0.4 × 0.4 × 17 m (2.72 m3 volume) parallel-walled column of rendered matrix and vuggy megaporosity indicates a high hydraulic conductivity of 53 m s−1. This value is similar to previous Lattice Boltzmann calculations of hydraulic conductivity in smaller limestone samples of the Biscayne aquifer. The development of simulation methods that reproduce dual-porosity systems with higher resolution and fidelity and that consider flow through horizontally longer renderings could provide improved estimates of the hydraulic conductivity and help to address questions about the importance of scale.
Binary nucleation beyond capillarity approximation
Kalikmanov, V.I.
2010-01-01
Large discrepancies between binary classical nucleation theory (BCNT) and experiments result from adsorption effects and inability of BCNT, based on the phenomenological capillarity approximation, to treat small clusters. We propose a model aimed at eliminating both of these deficiencies. Adsorption
Approximate Implicitization Using Linear Algebra
Directory of Open Access Journals (Sweden)
Oliver J. D. Barrowclough
2012-01-01
Full Text Available We consider a family of algorithms for approximate implicitization of rational parametric curves and surfaces. The main approximation tool in all of the approaches is the singular value decomposition, and they are therefore well suited to floating-point implementation in computer-aided geometric design (CAGD systems. We unify the approaches under the names of commonly known polynomial basis functions and consider various theoretical and practical aspects of the algorithms. We offer new methods for a least squares approach to approximate implicitization using orthogonal polynomials, which tend to be faster and more numerically stable than some existing algorithms. We propose several simple propositions relating the properties of the polynomial bases to their implicit approximation properties.
Weighted approximation with varying weight
Totik, Vilmos
1994-01-01
A new construction is given for approximating a logarithmic potential by a discrete one. This yields a new approach to approximation with weighted polynomials of the form w"n"(" "= uppercase)P"n"(" "= uppercase). The new technique settles several open problems, and it leads to a simple proof for the strong asymptotics on some L p(uppercase) extremal problems on the real line with exponential weights, which, for the case p=2, are equivalent to power- type asymptotics for the leading coefficients of the corresponding orthogonal polynomials. The method is also modified toyield (in a sense) uniformly good approximation on the whole support. This allows one to deduce strong asymptotics in some L p(uppercase) extremal problems with varying weights. Applications are given, relating to fast decreasing polynomials, asymptotic behavior of orthogonal polynomials and multipoint Pade approximation. The approach is potential-theoretic, but the text is self-contained.
Approximate maximizers of intricacy functionals
Buzzi, Jerome
2009-01-01
G. Edelman, O. Sporns, and G. Tononi introduced in theoretical biology the neural complexity of a family of random variables. This functional is a special case of intricacy, i.e., an average of the mutual information of subsystems whose weights have good mathematical properties. Moreover, its maximum value grows at a definite speed with the size of the system. In this work, we compute exactly this speed of growth by building "approximate maximizers" subject to an entropy condition. These approximate maximizers work simultaneously for all intricacies. We also establish some properties of arbitrary approximate maximizers, in particular the existence of a threshold in the size of subsystems of approximate maximizers: most smaller subsystems are almost equidistributed, most larger subsystems determine the full system. The main ideas are a random construction of almost maximizers with a high statistical symmetry and the consideration of entropy profiles, i.e., the average entropies of sub-systems of a given size. ...
The Initial Boundary Value Problem for the Boltzmann Equation with Soft Potential
Liu, Shuangqian; Yang, Xiongfeng
2016-08-01
Boundary effects are central to the dynamics of the dilute particles governed by the Boltzmann equation. In this paper, we study both the diffuse reflection and the specular reflection boundary value problems for the Boltzmann equation with a soft potential, in which the collision kernel is ruled by the inverse power law. For the diffuse reflection boundary condition, based on an L 2 argument and its interplay with intricate {L^∞} analysis for the linearized Boltzmann equation, we first establish the global existence and then obtain the exponential decay in {L^∞} space for the nonlinear Boltzmann equation in general classes of bounded domain. It turns out that the zero lower bound of the collision frequency and the singularity of the collision kernel lead to some new difficulties for achieving the a priori {L^∞} estimates and time decay rates of the solution. In the course of the proof, we capture some new properties of the probability integrals along the stochastic cycles and improve the {L^2-L^∞} theory to give a more direct approach to overcome those difficulties. As to the specular reflection condition, our key contribution is to develop a new time-velocity weighted {L^∞} theory so that we could deal with the greater difficulties stemming from the complicated velocity relations among the specular cycles and the zero lower bound of the collision frequency. From this new point, we are also able to prove that the solutions of the linearized Boltzmann equation tend to equilibrium exponentially in {L^∞} space with the aid of the L 2 theory and a bootstrap argument. These methods, in the latter case, can be applied to the Boltzmann equation with soft potential for all other types of boundary condition.
Reinforcement Learning via AIXI Approximation
Veness, Joel; Ng, Kee Siong; Hutter, Marcus; Silver, David
2010-01-01
This paper introduces a principled approach for the design of a scalable general reinforcement learning agent. This approach is based on a direct approximation of AIXI, a Bayesian optimality notion for general reinforcement learning agents. Previously, it has been unclear whether the theory of AIXI could motivate the design of practical algorithms. We answer this hitherto open question in the affirmative, by providing the first computationally feasible approximation to the AIXI agent. To deve...
Binary nucleation beyond capillarity approximation
Kalikmanov, V.I.
2010-01-01
Large discrepancies between binary classical nucleation theory (BCNT) and experiments result from adsorption effects and inability of BCNT, based on the phenomenological capillarity approximation, to treat small clusters. We propose a model aimed at eliminating both of these deficiencies. Adsorption is taken into account within Gibbsian approximation. Binary clusters are treated by means of statistical-mechanical considerations: tracing out the molecular degrees of freedom of the more volatil...
A Characteristic Non-Reflecting Boundary Treatment in Lattice Boltzmann Method
Institute of Scientific and Technical Information of China (English)
KIM Dehee; KIM Hyung Min; JHON Myung S.; VINAY Ⅲ Stephen J.; BUCHANAN John
2008-01-01
In lattice Boltzmann methods, disturbances develop at the initial stages of the simulation, the decay characteristics depend mainly on boundary treatment methods; open boundary conditions such as equilibrium and bounce-back schemes potentially generate uncontrollable disturbances. Excessive disturbances originate from non-physical reflecting waves at boundaries. Characteristic boundary conditions utilizing the signs of waves at boundaries which suppress these reflecting waves, as well as their implementation in the lattice Boltzmann method, are introduced herein. The performance of our novel boundary treatment method to effectively suppress excessive disturbances is verified by three different numerical experiments.
Lattice Boltzmann Model for The Volume-Averaged Navier-Stokes Equations
Zhang, Jingfeng; Ouyang, Jie
2014-01-01
A numerical method, based on discrete lattice Boltzmann equation, is presented for solving the volume-averaged Navier-Stokes equations. With a modified equilibrium distribution and an additional forcing term, the volume-averaged Navier-Stokes equations can be recovered from the lattice Boltzmann equation in the limit of small Mach number by the Chapman-Enskog analysis and Taylor expansion. Due to its advantages such as explicit solver and inherent parallelism, the method appears to be more competitive with traditional numerical techniques. Numerical simulations show that the proposed model can accurately reproduce both the linear and nonlinear drag effects of porosity in the fluid flow through porous media.
Lattice Boltzmann method for bosons and fermions and the fourth order Hermite polynomial expansion
Coelho, Rodrigo C V; Doria, M M; Pereira, R M; Aibe, Valter Yoshihiko
2013-01-01
The Boltzmann equation with the Bhatnagar-Gross-Krook collision operator is considered for the Bose-Einstein and Fermi-Dirac equilibrium distribution functions. We show that the expansion of the microscopic velocity in terms of Hermite polynomials must be carried until the fourth order to correctly describe the energy equation. The viscosity and thermal coefficients, previously obtained by J.Y. Yang et al through the Uehling-Uhlenbeck approach, are also derived here. Thus the construction of a lattice Boltzmann method for the quantum fluid is possible provided that the Bose-Einstein and Fermi-Dirac equilibrium distribution functions are expanded until fourth order in the Hermite polynomials.
From Newton's law to the linear Boltzmann equation without cut-off
Ayi, Nathalie
2016-01-01
We provide a rigorous derivation of the linear Boltzmann equation without cutoff 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. 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 combin...
Lattice Boltzmann equation calculation of internal, pressure-driven turbulent flow
Hammond, L A; 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 sup 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.
Numerical scheme for a spatially inhomogeneous matrix-valued quantum Boltzmann equation
Lu, Jianfeng
2014-01-01
We develop an efficient algorithm for a spatially inhomogeneous matrix-valued quantum Boltzmann equation derived from the Hubbard model. The distribution functions are 2 x 2 matrix-valued to accommodate the spin degree of freedom, and the scalar quantum Boltzmann equation is recovered as special case when all matrices are proportional to the identity. We use Fourier discretization and fast Fourier transform to efficiently evaluate the collision kernel with spectral accuracy, and numerically investigate periodic, Dirichlet and Maxwell boundary conditions. Model simulations quantify the convergence to local and global thermal equilibrium.
Simulation of Blood Flow at Vessel Bifurcation by Lattice Boltzmann Method
Kang, Xiu-Ying; Liu, Da-He; Zhou, Jing; Jin, Yong-Juan
2005-11-01
The application of the lattice Boltzmann method to the large vessel bifurcation blood flow is investigated in a wide range of Reynolds numbers. The velocity, shear stress and pressure distributions at the bifurcation are presented in detail. The flow separation zones revealed with increase of Reynolds number are located in the areas of the daughter branches distal to the outer corners of the bifurcation where some deposition of particular blood components might occur to form arteriosclerosis. The results also demonstrate that the lattice Boltzmann method is adaptive to simulating the flow in larger vessels under a high Reynolds number.
A Lattice Boltzmann Model for Fluid-Solid Coupling Heat Transfer in Fractal Porous Media
Institute of Scientific and Technical Information of China (English)
CAI Jun; HUAI Xiu-Lan
2009-01-01
We report a lattice Boltzmann model that can be used to simulate fluid-solid coupling heat transfer in fractal porous media.A numerical simulation is conducted to investigate the temperature evolution under different ratios of thermal conductivity of solid matrix of porous media to that of fluid.The accordance of our simulation results with the solutions from the conventional CFD method indicates the feasibility and the reliability for the developed lattice Boltzmann model to reveal the phenomena and rules of fluid-solid coupling heat transfer in complex porous structures.
An H Theorem for Boltzmann's Equation for the Yard-Sale Model of Asset Exchange
Boghosian, Bruce M.; Johnson, Merek; Marcq, Jeremy A.
2015-12-01
In recent work (Boghosian, Phys Rev E 89:042804-042825, 2014; Boghosian, Int J Mod Phys 25:1441008-1441015, 2014), Boltzmann and Fokker-Planck equations were derived for the "Yard-Sale Model" of asset exchange. For the version of the model without redistribution, it was conjectured, based on numerical evidence, that the time-asymptotic state of the model was oligarchy—complete concentration of wealth by a single individual. In this work, we prove that conjecture by demonstrating that the Gini coefficient, a measure of inequality commonly used by economists, is an H function of both the Boltzmann and Fokker-Planck equations for the model.
Liang, Jun; Liu, Yan-Chun; Zhu, Qiao
2014-02-01
In order to further explore the effects of non-Gaussian smeared mass distribution on the thermodynamical properties of noncommutative black holes, we consider noncommutative black holes based on Maxwell-Boltzmann smeared mass distribution in (2+1)-dimensional spacetime. The thermodynamical properties of the black holes are investigated, including Hawking temperature, heat capacity, entropy and free energy. We find that multiple black holes with the same temperature do not exist, while there exists a possible decay of the noncommutative black hole based on Maxwell-Boltzmann smeared mass distribution into the rotating (commutative) BTZ black hole.
International Nuclear Information System (INIS)
In order to further explore the effects of non-Gaussian smeared mass distribution on the thermodynamical properties of noncommutative black holes, we consider noncommutative black holes based on Maxwell-Boltzmann smeared mass distribution in (2+1)-dimensional spacetime. The thermodynamical properties of the black holes are investigated, including Hawking temperature, heat capacity, entropy and free energy. We find that multiple black holes with the same temperature do not exist, while there exists a possible decay of the noncommutative black hole based on Maxwell-Boltzmann smeared mass distribution into the rotating (commutative) BTZ black hole. (authors)
Wilson, Alan
2008-08-01
It is shown that Boltzmann's methods from statistical physics can be applied to a much wider range of systems, and in a variety of disciplines, than has been commonly recognized. A similar argument can be applied to the ecological models of Lotka and Volterra. Furthermore, it is shown that the two methodologies can be applied in combination to generate the Boltzmann, Lotka and Volterra (BLV) models. These techniques enable both spatial interaction and spatial structural evolution to be modelled, and it is argued that they potentially provide a much richer modelling methodology than that currently used in the analysis of 'scale-free' networks.
Lattice Boltzmann model for Coulomb-driven flows in dielectric liquids.
Luo, Kang; Wu, Jian; Yi, Hong-Liang; Tan, He-Ping
2016-02-01
In this paper, we developed a unified lattice Boltzmann model (LBM) to simulate electroconvection in a dielectric liquid induced by unipolar charge injection. Instead of solving the complex set of coupled Navier-Stokes equations, the charge conservation equation, and the Poisson equation of electric potential, three consistent lattice Boltzmann equations are formulated. Numerical results are presented for both strong and weak injection regimes, and different scenarios for the onset and evolution of instability, bifurcation, and chaos are tracked. All LBM results are found to be highly consistent with the analytical solutions and other numerical work.
Numerical scheme for a spatially inhomogeneous matrix-valued quantum Boltzmann equation
Lu, Jianfeng; Mendl, Christian B.
2015-06-01
We develop an efficient algorithm for a spatially inhomogeneous matrix-valued quantum Boltzmann equation derived from the Hubbard model. The distribution functions are 2 × 2 matrix-valued to accommodate the spin degree of freedom, and the scalar quantum Boltzmann equation is recovered as a special case when all matrices are proportional to the identity. We use Fourier discretization and fast Fourier transform to efficiently evaluate the collision kernel with spectral accuracy, and numerically investigate periodic, Dirichlet and Maxwell boundary conditions. Model simulations quantify the convergence to local and global thermal equilibrium.
Lattice Boltzmann model for Coulomb-driven flows in dielectric liquids
Luo, Kang; Wu, Jian; Yi, Hong-Liang; Tan, He-Ping
2016-02-01
In this paper, we developed a unified lattice Boltzmann model (LBM) to simulate electroconvection in a dielectric liquid induced by unipolar charge injection. Instead of solving the complex set of coupled Navier-Stokes equations, the charge conservation equation, and the Poisson equation of electric potential, three consistent lattice Boltzmann equations are formulated. Numerical results are presented for both strong and weak injection regimes, and different scenarios for the onset and evolution of instability, bifurcation, and chaos are tracked. All LBM results are found to be highly consistent with the analytical solutions and other numerical work.
Well-Posedness of the Cauchy Problem for a Space-Dependent Anyon Boltzmann Equation
Arkeryd, Leif; Nouri, Anne
2015-01-01
A fully non-linear kinetic Boltzmann equation for anyons is studied in a periodic 1d setting with large initial data. Strong L 1 solutions are obtained for the Cauchy problem. The main results concern global existence, uniqueness and stabililty. We use the Bony functional, the two-dimensional velocity frame specific for anyons, and an initial layer analysis that moves the solution away from a critical value. 1 Anyons and the Boltzmann equation. Let us first recall the definition of anyon. Con...
A Fokker-Planck model of the Boltzmann equation with correct Prandtl number
Mathiaud, J
2015-01-01
We propose an extension of the Fokker-Planck model of the Boltzmann equation to get a correct Prandtl number in the Compressible Navier-Stokes asymptotics. This is obtained by replacing the diffusion coefficient (which is the equilibrium temperature) by a non diagonal temperature tensor, like the Ellipsoidal-Statistical model (ES) is obtained from the Bathnagar-Gross-Krook model (BGK) of the Boltzmann equation. Our model is proved to satisfy the properties of conservation and a H-theorem. A Chapman-Enskog analysis and two numerical tests show that a correct Prandtl number of 2/3 can be obtained.
Twisted inhomogeneous Diophantine approximation and badly approximable sets
Harrap, Stephen
2010-01-01
For any real pair i, j geq 0 with i+j=1 let Bad(i, j) denote the set of (i, j)-badly approximable pairs. That is, Bad(i, j) consists of irrational vectors x:=(x_1, x_2) in R^2 for which there exists a positive constant c(x) such that max {||qx_1||^(-i), ||qx_2||^(-j)} > c(x)/q for all q in N. Building on a result of Kurzweil, a new characterization of the set Bad(i, j) in terms of `well-approximable' vectors in the area of `twisted' inhomogeneous Diophantine approximation is established. In addition, it is shown that Bad^x(i, j), the `twisted' inhomogeneous analogue of Bad(i, j), has full Hausdorff dimension 2 when x is chosen from the set Bad(i, j).
Wavelet Sparse Approximate Inverse Preconditioners
Chan, Tony F.; Tang, W.-P.; Wan, W. L.
1996-01-01
There is an increasing interest in using sparse approximate inverses as preconditioners for Krylov subspace iterative methods. Recent studies of Grote and Huckle and Chow and Saad also show that sparse approximate inverse preconditioner can be effective for a variety of matrices, e.g. Harwell-Boeing collections. Nonetheless a drawback is that it requires rapid decay of the inverse entries so that sparse approximate inverse is possible. However, for the class of matrices that, come from elliptic PDE problems, this assumption may not necessarily hold. Our main idea is to look for a basis, other than the standard one, such that a sparse representation of the inverse is feasible. A crucial observation is that the kind of matrices we are interested in typically have a piecewise smooth inverse. We exploit this fact, by applying wavelet techniques to construct a better sparse approximate inverse in the wavelet basis. We shall justify theoretically and numerically that our approach is effective for matrices with smooth inverse. We emphasize that in this paper we have only presented the idea of wavelet approximate inverses and demonstrated its potential but have not yet developed a highly refined and efficient algorithm.
The structure of approximate groups
Breuillard, Emmanuel; Tao, Terence
2011-01-01
Let K >= 1 be a parameter. A K-approximate group is a finite set A in a (local) group which contains the identity, is symmetric, and such that A^2 is covered by K left translates of A. The main result of this paper is a qualitative description of approximate groups as being essentially finite-by-nilpotent, answering a conjecture of H. Helfgott and E. Lindenstrauss. This may be viewed as a generalisation of the Freiman-Ruzsa theorem on sets of small doubling in the integers to arbitrary groups. We begin by establishing a correspondence principle between approximate groups and locally compact (local) groups that allows us to recover many results recently established in a fundamental paper of Hrushovski. In particular we establish that approximate groups can be approximately modeled by Lie groups. To prove our main theorem we apply some additional arguments essentially due to Gleason. These arose in the solution of Hilbert's fifth problem in the 1950s. Applications of our main theorem include a finitary refineme...
Relativistic regular approximations revisited: An infinite-order relativistic approximation
International Nuclear Information System (INIS)
The concept of the regular approximation is presented as the neglect of the energy dependence of the exact Foldy - Wouthuysen transformation of the Dirac Hamiltonian. Expansion of the normalization terms leads immediately to the zeroth-order regular approximation (ZORA) and first-order regular approximation (FORA) Hamiltonians as the zeroth- and first-order terms of the expansion. The expansion may be taken to infinite order by using an un-normalized Foldy - Wouthuysen transformation, which results in the ZORA Hamiltonian and a non-unit metric. This infinite-order regular approximation, IORA, has eigenvalues which differ from the Dirac eigenvalues by order E3/c4 for a hydrogen-like system, which is a considerable improvement over the ZORA eigenvalues, and similar to the non-variational FORA energies. A further perturbation analysis yields a third-order correction to the IORA energies, TIORA. Results are presented for several systems including the neutral U atom. The IORA eigenvalues for all but the 1s spinor of the neutral system are superior even to the scaled ZORA energies, which are exact for the hydrogenic system. The third-order correction reduces the IORA error for the inner orbitals to a very small fraction of the Dirac eigenvalue. copyright 1999 American Institute of Physics
Approximating Graphic TSP by Matchings
Mömke, Tobias
2011-01-01
We present a framework for approximating the metric TSP based on a novel use of matchings. Traditionally, matchings have been used to add edges in order to make a given graph Eulerian, whereas our approach also allows for the removal of certain edges leading to a decreased cost. For the TSP on graphic metrics (graph-TSP), the approach yields a 1.461-approximation algorithm with respect to the Held-Karp lower bound. For graph-TSP restricted to a class of graphs that contains degree three bounded and claw-free graphs, we show that the integrality gap of the Held-Karp relaxation matches the conjectured ratio 4/3. The framework allows for generalizations in a natural way and also leads to a 1.586-approximation algorithm for the traveling salesman path problem on graphic metrics where the start and end vertices are prespecified.
Approximation methods in probability theory
Čekanavičius, Vydas
2016-01-01
This book presents a wide range of well-known and less common methods used for estimating the accuracy of probabilistic approximations, including the Esseen type inversion formulas, the Stein method as well as the methods of convolutions and triangle function. Emphasising the correct usage of the methods presented, each step required for the proofs is examined in detail. As a result, this textbook provides valuable tools for proving approximation theorems. While Approximation Methods in Probability Theory will appeal to everyone interested in limit theorems of probability theory, the book is particularly aimed at graduate students who have completed a standard intermediate course in probability theory. Furthermore, experienced researchers wanting to enlarge their toolkit will also find this book useful.
Reinforcement Learning via AIXI Approximation
Veness, Joel; Hutter, Marcus; Silver, David
2010-01-01
This paper introduces a principled approach for the design of a scalable general reinforcement learning agent. This approach is based on a direct approximation of AIXI, a Bayesian optimality notion for general reinforcement learning agents. Previously, it has been unclear whether the theory of AIXI could motivate the design of practical algorithms. We answer this hitherto open question in the affirmative, by providing the first computationally feasible approximation to the AIXI agent. To develop our approximation, we introduce a Monte Carlo Tree Search algorithm along with an agent-specific extension of the Context Tree Weighting algorithm. Empirically, we present a set of encouraging results on a number of stochastic, unknown, and partially observable domains.
Concept Approximation between Fuzzy Ontologies
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
Fuzzy ontologies are efficient tools to handle fuzzy and uncertain knowledge on the semantic web; but there are heterogeneity problems when gaining interoperability among different fuzzy ontologies. This paper uses concept approximation between fuzzy ontologies based on instances to solve the heterogeneity problems. It firstly proposes an instance selection technology based on instance clustering and weighting to unify the fuzzy interpretation of different ontologies and reduce the number of instances to increase the efficiency. Then the paper resolves the problem of computing the approximations of concepts into the problem of computing the least upper approximations of atom concepts. It optimizes the search strategies by extending atom concept sets and defining the least upper bounds of concepts to reduce the searching space of the problem. An efficient algorithm for searching the least upper bounds of concept is given.
Lattice Boltzmann Hydrodynamic and Transport Modeling of Everglades Mangrove Estuaries
Sukop, M. C.; Engel, V.
2010-12-01
Lattice Boltzmann methods are being developed and applied to simulate groundwater and surface water flows, and heat, solute, and particle transport. Their ability to solve Navier-Stokes, St. Venant, or Darcy equations with closely coupled solute transport and density-dependent flow effects in geometrically complex domains is attractive for inverse modeling of tracer release data and forward modeling of carbon transport in mangrove estuaries under various future conditions. Key physical processes to be simulated include tidal cycles, storm surge, sea level change, variable upstream stage, subsurface groundwater inputs, and precipitation/recharge and their effects on estuary salinity and carbon transport in the estuaries and groundwater beneath the mangroves. Carbon sources and storage in the aquifer and exchanges at the mangrove-estuary interface and carbon transformations in the water column also need to be simulated. Everglades tidal mangrove estuaries are characterized by relatively high velocity (approaching 1 m s-1) tidal flows. The channels are generally less than 2 m in depth. Tidal fluctuations approach 2 m leading to significant areas of periodic inundation and emergence of oyster beds, shell beaches, mangrove root masses, and sandy beaches. Initial models are two-dimensional, although a three-dimensional model explicitly incorporating bathymetry, density-dependent flow, and wind-driven circulation could be developed. Preliminary work highlights some of the abilities of early models. A satellite image of a 64-km2 area surrounding a CO2 flux tower is used to provide the model geometry. Model resolution is 15 m per grid node. A sinusoidal tidal stage variation and constant, high salinity are applied to the Gulf side of the model while a constant stage (corresponding to mean tide), zero salinity boundary is applied on the inland side. The Navier-Stokes equations coupled with the advection-diffusion equation are solved in the open channels. The mangrove areas
Noronha, Jorge
2015-01-01
In this paper we obtain an analytical solution of the relativistic Boltzmann equation under the relaxation time approximation that describes the out-of-equilibrium dynamics of a radially expanding massless gas. This solution is found by mapping this expanding system in flat spacetime to a static flow in the curved spacetime $\\mathrm{AdS}_{2}\\otimes \\mathrm{S}_{2}$. We further derive explicit analytic expressions for the momentum dependence of the single particle distribution function as well as for the spatial dependence of its moments. We find that this dissipative system has the ability to flow as a perfect fluid even though its entropy density does not match the equilibrium form. The non-equilibrium contribution to the entropy density is shown to be due to higher order scalar moments (which possess no hydrodynamical interpretation) of the Boltzmann equation that can remain out of equilibrium but do not couple to the energy-momentum tensor of the system. Thus, in this system the slowly moving hydrodynamic d...
An Approximation Ratio for Biclustering
Puolamäki, Kai; Hanhijärvi, Sami; Garriga, Gemma C
2007-01-01
The problem of biclustering consists of the simultaneous clustering of rows and columns of a matrix such that each of the submatrices induced by a pair of row and column clusters is as uniform as possible. In this paper we approximate the optimal biclustering by applying one-way clustering algorithms independently on the rows and on the columns of the input matrix. We show that such a solution yields a worst-case approximation ratio of 1+sqrt(2) under L1-norm for 0-1 valued matrices, and of 2...
An Approximation Ratio for Biclustering
Puolamäki, Kai; Garriga, Gemma C
2007-01-01
The problem of biclustering consists of the simultaneous clustering of rows and columns of a matrix such that each of the submatrices induced by a pair of row and column clusters is as uniform as possible. In this paper we approximate the optimal biclustering by applying one-way clustering algorithms independently on the rows and on the columns of the input matrix. We show that such a solution yields a worst-case approximation ratio of 1+sqrt(2) under L1-norm for 0-1 valued matrices, and of 2 under L2-norm for real valued matrices.
Tree wavelet approximations with applications
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
[1]Baraniuk, R. G., DeVore, R. A., Kyriazis, G., Yu, X. M., Near best tree approximation, Adv. Comput. Math.,2002, 16: 357-373.[2]Cohen, A., Dahmen, W., Daubechies, I., DeVore, R., Tree approximation and optimal encoding, Appl. Comput.Harmonic Anal., 2001, 11: 192-226.[3]Dahmen, W., Schneider, R., Xu, Y., Nonlinear functionals of wavelet expansions-adaptive reconstruction and fast evaluation, Numer. Math., 2000, 86: 49-101.[4]DeVore, R. A., Nonlinear approximation, Acta Numer., 1998, 7: 51-150.[5]Davis, G., Mallat, S., Avellaneda, M., Adaptive greedy approximations, Const. Approx., 1997, 13: 57-98.[6]DeVore, R. A., Temlyakov, V. N., Some remarks on greedy algorithms, Adv. Comput. Math., 1996, 5: 173-187.[7]Kashin, B. S., Temlyakov, V. N., Best m-term approximations and the entropy of sets in the space L1, Mat.Zametki (in Russian), 1994, 56: 57-86.[8]Temlyakov, V. N., The best m-term approximation and greedy algorithms, Adv. Comput. Math., 1998, 8:249-265.[9]Temlyakov, V. N., Greedy algorithm and m-term trigonometric approximation, Constr. Approx., 1998, 14:569-587.[10]Hutchinson, J. E., Fractals and self similarity, Indiana. Univ. Math. J., 1981, 30: 713-747.[11]Binev, P., Dahmen, W., DeVore, R. A., Petruchev, P., Approximation classes for adaptive methods, Serdica Math.J., 2002, 28: 1001-1026.[12]Gilbarg, D., Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, Berlin: Springer-Verlag,1983.[13]Ciarlet, P. G., The Finite Element Method for Elliptic Problems, New York: North Holland, 1978.[14]Birman, M. S., Solomiak, M. Z., Piecewise polynomial approximation of functions of the class Wαp, Math. Sb.,1967, 73: 295-317.[15]DeVore, R. A., Lorentz, G. G., Constructive Approximation, New York: Springer-Verlag, 1993.[16]DeVore, R. A., Popov, V., Interpolation of Besov spaces, Trans. Amer. Math. Soc., 1988, 305: 397-414.[17]Devore, R., Jawerth, B., Popov, V., Compression of wavelet decompositions, Amer. J. Math., 1992, 114: 737-785.[18]Storozhenko, E
Developing extensible lattice-Boltzmann simulators for general-purpose graphics-processing units
Energy Technology Data Exchange (ETDEWEB)
Walsh, S C; Saar, M O
2011-12-21
Lattice-Boltzmann methods are versatile numerical modeling techniques capable of reproducing a wide variety of fluid-mechanical behavior. These methods are well suited to parallel implementation, particularly on the single-instruction multiple data (SIMD) parallel processing environments found in computer graphics processing units (GPUs). Although more recent programming tools dramatically improve the ease with which GPU programs can be written, the programming environment still lacks the flexibility available to more traditional CPU programs. In particular, it may be difficult to develop modular and extensible programs that require variable on-device functionality with current GPU architectures. This paper describes a process of automatic code generation that overcomes these difficulties for lattice-Boltzmann simulations. It details the development of GPU-based modules for an extensible lattice-Boltzmann simulation package - LBHydra. The performance of the automatically generated code is compared to equivalent purpose written codes for both single-phase, multiple-phase, and multiple-component flows. The flexibility of the new method is demonstrated by simulating a rising, dissolving droplet in a porous medium with user generated lattice-Boltzmann models and subroutines.
Developing extensible lattice-Boltzmann simulationsfor general-purpose graphics-programming units
Energy Technology Data Exchange (ETDEWEB)
Walsh, S C; Saar, M O
2011-10-27
Lattice-Boltzmann methods are versatile numerical modeling techniques capable of reproducing a wide variety of fluid-mechanical behavior. These methods are well suited to parallel implementation, particularly on the single-instruction multiple data (SIMD) parallel processing environments found in computer graphics processing units (GPUs). Although more recent programming tools dramatically improve the ease with which GPU programs can be written, the programming environment still lacks the flexibility available to more traditional CPU programs. In particular, it may be difficult to develop modular and extensible programs that require variable on-device functionality with current GPU architectures. This paper describes a process of automatic code generation that overcomes these difficulties for lattice-Boltzmann simulations. It details the development of GPU-based modules for an extensible lattice-Boltzmann simulation package - LBHydra. The performance of the automatically generated code is compared to equivalent purpose written codes for both single-phase, multiple-phase, and multiple-component flows. The flexibility of the new method is demonstrated by simulating a rising, dissolving droplet in a porous medium with user generated lattice-Boltzmann models and subroutines.
Liñán-García, Ernesto; Sánchez-Hernández, Juan Paulo; González-Barbosa, J. Javier; González-Flores, Carlos
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. PMID:27413369
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 ma
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
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…
Lattice-Boltzmann-based two-phase thermal model for simulating phase change
Kamali, M.R.; Gillissen, J.J.J.; Van den Akker, H.E.A.; Sundaresan, S.
2013-01-01
A lattice Boltzmann (LB) method is presented for solving the energy conservation equation in two phases when the phase change effects are included in the model. This approach employs multiple distribution functions, one for a pseudotemperature scalar variable and the rest for the various species. A
A New Regularization Mechanism for the Boltzmann Equation Without Cut-Off
Silvestre, Luis
2016-11-01
We apply recent results on regularity for general integro-differential equations to derive a priori estimates in Hölder spaces for the space homogeneous Boltzmann equation in the non cut-off case. We also show an a priori estimate in {L^∞} which applies in the space inhomogeneous case as well, provided that the macroscopic quantities remain bounded.
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. PMID:27413369
Shan, Ming-Lei; Zhu, Chang-Ping; Yao, Cheng; Yin, Cheng; Jiang, Xiao-Yan
2016-10-01
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. Project supported by the National Natural Science Foundation of China (Grant Nos. 11274092 and 1140040119) and the Natural Science Foundation of Jiangsu Province, China (Grant No. SBK2014043338).
DEFF Research Database (Denmark)
Ferraris, Chiara F; Geiker, Mette Rica; Martys, Nicos S;
2007-01-01
inapplicable here. This paper presents the analysis of a modified parallel plate rheometer for measuring cement mortar and propose a methodology for calibration using standard oils and numerical simulation of the flow. A lattice Boltzmann method was used to simulate the flow in the modified rheometer, thus...
Revised lattice Boltzmann model for traffic flow with equilibrium traffic pressure
Shi, Wei; Lu, Wei-Zhen; Xue, Yu; He, Hong-Di
2016-02-01
A revised lattice Boltzmann model concerning the equilibrium traffic pressure is proposed in this study to tackle the phase transition phenomena of traffic flow system. The traditional lattice Boltzmann model has limitation to investigate the complex traffic phase transitions due to its difficulty for modeling the equilibrium velocity distribution. Concerning this drawback, the equilibrium traffic pressure is taken into account to derive the equilibrium velocity distribution in the revised lattice Boltzmann model. In the proposed model, a three-dimensional velocity-space is assumed to determine the equilibrium velocity distribution functions and an alternative, new derivative approach is introduced to deduct the macroscopic equations with the first-order accuracy level from the lattice Boltzmann model. Based on the linear stability theory, the stability conditions of the corresponding macroscopic equations can be obtained. The outputs indicate that the stability curve is divided into three regions, i.e., the stable region, the neutral stability region, and the unstable region. In the stable region, small disturbance appears in the initial uniform flow and will vanish after long term evolution, while in the unstable region, the disturbance will be enlarged and finally leads to the traffic system entering the congested state. In the neutral stability region, small disturbance does not vanish with time and maintains its amplitude in the traffic system. Conclusively, the stability of traffic system is found to be enhanced as the equilibrium traffic pressure increases. Finally, the numerical outputs of the proposed model are found to be consistent with the recognized, theoretical results.
J.B.W. Geerdink; A.G. Hoekstra
2009-01-01
We compare the Lattice BGK, the Multiple Relaxation Times and the Entropic Lattice Boltzmann Methods for time harmonic flows. We measure the stability, speed and accuracy of the three models for Reynolds and Womersley numbers that are representative for human arteries. The Lattice BGK shows predicta
Podolsky Electromagnetism at Finite Temperature: Implications on Stefan-Boltzmann Law
Bonin, C. A.; Bufalo, R.; Pimentel, B. M.; Zambrano, G. E. R.
2009-01-01
In this work we study Podolsky electromagnetism in thermodynamic equilibrium. We show that a Podolsky mass-dependent modification to the Stefan-Boltzmann law is induced and we use experimental data to limit the possible values for this free parameter.
Castle, Karen J.
2007-01-01
In this undergraduate physical chemistry laboratory experiment, students acquire a high-resolution infrared absorption spectrum of carbon dioxide and use their data to show that the rotational-vibrational state populations follow a Boltzmann distribution. Data are acquired with a mid-infrared laser source and infrared detector. Appropriate…
Lattice Boltzmann simulation of 2D and 3D non-Brownian suspensions in Couette flow
Kromkamp, J.; Ende, van den D.; Kandhai, D.; Sman, van der R.G.M.; Boom, R.M.
2006-01-01
In this study, the Lattice Boltzmann (LB) method is applied for computer simulation of suspension flow in Couette systems. Typical aspects of Couette flow such as wall effects and non-zero Reynolds numbers can be studied well with the LB method because of its time-dependent character. Couette flow o
Linearized lattice Boltzmann method for micro- and nanoscale flow and heat transfer.
Shi, Yong; Yap, Ying Wan; Sader, John E
2015-07-01
Ability to characterize the heat transfer in flowing gases is important for a wide range of applications involving micro- and nanoscale devices. Gas flows away from the continuum limit can be captured using the Boltzmann equation, whose analytical solution poses a formidable challenge. An efficient and accurate numerical simulation of the Boltzmann equation is thus highly desirable. In this article, the linearized Boltzmann Bhatnagar-Gross-Krook equation is used to develop a hierarchy of thermal lattice Boltzmann (LB) models based on half-space Gaussian-Hermite (GH) quadrature ranging from low to high algebraic precision, using double distribution functions. Simplified versions of the LB models in the continuum limit are also derived, and are shown to be consistent with existing thermal LB models for noncontinuum heat transfer reported in the literature. Accuracy of the proposed LB hierarchy is assessed by simulating thermal Couette flows for a wide range of Knudsen numbers. Effects of the underlying quadrature schemes (half-space GH vs full-space GH) and continuum-limit simplifications on computational accuracy are also elaborated. The numerical findings in this article provide direct evidence of improved computational capability of the proposed LB models for modeling noncontinuum flows and heat transfer at small length scales.
Aerodynamic simulation of high-speed trains based on the Lattice Boltzmann Method (LBM)
Institute of Scientific and Technical Information of China (English)
2008-01-01
Aerodynamic simulation of high-speed trains has been carried out by using Lattice Boltzmann Method (LBM). Non-simplified train model was used and the number of space grids reached tens of millions. All results under different working conditions reflected the actual situation.
Podolsky electromagnetism at finite temperature: Implications on the Stefan-Boltzmann law
International Nuclear Information System (INIS)
In this work we study Podolsky electromagnetism in thermodynamic equilibrium. We show that a Podolsky mass-dependent modification to the Stefan-Boltzmann law is induced and we use experimental data to limit the possible values for this free parameter.
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...
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…
Parallel-plate rheometer calibration using oil and lattice Boltzmann simulation
DEFF Research Database (Denmark)
Ferraris, Chiara F; Geiker, Mette Rica; Martys, Nicos S.;
2007-01-01
compute the viscosity. This paper presents a modified parallel plate rheometer, and proposes means of calibration using standard oils and numerical simulation of the flow. A lattice Boltzmann method was used to simulate the flow in the modified rheometer, thus using an accurate numerical solution in place...
Calibrating the Shan-Chen lattice Boltzmann model for immiscible displacement in porous media
DEFF Research Database (Denmark)
Christensen, Britt Stenhøj Baun; Schaap, M.G.; Wildenschild, D.;
2006-01-01
The lattice Boltzmann (LB) modeling technique is increasingly being applied in a variety of fields where computational fluid dynamics are investigated. In our field of interest, environmentally related flow processes in porous media, the use of the LB method is still not common. For the LB...
Simulation of three-phase displacement mechanisms using a 2D lattice-Boltzmann model
Kats, F.M. van; Egberts, P.J.P.
1999-01-01
Using a numerical technique, known as the lattice-Boltzmann method, we study immiscible three-phase flow at the pore scale. An important phenomenon at this scale is the spreading of oil onto the gas-water interface. In this paper, we recognize from first principles how injected gas remobilizes initi
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 well
Sman, van der R.G.M.
2014-01-01
In this paper we present a novel numerical scheme for simulating the one-dimensional deformation of hydrogel material due to drying or rehydration. The scheme is based on the versatile Lattice Boltzmann method, which has been extended such that the computational grid (lattice) deforms due to shrinka
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...
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.
Implementation of the Lattice Boltzmann Method on Heterogeneous Hardware and Platforms using OpenCL
Directory of Open Access Journals (Sweden)
TEKIC, P. M.
2012-02-01
Full Text Available The Lattice Boltzmann method (LBM has become an alternative method for computational fluid dynamics with a wide range of applications. Besides its numerical stability and accuracy, one of the major advantages of LBM is its relatively easy parallelization and, hence, it is especially well fitted to many-core hardware as graphics processing units (GPU. The majority of work concerning LBM implementation on GPU's has used the CUDA programming model, supported exclusively by NVIDIA. Recently, the open standard for parallel programming of heterogeneous systems (OpenCL has been introduced. OpenCL standard matures and is supported on processors from most vendors. In this paper, we make use of the OpenCL framework for the lattice Boltzmann method simulation, using hardware accelerators - AMD ATI Radeon GPU, AMD Dual-Core CPU and NVIDIA GeForce GPU's. Application has been developed using a combination of Java and OpenCL programming languages. Java bindings for OpenCL have been utilized. This approach offers the benefits of hardware and operating system independence, as well as speeding up of lattice Boltzmann algorithm. It has been showed that the developed lattice Boltzmann source code can be executed without modification on all of the used hardware accelerators. Performance results have been presented and compared for the hardware accelerators that have been utilized.
Reprint of : The Boltzmann--Langevin approach: A simple quantum-mechanical derivation
Nagaev, K. E.
2016-08-01
We present a simple quantum-mechanical derivation of correlation function of Langevin sources in the semiclassical Boltzmann-Langevin equation. The specific case of electron-phonon scattering is considered. It is shown that the assumption of weak scattering leads to the Poisson nature of the scattering fluxes.
Fully coupled Lattice Boltzmann simulation of ﬁber reinforced self compacting concrete ﬂow
DEFF Research Database (Denmark)
Svec, Oldrich; Skocek, Jan; Stang, Henrik;
accurately the most important phenomena is introduced. A conventional Lattice Boltzmann method has been chosen as a ﬂuid dynamics solver of the non-Newtonian ﬂuid. A Mass Tracking Algorithm has been implemented to correctly represent a free surface and a modiﬁed Immersed Boundary Method (IBM) with direct...
Truthful approximations to range voting
DEFF Research Database (Denmark)
Filos-Ratsika, Aris; Miltersen, Peter Bro
We consider the fundamental mechanism design problem of approximate social welfare maximization under general cardinal preferences on a finite number of alternatives and without money. The well-known range voting scheme can be thought of as a non-truthful mechanism for exact social welfare...
Shearlets and Optimally Sparse Approximations
DEFF Research Database (Denmark)
Kutyniok, Gitta; Lemvig, Jakob; Lim, Wang-Q
2012-01-01
Multivariate functions are typically governed by anisotropic features such as edges in images or shock fronts in solutions of transport-dominated equations. One major goal both for the purpose of compression as well as for an efficient analysis is the provision of optimally sparse approximations...
Approximate Reasoning with Fuzzy Booleans
Broek, van den P.M.; Noppen, J.A.R.
2004-01-01
This paper introduces, in analogy to the concept of fuzzy numbers, the concept of fuzzy booleans, and examines approximate reasoning with the compositional rule of inference using fuzzy booleans. It is shown that each set of fuzzy rules is equivalent to a set of fuzzy rules with singleton crisp ante
Analytical Approximations to Galaxy Clustering
Mo, H. J.
1997-01-01
We discuss some recent progress in constructing analytic approximations to the galaxy clustering. We show that successful models can be constructed for the clustering of both dark matter and dark matter haloes. Our understanding of galaxy clustering and galaxy biasing can be greatly enhanced by these models.
Approximation properties of haplotype tagging
Directory of Open Access Journals (Sweden)
Dreiseitl Stephan
2006-01-01
Full Text Available Abstract Background Single nucleotide polymorphisms (SNPs are locations at which the genomic sequences of population members differ. Since these differences are known to follow patterns, disease association studies are facilitated by identifying SNPs that allow the unique identification of such patterns. This process, known as haplotype tagging, is formulated as a combinatorial optimization problem and analyzed in terms of complexity and approximation properties. Results It is shown that the tagging problem is NP-hard but approximable within 1 + ln((n2 - n/2 for n haplotypes but not approximable within (1 - ε ln(n/2 for any ε > 0 unless NP ⊂ DTIME(nlog log n. A simple, very easily implementable algorithm that exhibits the above upper bound on solution quality is presented. This algorithm has running time O((2m - p + 1 ≤ O(m(n2 - n/2 where p ≤ min(n, m for n haplotypes of size m. As we show that the approximation bound is asymptotically tight, the algorithm presented is optimal with respect to this asymptotic bound. Conclusion The haplotype tagging problem is hard, but approachable with a fast, practical, and surprisingly simple algorithm that cannot be significantly improved upon on a single processor machine. Hence, significant improvement in computatational efforts expended can only be expected if the computational effort is distributed and done in parallel.
Ultrafast Approximation for Phylogenetic Bootstrap
Bui Quang Minh, [No Value; Nguyen, Thi; von Haeseler, Arndt
2013-01-01
Nonparametric bootstrap has been a widely used tool in phylogenetic analysis to assess the clade support of phylogenetic trees. However, with the rapidly growing amount of data, this task remains a computational bottleneck. Recently, approximation methods such as the RAxML rapid bootstrap (RBS) and
Approximation by Penultimate Stable Laws
L.F.M. de Haan (Laurens); L. Peng (Liang); H. Iglesias Pereira
1997-01-01
textabstractIn certain cases partial sums of i.i.d. random variables with finite variance are better approximated by a sequence of stable distributions with indices \\\\alpha_n \\\\to 2 than by a normal distribution. We discuss when this happens and how much the convergence rate can be improved by using
Approximate Reanalysis in Topology Optimization
DEFF Research Database (Denmark)
Amir, Oded; Bendsøe, Martin P.; Sigmund, Ole
2009-01-01
In the nested approach to structural optimization, most of the computational effort is invested in the solution of the finite element analysis equations. In this study, the integration of an approximate reanalysis procedure into the framework of topology optimization of continuum structures...
Simulations of electron capture supernovae with approximate neutrino transport
Energy Technology Data Exchange (ETDEWEB)
Moeller, Heiko [TU Darmstadt (Germany); Fischer, Tobias [University of Wroclaw (Poland); Jones, Sam [Keele University (United Kingdom); Martinez-Pinedo, Gabriel [TU Darmstadt (Germany); GSI Helmholtzzentrum fuer Schwerionenforschung, Darmstadt (Germany)
2014-07-01
We have performed simulations of electron capture supernovae in a spherically symmetric general relativistic radiation hydrodynamics model with approximate neutrino treatment. We base our study on an 8.8 M {sub CircleDot} O-Ne-Mg core progenitor (Nomoto, 1984, 1987). We successfully obtain an explosion and compare our results with a reference run performed with an state-of-the-art three-flavor Boltzmann neutrino transport scheme implemented into the same hydrodynamic code. In general, we find good agreement in the the electron-flavor neutrino spectra. However, we find shorter explosion timescales and also significantly lower explosion energies of only 1.4 . 10{sup 48} erg. This result is in agreement with the explosion energy of SN 2008S as derived by Tominaga et al. (2013) based on light curve studies. Currently we are extending our simulations to the recently published super-AGB star progenitor models by Jones et al. (2013) with regard to their evolution towards an electron capture supernova. Our study also explores the role of weak interaction rates in determining the evolution and shaping the spectra of the emitted neutrinos.
Low Rank Approximation in $G_0W_0$ Approximation
Shao, Meiyue; Yang, Chao; Liu, Fang; da Jornada, Felipe H; Deslippe, Jack; Louie, Steven G
2016-01-01
The single particle energies obtained in a Kohn--Sham density functional theory (DFT) calculation are generally known to be poor approximations to electron excitation energies that are measured in transport, tunneling and spectroscopic experiments such as photo-emission spectroscopy. The correction to these energies can be obtained from the poles of a single particle Green's function derived from a many-body perturbation theory. From a computational perspective, the accuracy and efficiency of such an approach depends on how a self energy term that properly accounts for dynamic screening of electrons is approximated. The $G_0W_0$ approximation is a widely used technique in which the self energy is expressed as the convolution of a non-interacting Green's function ($G_0$) and a screened Coulomb interaction ($W_0$) in the frequency domain. The computational cost associated with such a convolution is high due to the high complexity of evaluating $W_0$ at multiple frequencies. In this paper, we discuss how the cos...
Temperature and Friction Accelerated Sampling of Boltzmann-Gibbs Distribution
Tao, Molei; Marsden, Jerrold E
2010-01-01
This paper is concerned with tuning friction and temperature in Langevin dynamics for fast sampling from the canonical ensemble. We show that near-optimal acceleration is achieved by choosing friction so that the local quadratic approximation of the Hamiltonian is a critical damped oscillator. The system is also over-heated and cooled down to its final temperature. The performances of different cooling schedules are analyzed as functions of total simulation time.
Energy Technology Data Exchange (ETDEWEB)
Shirdel-Havar, A. H., E-mail: Amir.hushang.shirdel@gmail.com; Masoudian Saadabad, R. [Department of Physics, University of Sistan and Baluchestan, Zahedan 98135-674 (Iran, Islamic Republic of)
2015-03-21
Based on ballistic-diffusive approximation, a method is presented to model heat transfer in nanocomposites containing metal nanoparticles. This method provides analytical expression for the temperature dynamics of metallic nanoparticles embedded in a dielectric medium. In this study, nanoparticles are considered as spherical shells, so that Boltzmann equation is solved using ballistic-diffusive approximation to calculate the electron and lattice thermal dynamics in gold nanoparticles, while thermal exchange between the particles is taken into account. The model was used to investigate the influence of particle size and metal concentration of the medium on the electron and lattice thermal dynamics. It is shown that these two parameters are crucial in determining the nanocomposite thermal behavior. Our results showed that the heat transfer rate from nanoparticles to the matrix decreases as the nanoparticle size increases. On the other hand, increasing the metal concentration of the medium can also decrease the heat transfer rate.
International Nuclear Information System (INIS)
Based on ballistic-diffusive approximation, a method is presented to model heat transfer in nanocomposites containing metal nanoparticles. This method provides analytical expression for the temperature dynamics of metallic nanoparticles embedded in a dielectric medium. In this study, nanoparticles are considered as spherical shells, so that Boltzmann equation is solved using ballistic-diffusive approximation to calculate the electron and lattice thermal dynamics in gold nanoparticles, while thermal exchange between the particles is taken into account. The model was used to investigate the influence of particle size and metal concentration of the medium on the electron and lattice thermal dynamics. It is shown that these two parameters are crucial in determining the nanocomposite thermal behavior. Our results showed that the heat transfer rate from nanoparticles to the matrix decreases as the nanoparticle size increases. On the other hand, increasing the metal concentration of the medium can also decrease the heat transfer rate
Klippenstein, Stephen J.; Babamov, Vasil K.; Marcus, R. A.
1986-01-01
Reactive transition probabilities and Boltzmann-averaged reactive transition probabilities for a slightly off-resonant model H-atom transfer system with an appreciable energy barrier are calculated using the approximate methods of Babamov et al. and of Crothers–Stückelberg. Both are compared with the corresponding quantities obtained from a numerical two-state treatment of the same model system. The method of Babamov et al. is seen to give more accurate results for the transition probabilitie...
Hydrogen Beyond the Classic Approximation
Scivetti, I
2003-01-01
The classical nucleus approximation is the most frequently used approach for the resolution of problems in condensed matter physics.However, there are systems in nature where it is necessary to introduce the nuclear degrees of freedom to obtain a correct description of the properties.Examples of this, are the systems with containing hydrogen.In this work, we have studied the resolution of the quantum nuclear problem for the particular case of the water molecule.The Hartree approximation has been used, i.e. we have considered that the nuclei are distinguishable particles.In addition, we have proposed a model to solve the tunneling process, which involves the resolution of the nuclear problem for configurations of the system away from its equilibrium position
Approximate Matching of Hierarchial Data
DEFF Research Database (Denmark)
Augsten, Nikolaus
formally proof that the pq-gram index can be incrementally updated based on the log of edit operations without reconstructing intermediate tree versions. The incremental update is independent of the data size and scales to a large number of changes in the data. We introduce windowed pq-grams for the......-gram based distance between streets, introduces a global greedy matching that guarantees stable pairs, and links addresses that are stored with different granularity. The connector has been successfully tested with public administration databases. Our extensive experiments on both synthetic and real world......The goal of this thesis is to design, develop, and evaluate new methods for the approximate matching of hierarchical data represented as labeled trees. In approximate matching scenarios two items should be matched if they are similar. Computing the similarity between labeled trees is hard as in...
Validity of the eikonal approximation
Kabat, D
1992-01-01
We summarize results on the reliability of the eikonal approximation in obtaining the high energy behavior of a two particle forward scattering amplitude. Reliability depends on the spin of the exchanged field. For scalar fields the eikonal fails at eighth order in perturbation theory, when it misses the leading behavior of the exchange-type diagrams. In a vector theory the eikonal gets the exchange diagrams correctly, but fails by ignoring certain non-exchange graphs which dominate the asymptotic behavior of the full amplitude. For spin--2 tensor fields the eikonal captures the leading behavior of each order in perturbation theory, but the sum of eikonal terms is subdominant to graphs neglected by the approximation. We also comment on the eikonal for Yang-Mills vector exchange, where the additional complexities of the non-abelian theory may be absorbed into Regge-type modifications of the gauge boson propagators.
Approximate Privacy: Foundations and Quantification
Feigenbaum, Joan; Schapira, Michael
2009-01-01
Increasing use of computers and networks in business, government, recreation, and almost all aspects of daily life has led to a proliferation of online sensitive data about individuals and organizations. Consequently, concern about the privacy of these data has become a top priority, particularly those data that are created and used in electronic commerce. There have been many formulations of privacy and, unfortunately, many negative results about the feasibility of maintaining privacy of sensitive data in realistic networked environments. We formulate communication-complexity-based definitions, both worst-case and average-case, of a problem's privacy-approximation ratio. We use our definitions to investigate the extent to which approximate privacy is achievable in two standard problems: the second-price Vickrey auction and the millionaires problem of Yao. For both the second-price Vickrey auction and the millionaires problem, we show that not only is perfect privacy impossible or infeasibly costly to achieve...
Hydrogen: Beyond the Classic Approximation
International Nuclear Information System (INIS)
The classical nucleus approximation is the most frequently used approach for the resolution of problems in condensed matter physics.However, there are systems in nature where it is necessary to introduce the nuclear degrees of freedom to obtain a correct description of the properties.Examples of this, are the systems with containing hydrogen.In this work, we have studied the resolution of the quantum nuclear problem for the particular case of the water molecule.The Hartree approximation has been used, i.e. we have considered that the nuclei are distinguishable particles.In addition, we have proposed a model to solve the tunneling process, which involves the resolution of the nuclear problem for configurations of the system away from its equilibrium position
Concentration Bounds for Stochastic Approximations
Frikha, Noufel
2012-01-01
We obtain non asymptotic concentration bounds for two kinds of stochastic approximations. We first consider the deviations between the expectation of a given function of the Euler scheme of some diffusion process at a fixed deterministic time and its empirical mean obtained by the Monte-Carlo procedure. We then give some estimates concerning the deviation between the value at a given time-step of a stochastic approximation algorithm and its target. Under suitable assumptions both concentration bounds turn out to be Gaussian. The key tool consists in exploiting accurately the concentration properties of the increments of the schemes. For the first case, as opposed to the previous work of Lemaire and Menozzi (EJP, 2010), we do not have any systematic bias in our estimates. Also, no specific non-degeneracy conditions are assumed.
Variance approximation under balanced sampling
Deville, Jean-Claude; Tillé, Yves
2016-01-01
A balanced sampling design has the interesting property that Horvitz–Thompson estimators of totals for a set of balancing variables are equal to the totals we want to estimate, therefore the variance of Horvitz–Thompson estimators of variables of interest are reduced in function of their correlations with the balancing variables. Since it is hard to derive an analytic expression for the joint inclusion probabilities, we derive a general approximation of variance based on a residual technique....
Approximate maximizers of intricacy functionals
Buzzi, Jerome; Zambotti, Lorenzo
2009-01-01
G. Edelman, O. Sporns, and G. Tononi introduced in theoretical biology the neural complexity of a family of random variables. This functional is a special case of intricacy, i.e., an average of the mutual information of subsystems whose weights have good mathematical properties. Moreover, its maximum value grows at a definite speed with the size of the system. In this work, we compute exactly this speed of growth by building "approximate maximizers" subject to an entropy condition. These appr...
Approximating Metal-Insulator Transitions
Danieli, C.; Rayanov, K.; Pavlov, B.; Martin, G.; Flach, S
2014-01-01
We consider quantum wave propagation in one-dimensional quasiperiodic lattices. We propose an iterative construction of quasiperiodic potentials from sequences of potentials with increasing spatial period. At each finite iteration step the eigenstates reflect the properties of the limiting quasiperiodic potential properties up to a controlled maximum system size. We then observe approximate metal-insulator transitions (MIT) at the finite iteration steps. We also report evidence on mobility ed...
The role of the Stefan-Boltzmann law in the thermodynamic optimization of an n-Müser engine
Ramírez-Moreno, M. A.; González-Hernández, S.; Angulo-Brown, F.
2016-02-01
A Müser-type engine model can be taken as a particular case of a Curzon-Ahlborn thermal cycle, when the upper thermal conductance is finite and the lower one is infinite. In addition, the upper heat exchange is given by the Stefan-Boltzmann law. That model is suitable to thermodynamically describe some aspects of energy converters as solar cells and photosynthetic systems. In the present article, we call n-Müser engine to an engine of the Müser type in which the T4 heat transfer law is substituted by a Tn-law, being n > 0 a real number. Here, we show that if we use the so-called ecological criterion of merit to optimize finite-time heat engines to compare the thermodynamic performance of the n-Müser engines under approximate terrestrial conditions (see below), we obtain that n = 4 accomplishes the best performance. This same result was obtained by using data from the rest of planets of the solar system.
Coarse-grained transport of a turbulent flow via moments of the Reynolds-averaged Boltzmann equation
Abramov, Rafail V
2015-01-01
Here we introduce new coarse-grained variables for a turbulent flow in the form of moments of its Reynolds-averaged Boltzmann equation. With the exception of the collision moments, the transport equations for the new variables are identical to the usual moment equations, and thus naturally lend themselves to the variety of already existing closure methods. Under the anelastic turbulence approximation, we derive equations for the Reynolds-averaged turbulent fluctuations around the coarse-grained state. We show that the global relative entropy of the coarse-grained state is bounded from above by the Reynolds average of the fine-grained global relative entropy, and thus obeys the time decay bound of Desvillettes and Villani. This is similar to what is observed in the rarefied gas dynamics, which makes the Grad moment closure a good candidate for truncating the hierarchy of the coarse-grained moment equations. We also show that, under additional assumptions on the form of the coarse-grained collision terms, one a...
Energy Technology Data Exchange (ETDEWEB)
Sloan, D.P.
1983-05-01
Morel (1981) has developed multigroup Legendre cross sections suitable for input to standard discrete ordinates transport codes for performing charged-particle Fokker-Planck calculations in one-dimensional slab and spherical geometries. Since the Monte Carlo neutron transport code, MORSE, uses the same multigroup cross section data that discrete ordinates codes use, it was natural to consider whether Fokker-Planck calculations could be performed with MORSE. In order to extend the unique three-dimensional forward or adjoint capability of MORSE to Fokker-Planck calculations, the MORSE code was modified to correctly treat the delta-function scattering of the energy operator, and a new set of physically acceptable cross sections was derived to model the angular operator. Morel (1979) has also developed multigroup Legendre cross sections suitable for input to standard discrete ordinates codes for performing electron Boltzmann calculations. These electron cross sections may be treated in MORSE with the same methods developed to treat the Fokker-Planck cross sections. The large magnitude of the elastic scattering cross section, however, severely increases the computation or run time. It is well-known that approximate elastic cross sections are easily obtained by applying the extended transport (or delta function) correction to the Legendre coefficients of the exact cross section. An exact method for performing the extended transport cross section correction produces cross sections which are physically acceptable. Sample calculations using electron cross sections have demonstrated this new technique to be very effective in decreasing the large magnitude of the cross sections.
Wind-Driven, Double-Gyre, Ocean Circulation in a Reduced-Gravity, 2.5-Layer, Lattice Boltzmann Model
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
A coupled lattice Boltzmann (LB) model with second-order accuracy is applied to the reduced-gravity,shallow water, 2.5-layer model for wind-driven double-gyre ocean circulation. By introducing the secondorder integral approximation for the collision operator, the model becomes fully explicit. The Coriolis force and other external forces are included in the model with second-order accuracy, which is consistent with the discretization accuracy of the LB equation. The feature of the multiple equilibria solutions is found in the numerical experiments under different Reynolds numbers based on this LB scheme. With the Reynolds number increasing from 3000 to 4000, the solution of this model is destabilized from the anti-symmetric double-gyre solution to the subtropic gyre solution and then to the subpolar gyre solution. The transitions between these equilibria states are also found in some parameter ranges. The time-dependent variability of the circulation based on this LB simulation is also discussed for varying viscosity regimes. The flow of this model exhibits oscillations with different timescales varying from subannual to interannual. The corresponding statistical oscillation modes are obtained by spectral analysis. By analyzing the spatiotemporal structures of these modes, it is found that the subannual oscillation with a 9-month period originates from the barotropic Rossby basin mode, and the interannual oscillations with periods ranging from 1.5 years to 4.6 years originate from the recirculation gyre modes, which include the barotropic and the baroclinic recirculation gyre modes.
Product Approximation of Grade and Precision
Institute of Scientific and Technical Information of China (English)
ZHANG Xian-yong; MO Zhi-wen
2005-01-01
The normal graded approximation and variable precision approximation are defined in approximate space. The relationship between graded approximation and variable precision approximation is studied, and an important formula of conversion between them is achieved. The product approximation of gradeand precision is defined and its basic properties are studied.
Khali, S; Nebbali, R; Ameziani, D E; Bouhadef, K
2013-05-01
In this work the instability of the Taylor-Couette flow for Newtonian and non-Newtonian fluids (dilatant and pseudoplastic fluids) is investigated for cases of finite aspect ratios. The study is conducted numerically using the lattice Boltzmann method (LBM). In many industrial applications, the apparatuses and installations drift away from the idealized case of an annulus of infinite length, and thus the end caps effect can no longer be ignored. The inner cylinder is rotating while the outer one and the end walls are maintained at rest. The lattice two-dimensional nine-velocity (D2Q9) Boltzmann model developed from the Bhatnagar-Gross-Krook approximation is used to obtain the flow field for fluids obeying the power-law model. The combined effects of the Reynolds number, the radius ratio, and the power-law index n on the flow characteristics are analyzed for an annular space of finite aspect ratio. Two flow modes are obtained: a primary Couette flow (CF) mode and a secondary Taylor vortex flow (TVF) mode. The flow structures so obtained are different from one mode to another. The critical Reynolds number Re(c) for the passage from the primary to the secondary mode exhibits the lowest value for the pseudoplastic fluids and the highest value for the dilatant fluids. The findings are useful for studies of the swirling flow of non-Newtonians fluids in axisymmetric geometries using LBM. The flow changes from the CF to TVF and its structure switches from the two-cells to four-cells regime for both Newtonian and dilatant fluids. Contrariwise for pseudoplastic fluids, the flow exhibits 2-4-2 structure passing from two-cells to four cells and switches again to the two-cells configuration. Furthermore, the critical Reynolds number presents a monotonic increase with the power-law index n of the non-Newtonian fluid, and as the radius ratio grows, the transition flow regimes tend to appear for higher critical Reynolds numbers. PMID:23767615
Khali, S.; Nebbali, R.; Ameziani, D. E.; Bouhadef, K.
2013-05-01
In this work the instability of the Taylor-Couette flow for Newtonian and non-Newtonian fluids (dilatant and pseudoplastic fluids) is investigated for cases of finite aspect ratios. The study is conducted numerically using the lattice Boltzmann method (LBM). In many industrial applications, the apparatuses and installations drift away from the idealized case of an annulus of infinite length, and thus the end caps effect can no longer be ignored. The inner cylinder is rotating while the outer one and the end walls are maintained at rest. The lattice two-dimensional nine-velocity (D2Q9) Boltzmann model developed from the Bhatnagar-Gross-Krook approximation is used to obtain the flow field for fluids obeying the power-law model. The combined effects of the Reynolds number, the radius ratio, and the power-law index n on the flow characteristics are analyzed for an annular space of finite aspect ratio. Two flow modes are obtained: a primary Couette flow (CF) mode and a secondary Taylor vortex flow (TVF) mode. The flow structures so obtained are different from one mode to another. The critical Reynolds number Rec for the passage from the primary to the secondary mode exhibits the lowest value for the pseudoplastic fluids and the highest value for the dilatant fluids. The findings are useful for studies of the swirling flow of non-Newtonians fluids in axisymmetric geometries using LBM. The flow changes from the CF to TVF and its structure switches from the two-cells to four-cells regime for both Newtonian and dilatant fluids. Contrariwise for pseudoplastic fluids, the flow exhibits 2-4-2 structure passing from two-cells to four cells and switches again to the two-cells configuration. Furthermore, the critical Reynolds number presents a monotonic increase with the power-law index n of the non-Newtonian fluid, and as the radius ratio grows, the transition flow regimes tend to appear for higher critical Reynolds numbers.
Rollout Sampling Approximate Policy Iteration
Dimitrakakis, Christos
2008-01-01
Several researchers have recently investigated the connection between reinforcement learning and classification. We are motivated by proposals of approximate policy iteration schemes without value functions which focus on policy representation using classifiers and address policy learning as a supervised learning problem. This paper proposes variants of an improved policy iteration scheme which addresses the core sampling problem in evaluating a policy through simulation as a multi-armed bandit machine. The resulting algorithm offers comparable performance to the previous algorithm achieved, however, with significantly less computational effort. An order of magnitude improvement is demonstrated experimentally in two standard reinforcement learning domains: inverted pendulum and mountain-car.
Quantum Tunneling Beyond Semiclassical Approximation
Banerjee, Rabin
2008-01-01
Hawking radiation as tunneling by Hamilton-Jacobi method beyond semiclassical approximation is analysed. We compute all quantum corrections in the single particle action revealing that these are proportional to the usual semiclassical contribution. We show that a simple choice of the proportionality constants reproduces the one loop back reaction effect in the spacetime, found by conformal field theory methods, which modifies the Hawking temperature of the black hole. Using the law of black hole mechanics we give the corrections to the Bekenstein-Hawking area law following from the modified Hawking temperature. Some examples are explicitly worked out.
Quantum tunneling beyond semiclassical approximation
Banerjee, Rabin; Ranjan Majhi, Bibhas
2008-06-01
Hawking radiation as tunneling by Hamilton-Jacobi method beyond semiclassical approximation is analysed. We compute all quantum corrections in the single particle action revealing that these are proportional to the usual semiclassical contribution. We show that a simple choice of the proportionality constants reproduces the one loop back reaction effect in the spacetime, found by conformal field theory methods, which modifies the Hawking temperature of the black hole. Using the law of black hole mechanics we give the corrections to the Bekenstein-Hawking area law following from the modified Hawking temperature. Some examples are explicitly worked out.
Fermion Tunneling Beyond Semiclassical Approximation
Majhi, Bibhas Ranjan
2008-01-01
Applying the Hamilton-Jacobi method beyond the semiclassical approximation prescribed in \\cite{Majhi3} for the scalar particle, Hawking radiation as tunneling of Dirac particle through an event horizon is analysed. We show that, as before, all quantum corrections in the single particle action are proportional to the usual semiclassical contribution. We also compute the modifications to the Hawking temperature and Bekenstein-Hawking entropy for the Schwarzschild black hole. Finally, the coefficient of the logarithmic correction to entropy is shown to be related with the trace anomaly.
Fermion tunneling beyond semiclassical approximation
Majhi, Bibhas Ranjan
2009-02-01
Applying the Hamilton-Jacobi method beyond the semiclassical approximation prescribed in R. Banerjee and B. R. Majhi, J. High Energy Phys.JHEPFG1029-8479 06 (2008) 09510.1088/1126-6708/2008/06/095 for the scalar particle, Hawking radiation as tunneling of the Dirac particle through an event horizon is analyzed. We show that, as before, all quantum corrections in the single particle action are proportional to the usual semiclassical contribution. We also compute the modifications to the Hawking temperature and Bekenstein-Hawking entropy for the Schwarzschild black hole. Finally, the coefficient of the logarithmic correction to entropy is shown to be related with the trace anomaly.
The distorted wave Glauber approximation
International Nuclear Information System (INIS)
A solution of the Pauli equation with non-zero potentials defines quantum scalar and vector potentials and magnetic fields and quantum trajectories. If a line integral of perturbing potentials and fields along these quantum trajectories is added to the phase of this solution, an approximate solution of the perturbed equation is found. Glauber theory is a special case and the conditions of applicability are similar. Applications given start from the harmonic oscillator and from a homogeneous magnetic field and add a perturbation. (author)
Approximation of Surfaces by Cylinders
DEFF Research Database (Denmark)
Randrup, Thomas
1998-01-01
We present a new method for approximation of a given surface by a cylinder surface. It is a constructive geometric method, leading to a monorail representation of the cylinder surface. By use of a weighted Gaussian image of the given surface, we determine a projection plane. In the orthogonal...... projection of the surface onto this plane, a reference curve is determined by use of methods for thinning of binary images. Finally, the cylinder surface is constructed as follows: the directrix of the cylinder surface is determined by a least squares method minimizing the distance to the points...
Energy Technology Data Exchange (ETDEWEB)
Horsten, N., E-mail: niels.horsten@kuleuven.be; Baelmans, M. [KU Leuven, Department of Mechanical Engineering, Celestijnenlaan 300A, 3001 Leuven (Belgium); Dekeyser, W. [ITER Organization, route de Vinon-sur-Verdon, 13067 St. Paul lez Durance Cedex (France); Samaey, G. [KU Leuven, Department of Computer Science, Celestijnenlaan 200A, 3001 Leuven (Belgium)
2016-01-15
We derive fluid neutral approximations for a simplified 1D edge plasma model, suitable to study the neutral behavior close to the target of a nuclear fusion divertor, and compare its solutions to the solution of the corresponding kinetic Boltzmann equation. The plasma is considered as a fixed background extracted from a detached 2D simulation. We show that the Maxwellian equilibrium distribution is already obtained very close to the target, justifying the use of a fluid approximation. We compare three fluid neutral models: (i) a diffusion model; (ii) a pressure-diffusion model (i.e., a combination of a continuity and momentum equation) assuming equal neutral and ion temperatures; and (iii) the pressure-diffusion model coupled to a neutral energy equation taking into account temperature differences between neutrals and ions. Partial reflection of neutrals reaching the boundaries is included in both the kinetic and fluid models. We propose two methods to obtain an incident neutral flux boundary condition for the fluid models: one based on a diffusion approximation and the other assuming a truncated Chapman-Enskog distribution. The pressure-diffusion model predicts the plasma sources very well. The diffusion boundary condition gives slightly better results overall. Although including an energy equation still improves the results, the assumption of equal ion and neutral temperature already gives a very good approximation.
International Nuclear Information System (INIS)
We derive fluid neutral approximations for a simplified 1D edge plasma model, suitable to study the neutral behavior close to the target of a nuclear fusion divertor, and compare its solutions to the solution of the corresponding kinetic Boltzmann equation. The plasma is considered as a fixed background extracted from a detached 2D simulation. We show that the Maxwellian equilibrium distribution is already obtained very close to the target, justifying the use of a fluid approximation. We compare three fluid neutral models: (i) a diffusion model; (ii) a pressure-diffusion model (i.e., a combination of a continuity and momentum equation) assuming equal neutral and ion temperatures; and (iii) the pressure-diffusion model coupled to a neutral energy equation taking into account temperature differences between neutrals and ions. Partial reflection of neutrals reaching the boundaries is included in both the kinetic and fluid models. We propose two methods to obtain an incident neutral flux boundary condition for the fluid models: one based on a diffusion approximation and the other assuming a truncated Chapman-Enskog distribution. The pressure-diffusion model predicts the plasma sources very well. The diffusion boundary condition gives slightly better results overall. Although including an energy equation still improves the results, the assumption of equal ion and neutral temperature already gives a very good approximation
Wavelet Approximation in Data Assimilation
Tangborn, Andrew; Atlas, Robert (Technical Monitor)
2002-01-01
Estimation of the state of the atmosphere with the Kalman filter remains a distant goal because of high computational cost of evolving the error covariance for both linear and nonlinear systems. Wavelet approximation is presented here as a possible solution that efficiently compresses both global and local covariance information. We demonstrate the compression characteristics on the the error correlation field from a global two-dimensional chemical constituent assimilation, and implement an adaptive wavelet approximation scheme on the assimilation of the one-dimensional Burger's equation. In the former problem, we show that 99%, of the error correlation can be represented by just 3% of the wavelet coefficients, with good representation of localized features. In the Burger's equation assimilation, the discrete linearized equations (tangent linear model) and analysis covariance are projected onto a wavelet basis and truncated to just 6%, of the coefficients. A nearly optimal forecast is achieved and we show that errors due to truncation of the dynamics are no greater than the errors due to covariance truncation.
Plasma Physics Approximations in Ares
Energy Technology Data Exchange (ETDEWEB)
Managan, R. A.
2015-01-08
Lee & More derived analytic forms for the transport properties of a plasma. Many hydro-codes use their formulae for electrical and thermal conductivity. The coefficients are complex functions of Fermi-Dirac integrals, F_{n}( μ/θ ), the chemical potential, μ or ζ = ln(1+e^{ μ/θ} ), and the temperature, θ = kT. Since these formulae are expensive to compute, rational function approximations were fit to them. Approximations are also used to find the chemical potential, either μ or ζ . The fits use ζ as the independent variable instead of μ/θ . New fits are provided for A^{α} (ζ ),A^{β} (ζ ), ζ, f(ζ ) = (1 + e^{-μ/θ})F_{1/2}(μ/θ), F_{1/2}'/F_{1/2}, F_{c}^{α}, and F_{c}^{β}. In each case the relative error of the fit is minimized since the functions can vary by many orders of magnitude. The new fits are designed to exactly preserve the limiting values in the non-degenerate and highly degenerate limits or as ζ→ 0 or ∞. The original fits due to Lee & More and George Zimmerman are presented for comparison.
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)
Numerical simulation of ski-jump jet motion using lattice Boltzmann method
Institute of Scientific and Technical Information of China (English)
无
2011-01-01
Based on the lattice Boltzmann method,a lattice Boltzmann(LB) model of the ski-jump jet two-phase flow is developed first and the corresponding boundary conditions are studied.A simple case study of a droplet horizontal movement calculation is carried out to test and verify the model,where level set method is used to track and reconstruct the moving droplet free surface. Then,we numerically simulate a two dimensional flow field of the ski-jump jet with the LB model,derive the moving surface and velocity vector field of the jet flow.The simulation results are very consistent with the physical mechanisms.The effectiveness and reliability of the model are demonstrated by the numerical examples.
Why Boltzmann Brains Don't Fluctuate Into Existence From the De Sitter Vacuum
Boddy, Kimberly K; Pollack, Jason
2015-01-01
Many modern cosmological scenarios feature large volumes of spacetime in a de Sitter vacuum phase. Such models are said to be faced with a "Boltzmann Brain problem" - the overwhelming majority of observers with fixed local conditions are random fluctuations in the de Sitter vacuum, rather than arising via thermodynamically sensible evolution from a low-entropy past. We argue that this worry can be straightforwardly avoided in the Many-Worlds (Everett) approach to quantum mechanics, as long as the underlying Hilbert space is infinite-dimensional. In that case, de Sitter settles into a truly stationary quantum vacuum state. While there would be a nonzero probability for observing Boltzmann-Brain-like fluctuations in such a state, "observation" refers to a specific kind of dynamical process that does not occur in the vacuum (which is, after all, time-independent). Observers are necessarily out-of-equilibrium physical systems, which are absent in the vacuum. Hence, the fact that projection operators corresponding...
Prediction of sound absorption in rigid porous media with the lattice Boltzmann method
International Nuclear Information System (INIS)
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)
Study of acoustic bubble cluster dynamics using a lattice Boltzmann model
Institute of Scientific and Technical Information of China (English)
Mahdi Daemi; Mohammad Taeibi-Rahni; Hamidreza Massah
2015-01-01
Search for the development of a reliable mathematical model for understanding bubble dynamics behavior is an ongoing endeavor. A long list of complex phenomena underlies physics of this problem. In the past decades, the lattice Boltzmann (LB) method has emerged as a promising tool to address such complexities. In this regard, we have applied a 121-velocity multiphase lattice Boltzmann model (LBM) to an asymmetric cluster of bubbles in an acoustic field. A problem as a benchmark is studied to check the consistency and applicability of the model. The problem of interest is to study the deformation and coalescence phenomena in bubble cluster dynamics, and the screening effect on an acoustic multi-bubble medium. It has been observed that the LB model is able to simulate the combination of the three aforementioned phenomena for a bubble cluster as a whole and for every individual bubble in the cluster.
Simulation of Rarefied Gas Flow in Slip and Transitional Regimes by the Lattice Boltzmann Method
Directory of Open Access Journals (Sweden)
S Abdullah
2010-07-01
Full Text Available In this paper, a lattice Boltzmann method (LBM based simulation of microscale flow has been carried out, for various values of Knudsen number. The details in determining the parameters critical for LBM applications in microscale flow are provided. Pressure distributions in the slip flow regime are compared with the analytical solution based on the Navier-Stokes equationwith slip-velocity boundary condition. Satisfactory agreements have been achieved. Simulations are then extended to transition regime (Kn = 0.15 and compared with the same analytical solution. The results show some deviation from the analytical solution due to the breakdown of continuum assumption. From this study, we may conclude that the lattice Boltzmann method is an efficient approach for simulation of microscale flow.
On the Stability of the Finite Difference based Lattice Boltzmann Method
El-Amin, M.F.
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.
Held, M
2015-01-01
A lattice Boltzmann method (LBM) approach to the Charney-Hasegawa-Mima (CHM) model for adiabatic drift wave turbulence in magnetised plasmas, is implemented. The CHM-LBM model contains a barotropic equation of state for the potential, a force term including a cross-product analogous to the Coriolis force in quasigeostrophic models, and a density gradient source term. Expansion of the resulting lattice Boltzmann model equations leads to cold-ion fluid continuity and momentum equations, which resemble CHM dynamics under drift ordering. The resulting numerical solutions of standard test cases (monopole propagation, stable drift modes and decaying turbulence) are compared to results obtained by a conventional finite difference scheme that directly discretizes the CHM equation. The LB scheme resembles characteristic CHM dynamics apart from an additional shear in the density gradient direction. The occuring shear reduces with the drift ratio and is ascribed to the compressible limit of the underlying LBM.
Pore-scale lattice Boltzmann simulation of laminar and turbulent flow through a sphere pack
Fattahia, Ehsan; Wohlmuth, Barbara; Rüde, Ulrich; Manhart, Michael; Helmig, Rainer
2015-01-01
The lattice Boltzmann method can be used to simulate flow through porous media with full geometrical resolution. With such a direct numerical simulation, it becomes possible to study fundamental effects which are difficult to assess either by developing macroscopic mathematical models or experiments. We first evaluate the lattice Boltzmann method with various boundary handling of the solid-wall and various collision operators to assess their suitability for large scale direct numerical simulation of porous media flow. A periodic pressure drop boundary condition is used to mimic the pressure driven flow through the simple sphere pack in a periodic domain. The evaluation of the method is done in the Darcy regime and the results are compared to a semi-analytic solution. Taking into account computational cost and accuracy, we choose the most efficient combination of the solid boundary condition and collision operator. We apply this method to perform simulations for a wide range of Reynolds numbers from Stokes flo...
Institute of Scientific and Technical Information of China (English)
Jiang Ji-Jian; Meng Qing-Miao; Wang Shuai
2009-01-01
Using entropy density of Dirac field near the event horizon of a rectilinear non-uniformly accelerating Kinnersley black hole, the law for the thermal radiation of black hole is studied and the instantaneous radiation energy density is obtained. It is found that the instantaneous radiation energy density of a black hole is always proportional to the quartic of the temperature on event horizon in the same direction. That is to say, the thermal radiation of a black hole always satisfies the generalized Stefan Boltzmann law. In addition, the derived generalized Stefan-Boltzmann coefficient is no longer a constant, but a dynamic coefficient related to the space-time metric near the event horizon and the changing rate of the event horizon in black holes.
Thermal Lattice Boltzmann Simulations for Vapor-Liquid Two-Phase Flows in Two Dimensions
Wei, Yikun; Qian, Yuehong
2011-11-01
A lattice Boltzmann model with double distribution functions is developed to simulate thermal vapor-liquid two-phase flows. In this model, the so-called mesoscopic inter-particle pseudo-potential for the single component multi-phase lattice Boltzmann model is used to simulate the fluid dynamics and the internal energy field is simulated by using a energy distribution function. Theoretical results for large-scale dynamics including the internal energy equation can be derived and numerical results for the coexistence curve of vapor-liquid systems are in good agreement with the theoretical predictions. It is shown from numerical simulations that the model has the ability to mimic phase transitions, bubbly flows and slugging flows. This research is support in part by the grant of Education Ministry of China IRT0844 and the grant of Shanghai CST 11XD1402300.
DEFF Research Database (Denmark)
van Tulder, Gijs; de Bruijne, Marleen
2016-01-01
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 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 machines may...... 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...
Evaluation of the Performance of the Hybrid Lattice Boltzmann Based Numerical Flux
Zheng, H. W.; Shu, C.
2016-06-01
It is well known that the numerical scheme is a key factor to the stability and accuracy of a Navier-Stokes solver. Recently, a new hybrid lattice Boltzmann numerical flux (HLBFS) is developed by Shu's group. It combines two different LBFS schemes by a switch function. It solves the Boltzmann equation instead of the Euler equation. In this article, the main object is to evaluate the ability of this HLBFS scheme by our in-house cell centered hybrid mesh based Navier-Stokes code. Its performance is examined by several widely-used bench-mark test cases. The comparisons on results between calculation and experiment are conducted. They show that the scheme can capture the shock wave as well as the resolving of boundary layer.
Interpolation methods and the accuracy of lattice-Boltzmann mesh refinement
Energy Technology Data Exchange (ETDEWEB)
Guzik, Stephen M. [Colorado State Univ., Fort Collins, CO (United States). Dept. of Mechanical Engineering; Weisgraber, Todd H. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Colella, Phillip [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Alder, Berni J. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
2013-12-10
A lattice-Boltzmann model to solve the equivalent of the Navier-Stokes equations on adap- tively refined grids is presented. A method for transferring information across interfaces between different grid resolutions was developed following established techniques for finite- volume representations. This new approach relies on a space-time interpolation and solving constrained least-squares problems to ensure conservation. The effectiveness of this method at maintaining the second order accuracy of lattice-Boltzmann is demonstrated through a series of benchmark simulations and detailed mesh refinement studies. These results exhibit smaller solution errors and improved convergence when compared with similar approaches relying only on spatial interpolation. Examples highlighting the mesh adaptivity of this method are also provided.