#### Sample records for boltzmann factor

1. 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...

2. 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.

3. Boltzmann

This paper reports the development of an object-oriented programming methodology for particle simulations. It is established on the [m reductionist] view that many physical phenomena cana be reduced to many-body problems. By doing the reduction, many seemly unrelated physical phenomena can be simulated in a systematic way and a high-level programming system can be constructed to facilitate the programming and the solution of the simulations. In the object-oriented particle simulation methodology, a hierarchy of abstract particles is defined to represent a variety of characteristics in physical system simulations. A simulation program is constructed from particles derived from the abstract particles. The object- oriented particle simulation methodology provides a unifying modeling and simulation framework for a variety of simulation applications with the use of particle methods. It allows easy composition of simulation programs from predefined software modules and facilitates software reusability. It greatly increase the productivity of simulation program constructions. Boltzmann (after Ludwig Boltzmann, 1844-1906) is a prototype programming system in the object-oriented particle simulation methodology. Boltzmann is implemented in C++ and the X Window System. It contains a library of data types and functions that support simulations in particle methods. Moreover, it provides a visualization window to support friendly user-computer interaction. Examples of the application of the Boltzmann programming system are presented. The effectiveness of the object-oriented particle simulation methodology is demonstrated. A user's manual is included in the appendix

4. 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…

5. A Pedagogical Approach to the Boltzmann Factor through Experiments and Simulations

Battaglia, O. R.; Bonura, A.; Sperandeo-Mineo, R. M.

2009-01-01

The Boltzmann factor is the basis of a huge amount of thermodynamic and statistical physics, both classical and quantum. It governs the behaviour of all systems in nature that are exchanging energy with their environment. To understand why the expression has this specific form involves a deep mathematical analysis, whose flow of logic is hard to…

6. Student learning of upper-level thermal and statistical physics: The derivation and use of the Boltzmann factor

Thompson, John

2015-04-01

As the Physical Review Focused Collection demonstrates, recent frontiers in physics education research include systematic investigations at the upper division. As part of a collaborative project, we have examined student understanding of several topics in upper-division thermal and statistical physics. A fruitful context for research is the Boltzmann factor in statistical mechanics: the standard derivation involves several physically justified mathematical steps as well as the invocation of a Taylor series expansion. We have investigated student understanding of the physical significance of the Boltzmann factor as well as its utility in various circumstances, and identified various lines of student reasoning related to the use of the Boltzmann factor. Results from written data as well as teaching interviews suggest that many students do not use the Boltzmann factor when answering questions related to probability in applicable physical situations, even after lecture instruction. We designed an inquiry-based tutorial activity to guide students through a derivation of the Boltzmann factor and to encourage deep connections between the physical quantities involved and the mathematics. Observations of students working through the tutorial suggest that many students at this level can recognize and interpret Taylor series expansions, but they often lack fluency in creating and using Taylor series appropriately, despite previous exposure in both calculus and physics courses. Our findings also suggest that tutorial participation not only increases the prevalence of relevant invocation of the Boltzmann factor, but also helps students gain an appreciation of the physical implications and meaning of the mathematical formalism behind the formula. Supported in part by NSF Grants DUE-0817282, DUE-0837214, and DUE-1323426.

7. Demand Forecasting at Low Aggregation Levels using Factored Conditional Restricted Boltzmann Machine

Mocanu, Elena; Nguyen, Phuong H.; Gibescu, Madeleine;

2016-01-01

approaches have been proposed in the literature. As an evolution of neural network-based prediction methods, deep learning techniques are expected to increase the prediction accuracy by allowing stochastic formulations and bi-directional connections between neurons. In this paper, we investigate a newly...... developed deep learning model for time series prediction, namely Factored Conditional Restricted Boltzmann Machine (FCRBM), and extend it for electrical demand forecasting. The assessment is made on the EcoGrid dataset, originating from the Bornholm island experiment in Denmark, consisting of aggregated......The electrical demand forecasting problem can be regarded as a nonlinear time series prediction problem depending on many complex factors since it is required at various aggregation levels and at high temporal resolution. To solve this challenging problem, various time series and machine learning...

8. A quantitative method to analyse an open-ended questionnaire: A case study about the Boltzmann Factor

Rosario Battaglia, Onofrio; Di Paola, Benedetto

2016-05-01

This paper describes a quantitative method to analyse an open-ended questionnaire. Student responses to a specially designed written questionnaire are quantitatively analysed by not hierarchical clustering called k -means method. Through this we can characterise behaviour students with respect their expertise to formulate explanations for phenomena or processes and/or use a given model in the different context. The physics topic is about the Boltzmann Factor, which allows the students to have a unifying view of different phenomena in different contexts.

9. 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...

10. Temperature based Restricted Boltzmann Machines

Li, Guoqi; Deng, Lei; Xu, Yi; Wen, Changyun; Wang, Wei; Pei, Jing; Shi, Luping

2016-01-01

Restricted Boltzmann machines (RBMs), which apply graphical models to learning probability distribution over a set of inputs, have attracted much attention recently since being proposed as building blocks of multi-layer learning systems called deep belief networks (DBNs). Note that temperature is a key factor of the Boltzmann distribution that RBMs originate from. However, none of existing schemes have considered the impact of temperature in the graphical model of DBNs. In this work, we propose temperature based restricted Boltzmann machines (TRBMs) which reveals that temperature is an essential parameter controlling the selectivity of the firing neurons in the hidden layers. We theoretically prove that the effect of temperature can be adjusted by setting the parameter of the sharpness of the logistic function in the proposed TRBMs. The performance of RBMs can be improved by adjusting the temperature parameter of TRBMs. This work provides a comprehensive insights into the deep belief networks and deep learning architectures from a physical point of view.

11. Ludwig Boltzmann: Atomic genius

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.)

12. Deriving thermal lattice-Boltzmann models from the continuous Boltzmann equation: theoretical aspects

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...

13. Training Restricted Boltzmann Machines

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...

14. 玻尔兹曼因子等温线对气液相变与临界系数的数学描述%Mathematical Description on Gas Liquid Phase Transition and Critical Phenomena with Boltzmann Factor Isotherm

吴义彬

2014-01-01

应用气体的玻尔兹曼因子方程和液体的玻尔兹曼因子方程共同描述等温压缩二氧化碳气体的全过程，可以得到与安德鲁斯实验实测等温线完全一致的玻尔兹曼因子等温线。特别是，在低于临界温度时，玻尔兹曼因子等温线可以如实地描述从气体体系到气体与液体共存的非均匀体系，再到液体体系的等温压缩全过程；客观地表述了气、液相变与临界系数的自然变化规律；彻底地解决了“范氏方程给出的曲线不包含气、液共存的信息”等缺陷问题。%In this article , the Boltzmann factor equation for both gas and liquid are used to illustrate the isothermal compression process for CO2 ;the isothermal curves of the Boltzmann factor are obtainedcompletely consistent with the Andrews experimental results .It is especially true when the temperature goes below critical point ;the isothermal curves of the Boltzmann factor can describe the whole isothermal compression process including the ones from gas to gas liquid , furthermore to liquid system .The naturalvariation law of The gas liquid phase transition and critical phenomena can be objectively depicted which completely eliminates the recognized defects of Van der Waals equation :the curves do not provide information about gas liquid system .

15. An extension of the Boltzmann relation to collisionless magnetized plasma

The neutralization of positive space charge is studied for density perturbations of limited spatial extent in a collisionfree magnetized plasma. It is found that a local density maximum gets a positive potential which depends only on the ambient electron temperature Te and the relative increase in density ne/ne0. For small density increases, below 5%, the resulting relation between potential and plasma density agrees closely with the Boltzmann relation, which applies in the presence of collisions. For larger density increases, the difference from the Boltzmann relation rapidly becomes large, e.g. a factor 2 for a 50% density increase, and a factor 3 for a 100% density increase. The result constitutes (1) a justification for using the Boltzmann relation also in collisionless magnetized plasma, provided that the density perturbations are small, and (2) a general relation which replaces the Boltzmann relation for large-amplitude perturbations. (au)

16. 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 ...

17. On the full Boltzmann equations for Leptogenesis

Garayoa, J; Pinto, T; Rius, N; Vives, O

2009-01-01

We consider the full Boltzmann equations for standard and soft leptogenesis, instead of the usual integrated Boltzmann equations which assume kinetic equilibrium for all species. Decays and inverse decays may be inefficient for thermalising the heavy-(s)neutrino distribution function, leading to significant deviations from kinetic equilibrium. We analyse the impact of using the full kinetic equations in the case of a previously generated lepton asymmetry, and find that the washout of this initial asymmetry due to the interactions of the right-handed neutrino is larger than when calculated via the integrated equations. We also solve the full Boltzmann equations for soft leptogenesis, where the lepton asymmetry induced by the soft SUSY-breaking terms in sneutrino decays is a purely thermal effect, since at T=0 the asymmetry in leptons cancels the one in sleptons. In this case, we obtain that in the weak washout regime (K ~< 1) the final lepton asymmetry can change up to a factor four with respect to previous...

18. 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

19. 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.

20. A Note on Boltzmann Brains

Nomura, Yasunori

2015-01-01

Understanding the observed arrow of time is equivalent, under general assumptions, to explaining why Boltzmann brains do not overwhelm ordinary observers. It is usually thought that this provides a condition on the decay rate of every cosmologically accessible de Sitter vacuum, and that this condition is determined by the production rate of Boltzmann brains calculated using semiclassical theory built on each such vacuum. We argue, based on a recently developed picture of microscopic quantum gravitational degrees of freedom, that this thinking needs to be modified. In particular, depending on the structure of the fundamental theory, the decay rate of a de Sitter vacuum may not have to satisfy any condition except possibly the one imposed by the Poincare recurrence. The framework discussed here also addresses the question of whether a Minkowski vacuum may produce Boltzmann brains.

1. A note on Boltzmann brains

Nomura, Yasunori

2015-10-01

Understanding the observed arrow of time is equivalent, under general assumptions, to explaining why Boltzmann brains do not overwhelm ordinary observers. It is usually thought that this provides a condition on the decay rate of every cosmologically accessible de Sitter vacuum, and that this condition is determined by the production rate of Boltzmann brains calculated using semiclassical theory built on each such vacuum. We argue, based on a recently developed picture of microscopic quantum gravitational degrees of freedom, that this thinking needs to be modified. In particular, depending on the structure of the fundamental theory, the decay rate of a de Sitter vacuum may not have to satisfy any condition except for the one imposed by the Poincaré recurrence. The framework discussed here also addresses the question of whether a Minkowski vacuum may produce Boltzmann brains.

2. Exact solutions to the Boltzmann equation by mapping the scattering integral into a differential operator

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)

3. 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.

4. 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...

5. Quantum corrections for Boltzmann equation

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.

6. On Training Deep Boltzmann Machines

Desjardins, Guillaume; Courville, Aaron; Bengio, Yoshua

2012-01-01

The deep Boltzmann machine (DBM) has been an important development in the quest for powerful "deep" probabilistic models. To date, simultaneous or joint training of all layers of the DBM has been largely unsuccessful with existing training methods. We introduce a simple regularization scheme that encourages the weight vectors associated with each hidden unit to have similar norms. We demonstrate that this regularization can be easily combined with standard stochastic maximum likelihood to yie...

7. Boltzmann equation and hydrodynamic fluctuations.

Colangeli, Matteo; Kröger, Martin; Ottinger, Hans Christian

2009-11-01

We apply the method of invariant manifolds to derive equations of generalized hydrodynamics from the linearized Boltzmann equation and determine exact transport coefficients, obeying Green-Kubo formulas. Numerical calculations are performed in the special case of Maxwell molecules. We investigate, through the comparison with experimental data and former approaches, the spectrum of density fluctuations and address the regime of finite Knudsen numbers and finite frequencies hydrodynamics. PMID:20364972

8. Boltzmann equation and hydrodynamic fluctuations

Colangeli, M.; Kroger, M.; Ottinger, H. C.

2009-01-01

We apply the method of invariant manifolds to derive equations of generalized hydrodynamics from the linearized Boltzmann equation and determine exact transport coefficients, obeying Green-Kubo formulas. Numerical calculations are performed in the special case of Maxwell molecules. We investigate, through the comparison with experimental data and former approaches, the spectrum of density fluctuations and address the regime of finite Knudsen numbers and finite frequencies hydrodynamics.

9. Pruning Boltzmann networks and hidden Markov models

Pedersen, Morten With; Stork, D.

1996-01-01

Boltzmann chains and hidden Markov models (HMMs), we argue that our method can be applied to HMMs as well. We illustrate pruning on Boltzmann zippers, which are equivalent to two HMMs with cross-connection links. We verify that our second-order approximation preserves the rank ordering of weight saliencies...

10. 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

11. Relativistic Boltzmann theory for a plasma

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.)

12. Relativistic Entropy and Related Boltzmann Kinetics

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...

13. 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

14. Thermal Lattice Boltzmann Model for Compressible Fluid

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.

15. A Viscosity Adaptive Lattice Boltzmann Method

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...

16. 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

17. 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.

18. Boltzmann map for quantum oscillators

The authors define a map tau on the space of quasifree states of the CCR or CAR of more than one harmonic oscillator which increases entropy except at fixed points of tau. The map tau is the composition of a double stochastic map T*, and the quasifree reduction Q. Under mixing conditions on T, iterates of tau take any initial state to the Gibbs states, provided that the oscillator frequencies are mutually rational. They give an example of a system with three degrees of freedom with energies omega1, omega2, and omega3 mutually irrational, but obeying a relation n1omega1 + n2omega2 = n3omega3, n/sub i/epsilon Z. The iterated Boltzmann map converges from an initial state rho to independent Gibbs states of the three oscillators at betas (inverse temperatures) β1, β2, β3 obeying the equation n1omega1β1 + n2omega3β1number. The equilibrium state can be rewritten as a grand canonical state. They show that for two, three, or four fermions we can get the usual rate equations as a special case

19. 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 $tBoltzmann brains problem doesn't appear. The bounds on model parameters derived from such requirement don't contradict to allowable range from observational data. 20. Boltzmann-Electron Model in Aleph. Hughes, Thomas Patrick; Hooper, Russell 2014-11-01 We apply the Boltzmann-electron model in the electrostatic, particle-in-cell, finite- element code Aleph to a plasma sheath. By assuming a Boltzmann energy distribution for the electrons, the model eliminates the need to resolve the electron plasma fre- quency, and avoids the numerical %22grid instability%22 that can cause unphysical heating of electrons. This allows much larger timesteps to be used than with kinetic electrons. Ions are treated with the standard PIC algorithm. The Boltzmann-electron model re- quires solution of a nonlinear Poisson equation, for which we use an iterative Newton solver (NOX) from the Trilinos Project. Results for the spatial variation of density and voltage in the plasma sheath agree well with an analytic model 1. 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. 2. 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 3. 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. 4. Rigorous Navier-Stokes Limit of the Lattice Boltzmann Equation Junk, Michael; Yong, Wen-An 2001-01-01 Here we riqorously investigate the diffusive limit of a velocity-discrete Boltzmann equation which is used in the lattice Boltzmann method to construct approximate solutions of the incompressible Navier-Stokes equation. 5. 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. 6. 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. 7. 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. 8. Pair Production in the Quantum Boltzmann Equation Rau, Jochen 1994-01-01 A source term in the quantum Boltzmann equation, which accounts for the spontaneous creation of$e^+e^-$-pairs in external electric fields, is derived from first principles and evaluated numerically. Careful analysis of time scales reveals that this source term is generally non-Markovian. This implies in particular that there may be temporary violations of the$H$-theorem. 9. The Quantum Boltzmann Equation in Semiconductor Physics Snoke, D. W. 2010-01-01 The quantum Boltzmann equation, or Fokker-Planck equation, has been used to successfully explain a number of experiments in semiconductor optics in the past two decades. This paper reviews some of the developments of this work, including models of excitons in bulk materials, electron-hole plasmas, and polariton gases. 10. 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. 11. Boltzmann und das Ende des mechanistischen Weltbildes Renn, Jürgen 2007-01-01 Der Wissenschaftshistoriker und Physiker Jürgen Renn untersucht die Rolle des österreichischen Physikers und Philosophen Ludwig Boltzmann (18441906) bei der Entwicklung der modernen Physik. Boltzmann war einer der letzen Vertreter des mechanistischen Weltbildes und stand somit am Ende eines Zeitalters. Renn porträtiert den Wissenschaftler aber als einen Pionier der modernen Physik, dessen Beschäftigung mit den inneren Spannungen der klassischen Physik ihn visionär zukünftige Fragestellungen aufgreifen ließ. So befasste sich Boltzmann etwa mit den Grenzproblemen zwischen Mechanik und Thermodynamik, die ihn zur Entwicklung immer raffinierterer Instrumente der statistischen Physik antrieb, die schließlich zu Schlüsselinstrumenten der modernen Physik wurden. Boltzmanns Werk steht somit am Übergang vom mechanistischen Weltbild zur Relativitäts- und Quantentheorie. Der Aussage des viel bekannteren Physikers Albert Einstein, dass Fantasie wichtiger sei als Wissen, hält Jürgen Renn im Hinblick auf Leben ... 12. The Non-Classical Boltzmann Equation, and Diffusion-Based Approximations to the Boltzmann Equation Frank, Martin; Larsen, Edward W; Vasques, Richard 2014-01-01 We show that several diffusion-based approximations (classical diffusion or SP1, SP2, SP3) to the linear Boltzmann equation can (for an infinite, homogeneous medium) be represented exactly by a non-classical transport equation. As a consequence, we indicate a method to solve diffusion-based approximations to the Boltzmann equation via Monte Carlo, with only statistical errors - no truncation errors. 13. 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. 14. 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. 15. Preliminary Results in the Use of Energy-Dependent Octagonal Lattices for Thermal Lattice Boltzmann Simulations Pavlo, Pavol; Vahala, G.; Vahala, L. 2002-01-01 Roč. 107, 1/2 (2002), s. 499-519. ISSN 0022-4715 R&D Projects: GA ČR GA202/00/1216 Institutional research plan: CEZ:AV0Z2043910 Keywords : thermal lattice Boltzmann simulations Subject RIV: BL - Plasma and Gas Discharge Physics Impact factor: 1.537, year: 2002 16. 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 17. Lattice Boltzmann model for wave propagation. Zhang, Jianying; Yan, Guangwu; Shi, Xiubo 2009-08-01 A lattice Boltzmann model for two-dimensional wave equation is proposed by using the higher-order moment method. The higher-order moment method is based on the solution of a series of partial differential equations obtained by using multiscale technique and Chapman-Enskog expansion. In order to obtain the lattice Boltzmann model for the wave equation with higher-order accuracy of truncation errors, we removed the second-order dissipation term and the third-order dispersion term by employing the moments up to fourth order. The reversibility in time appears owing to the absence of the second-order dissipation term and the third-order dispersion term. As numerical examples, some classical examples, such as interference, diffraction, and wave passing through a convex lens, are simulated. The numerical results show that this model can be used to simulate wave propagation. PMID:19792280 18. Classical non-Markovian Boltzmann equation Alexanian, Moorad, E-mail: alexanian@uncw.edu [Department of Physics and Physical Oceanography, University of North Carolina Wilmington, Wilmington, North Carolina 28403-5606 (United States) 2014-08-01 The modeling of particle transport involves anomalous diffusion, (x²(t) ) ∝ t{sup α} with α ≠ 1, with subdiffusive transport corresponding to 0 < α < 1 and superdiffusive transport to α > 1. These anomalies give rise to fractional advection-dispersion equations with memory in space and time. The usual Boltzmann equation, with only isolated binary collisions, is Markovian and, in particular, the contributions of the three-particle distribution function are neglected. We show that the inclusion of higher-order distribution functions give rise to an exact, non-Markovian Boltzmann equation with resulting transport equations for mass, momentum, and kinetic energy with memory in both time and space. The two- and the three-particle distribution functions are considered under the assumption that the two- and the three-particle correlation functions are translationally invariant that allows us to obtain advection-dispersion equations for modeling transport in terms of spatial and temporal fractional derivatives. 19. 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 ... 20. 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... 1. Lattice Boltzmann Model and Geophysical Hydrodynamic Equation 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. 2. The Boltzmann equation in the difference formulation 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. 3. The Boltzmann equation in the difference formulation 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. 4. The Milne problem for the Boltzmann equation Existence, uniqueness and asymptotic properties are proved for the solution of the Milne problem for the Boltzmann equation, in which the incoming velocity distribution and the total mass flux are specified arbitrarily. The collision law corresponds to a hard sphere gas. The solution uses energy estimates and is similar to that of Bardos, Santos and Sentis for neutron transport. From the Milne problem one can then easily deduce the solution of the Kramers problem 5. Nonlocal Boltzmann theory of plasma channels Yu, S.S.; Melendez, R.E. 1983-01-03 The mathematical framework for the LLNL code NUTS is developed. This code is designed to study the evolution of an electron-beam-generated plasma channel at all pressures. The Boltzmann treatment of the secondary electrons presented include all inertial, nonlocal, electric and magnetic effects, as well as effects of atomic collisions. Field equations are advanced simultaneously and self-consistently with the evolving plasma currents. 6. Nonlocal Boltzmann theory of plasma channels The mathematical framework for the LLNL code NUTS is developed. This code is designed to study the evolution of an electron-beam-generated plasma channel at all pressures. The Boltzmann treatment of the secondary electrons presented include all inertial, nonlocal, electric and magnetic effects, as well as effects of atomic collisions. Field equations are advanced simultaneously and self-consistently with the evolving plasma currents 7. Improved learning algorithms for restricted Boltzmann machines Cho, Kyunghyun 2011-01-01 A restricted Boltzmann machine (RBM) is often used as a building block for constructing deep neural networks and deep generative models which have gained popularity recently as one way to learn complex and large probabilistic models. In these deep models, it is generally known that the layer-wise pretraining of RBMs facilitates finding a more accurate model for the data. It is, hence, important to have an efficient learning method for RBM. The conventional learning is mostly performed us... 8. An efficient annealing in Boltzmann machine in Hopfield neural network Kin, Teoh Yeong; Hasan, Suzanawati Abu; Bulot, Norhisam; Ismail, Mohammad Hafiz 2012-09-01 This paper proposes and implements Boltzmann machine in Hopfield neural network doing logic programming based on the energy minimization system. The temperature scheduling in Boltzmann machine enhancing the performance of doing logic programming in Hopfield neural network. The finest temperature is determined by observing the ratio of global solution and final hamming distance using computer simulations. The study shows that Boltzmann Machine model is more stable and competent in term of representing and solving difficult combinatory problems. 9. 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. 10. 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). 11. On the Three-dimensional Central Moment Lattice Boltzmann Method Premnath, Kannan N; 10.1007/s10955-011-0208-9 2012-01-01 A three-dimensional (3D) lattice Boltzmann method based on central moments is derived. Two main elements are the local attractors in the collision term and the source terms representing the effect of external and/or self-consistent internal forces. For suitable choices of the orthogonal moment basis for the three-dimensional, twenty seven velocity (D3Q27), and, its subset, fifteen velocity (D3Q15) lattice models, attractors are expressed in terms of factorization of lower order moments as suggested in an earlier work; the corresponding source terms are specified to correctly influence lower order hydrodynamic fields, while avoiding aliasing effects for higher order moments. These are achieved by successively matching the corresponding continuous and discrete central moments at various orders, with the final expressions written in terms of raw moments via a transformation based on the binomial theorem. Furthermore, to alleviate the discrete effects with the source terms, they are treated to be temporally semi-... 12. Numerical investigations of low-density nozzle flow by solving the Boltzmann equation Deng, Zheng-Tao; Liaw, Goang-Shin; Chou, Lynn Chen 1995-01-01 A two-dimensional finite-difference code to solve the BGK-Boltzmann equation has been developed. The solution procedure consists of three steps: (1) transforming the BGK-Boltzmann equation into two simultaneous partial differential equations by taking moments of the distribution function with respect to the molecular velocity u(sub z), with weighting factors 1 and u(sub z)(sup 2); (2) solving the transformed equations in the physical space based on the time-marching technique and the four-stage Runge-Kutta time integration, for a given discrete-ordinate. The Roe's second-order upwind difference scheme is used to discretize the convective terms and the collision terms are treated as source terms; and (3) using the newly calculated distribution functions at each point in the physical space to calculate the macroscopic flow parameters by the modified Gaussian quadrature formula. Repeating steps 2 and 3, the time-marching procedure stops when the convergent criteria is reached. A low-density nozzle flow field has been calculated by this newly developed code. The BGK Boltzmann solution and experimental data show excellent agreement. It demonstrated that numerical solutions of the BGK-Boltzmann equation are ready to be experimentally validated. 13. Boltzmann equation integration in thermionic converter conditions. Part II. Terms in Boltzmann equation 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. 14. Privacy-Preserving Restricted Boltzmann Machine 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. 15. 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 16. Application of lattice Boltzmann scheme to nanofluids 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. 17. Lattice Boltzmann methods for moving boundary flows Inamuro, Takaji, E-mail: inamuro@kuaero.kyoto-u.ac.jp [Department of Aeronautics and Astronautics, and Advanced Research Institute of Fluid Science and Engineering, Graduate School of Engineering, Kyoto University, Kyoto 606-8501 (Japan) 2012-04-01 The lattice Boltzmann methods (LBMs) for moving boundary flows are presented. The LBM for two-phase fluid flows with the same density and the LBM combined with the immersed boundary method are described. In addition, the LBM on a moving multi-block grid is explained. Three numerical examples (a droplet moving in a constricted tube, the lift generation of a flapping wing and the sedimentation of an elliptical cylinder) are shown in order to demonstrate the applicability of the LBMs to moving boundary problems. (invited review) 18. Lattice Boltzmann methods for moving boundary flows The lattice Boltzmann methods (LBMs) for moving boundary flows are presented. The LBM for two-phase fluid flows with the same density and the LBM combined with the immersed boundary method are described. In addition, the LBM on a moving multi-block grid is explained. Three numerical examples (a droplet moving in a constricted tube, the lift generation of a flapping wing and the sedimentation of an elliptical cylinder) are shown in order to demonstrate the applicability of the LBMs to moving boundary problems. (invited review) 19. Scattering theory of the linear Boltzmann operator In time dependent scattering theory we know three important examples: the wave equation around an obstacle, the Schroedinger and the Dirac equation with a scattering potential. In this paper another example from time dependent linear transport theory is added and considered in full detail. First the linear Boltzmann operator in certain Banach spaces is rigorously defined, and then the existence of the Moeller operators is proved by use of the theorem of Cook-Jauch-Kuroda, that is generalized to the case of a Banach space. (orig.) 20. 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. 1. 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. 2. Thermal equation of state for lattice Boltzmann gases 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. 3. 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. 4. 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. 5. 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. 6. 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. 7. The possible resolution of Boltzmann brains problem in phantom cosmology Astashenok, Artyom V.; Yurov, 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 and big rip 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

8. Polarizable atomic multipole solutes in a Poisson-Boltzmann continuum

Schnieders, Michael J.; Baker, Nathan A.; Ren, Pengyu; Ponder, Jay W.

2007-03-01

Modeling the change in the electrostatics of organic molecules upon moving from vacuum into solvent, due to polarization, has long been an interesting problem. In vacuum, experimental values for the dipole moments and polarizabilities of small, rigid molecules are known to high accuracy; however, it has generally been difficult to determine these quantities for a polar molecule in water. A theoretical approach introduced by Onsager [J. Am. Chem. Soc. 58, 1486 (1936)] used vacuum properties of small molecules, including polarizability, dipole moment, and size, to predict experimentally known permittivities of neat liquids via the Poisson equation. Since this important advance in understanding the condensed phase, a large number of computational methods have been developed to study solutes embedded in a continuum via numerical solutions to the Poisson-Boltzmann equation. Only recently have the classical force fields used for studying biomolecules begun to include explicit polarization in their functional forms. Here the authors describe the theory underlying a newly developed polarizable multipole Poisson-Boltzmann (PMPB) continuum electrostatics model, which builds on the atomic multipole optimized energetics for biomolecular applications (AMOEBA) force field. As an application of the PMPB methodology, results are presented for several small folded proteins studied by molecular dynamics in explicit water as well as embedded in the PMPB continuum. The dipole moment of each protein increased on average by a factor of 1.27 in explicit AMOEBA water and 1.26 in continuum solvent. The essentially identical electrostatic response in both models suggests that PMPB electrostatics offers an efficient alternative to sampling explicit solvent molecules for a variety of interesting applications, including binding energies, conformational analysis, and pKa prediction. Introduction of 150mM salt lowered the electrostatic solvation energy between 2 and 13kcal /mole, depending on

9. Monte Carlo variance reduction approaches for non-Boltzmann tallies

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

10. Training Restricted Boltzmann Machines on Word Observations

Dahl, George E; Larochelle, Hugo

2012-01-01

The restricted Boltzmann machine (RBM) is a flexible tool for modeling complex data, however there have been significant computational difficulties in using RBMs to model high-dimensional multinomial observations. In natural language processing applications, words are naturally modeled by K-ary discrete distributions, where K is determined by the vocabulary size and can easily be in the hundred thousands. The conventional approach to training RBMs on word observations is limited because it requires sampling the states of K-way softmax visible units during block Gibbs updates, an operation that takes time linear in K. In this work, we address this issue by employing a more general class of Markov chain Monte Carlo operators on the visible units, yielding updates with computational complexity independent of K. We demonstrate the success of our approach by training RBMs on hundreds of millions of word n-grams using larger vocabularies than previously feasible with RBMs and using the learned features to improve p...

11. 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.

12. Ordinal Boltzmann Machines for Collaborative Filtering

Truyen, Tran The; Venkatesh, Svetha

2012-01-01

Collaborative filtering is an effective recommendation technique wherein the preference of an individual can potentially be predicted based on preferences of other members. Early algorithms often relied on the strong locality in the preference data, that is, it is enough to predict preference of a user on a particular item based on a small subset of other users with similar tastes or of other items with similar properties. More recently, dimensionality reduction techniques have proved to be equally competitive, and these are based on the co-occurrence patterns rather than locality. This paper explores and extends a probabilistic model known as Boltzmann Machine for collaborative filtering tasks. It seamlessly integrates both the similarity and co-occurrence in a principled manner. In particular, we study parameterisation options to deal with the ordinal nature of the preferences, and propose a joint modelling of both the user-based and item-based processes. Experiments on moderate and large-scale movie recomm...

13. 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.

14. Boltzmann babies in the proper time measure

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.

15. 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.

16. 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...

17. 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

18. Lattice Boltzmann Model for Numerical Relativity

Ilseven, E

2015-01-01

In the Bona-Masso 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. Our model is validated with well-established tests, showing good agreement with analytical solutions. Furthermore, we show 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 improves. Finally, in order to show the potential of our approach a linear scaling law for parallelisation with respect to number of CPU cores is demonstrated. Our model represents the first step in using lattice kinetic theory to solve gravitational problems.

19. Scattering theory for the linearized Boltzmann equation

Scattering theory for a cloud of mutually non-interacting particles that in its passage through R3 undergoes absorption and production in a region D is contained in R3 through interaction with the medium in D is investigated. The motion for such a model is given by the linearized Boltzmann equation. Let n(x,v,t) denote the particle density in phase space at time t. The dynamics is described by a 1-parameter semigroup, W(t), which is in general not isometric. The existence of the wave operators Ω/sub +/ = s - lim W0(-t)W(t) (t →+infinity) and Ω/sub -/ = s - lim W(-t)W0(t) (t →-infinity), where W0(t) is the free dynamics, is examined at length

20. 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.

1. Boltzmann-Gaussian transition under specific noise effect

It is observed that a short time data set of market returns presents almost symmetric Boltzmann distribution whereas a long time data set tends to show a Gaussian distribution. To understand this universal phenomenon, many hypotheses which are spreading in a wide range of interdisciplinary research were proposed. In current work, the effects of background fluctuations on symmetric Boltzmann distribution is investigated. The numerical calculation is performed to show that the Gaussian noise may cause the transition from initial Boltzmann distribution to Gaussian one. The obtained results would reflect non-dynamic nature of the transition under consideration.

2. Fermion particle production in semiclassical Boltzmann-Vlasov transport theory

We present numerical solutions of the semiclassical Boltzmann-Vlasov equation for fermion particle-antiparticle production by strong electric fields in boost-invariant coordinates in (1+1) and (3+1) dimensional QED. We compare the Boltzmann-Vlasov results with those of recent quantum field theory calculations and find good agreement. We conclude that extending the Boltzmann-Vlasov approach to the case of QCD should allow us to do a thorough investigation of how backreaction affects recent results on the dependence of the transverse momentum distribution of quarks and antiquarks on a second Casimir invariant of color SU(3).

3. 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.

4. 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.

5. 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...

6. Second-order Boltzmann equation: gauge dependence and gauge invariance

In the context of cosmological perturbation theory, we derive the second-order Boltzmann equation describing the evolution of the distribution function of radiation without a specific gauge choice. The essential steps in deriving the Boltzmann equation are revisited and extended given this more general framework: (i) the polarization of light is incorporated in this formalism by using a tensor-valued distribution function; (ii) the importance of a choice of the tetrad field to define the local inertial frame in the description of the distribution function is emphasized; (iii) we perform a separation between temperature and spectral distortion, both for the intensity and polarization for the first time; (iv) the gauge dependence of all perturbed quantities that enter the Boltzmann equation is derived, and this enables us to check the correctness of the perturbed Boltzmann equation by explicitly showing its gauge-invariance for both intensity and polarization. We finally discuss several implications of the gauge dependence for the observed temperature. (paper)

7. 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.

8. Second order Boltzmann equation : gauge dependence and gauge invariance

Naruko, Atsushi; Koyama, Kazuya; Sasaki, Misao

2013-01-01

In the context of cosmological perturbation theory, we derive the second order Boltzmann equation describing the evolution of the distribution function of radiation without a specific gauge choice. The essential steps in deriving the Boltzmann equation are revisited and extended given this more general framework: i) the polarisation of light is incorporated in this formalism by using a tensor-valued distribution function; ii) the importance of a choice of the tetrad field to define the local inertial frame in the description of the distribution function is emphasized; iii) we perform a separation between temperature and spectral distortion, both for the intensity and for polarisation for the first time; iv) the gauge dependence of all perturbed quantities that enter the Boltzmann equation is derived, and this enables us to check the correctness of the perturbed Boltzmann equation by explicitly showing its gauge-invariance for both intensity and polarization. We finally discuss several implications of the gaug...

9. Poisson-Boltzmann-Nernst-Planck model

The Poisson-Nernst-Planck (PNP) model is based on a mean-field approximation of ion interactions and continuum descriptions of concentration and electrostatic potential. It provides qualitative explanation and increasingly quantitative predictions of experimental measurements for the ion transport problems in many areas such as semiconductor devices, nanofluidic systems, and biological systems, despite many limitations. While the PNP model gives a good prediction of the ion transport phenomenon for chemical, physical, and biological systems, the number of equations to be solved and the number of diffusion coefficient profiles to be determined for the calculation directly depend on the number of ion species in the system, since each ion species corresponds to one Nernst-Planck equation and one position-dependent diffusion coefficient profile. In a complex system with multiple ion species, the PNP can be computationally expensive and parameter demanding, as experimental measurements of diffusion coefficient profiles are generally quite limited for most confined regions such as ion channels, nanostructures and nanopores. We propose an alternative model to reduce number of Nernst-Planck equations to be solved in complex chemical and biological systems with multiple ion species by substituting Nernst-Planck equations with Boltzmann distributions of ion concentrations. As such, we solve the coupled Poisson-Boltzmann and Nernst-Planck (PBNP) equations, instead of the PNP equations. The proposed PBNP equations are derived from a total energy functional by using the variational principle. We design a number of computational techniques, including the Dirichlet to Neumann mapping, the matched interface and boundary, and relaxation based iterative procedure, to ensure efficient solution of the proposed PBNP equations. Two protein molecules, cytochrome c551 and Gramicidin A, are employed to validate the proposed model under a wide range of bulk ion concentrations and external

10. 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...

11. 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...

12. Linearized Boltzmann Equation and Hydrodynamics for Granular Gases

Brey, J. Javier; Dufty, James W.; Ruiz-Montero, M. J.

2003-01-01

The linearized Boltzmann equation is considered to describe small spatial perturbations of the homogeneous cooling state. The corresponding macroscopic balance equations for the density, temperature, and flow velocity are derived from it as the basis for a hydrodynamic description. Hydrodynamics is defined in terms of the spectrum of the generator for the dynamics of the linearized Boltzmann equation. The hydrodynamic eigenfunctions and eigenvalues are calculated in the long wavelength limit....

13. A new lattice Boltzmann model for incompressible magnetohydrodynamics

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.

14. High energy ion range and deposited energy calculation using the Boltzmann-Fokker-Planck splitting of the Boltzmann transport equation

We consider the splitting of the straight-ahead Boltzmann transport equation in the Boltzmann-Fokker-Planck equation, decomposing the differential cross-section into a singular part, corresponding to small energy transfer events, and in a regular one, which corresponds to large energy transfer. The convergence of implantation profile, nuclear and electronic energy depositions, calculated from the Boltzmann-Fokker-Planck equation, to the respective exact distributions, calculated from Monte-Carlo method, was exanimate in a large-energy interval for various values of splitting parameter and for different ion-target mass relations. It is shown that for the universal potential there exists an optimal value of splitting parameter, for which range and deposited energy distributions, calculated from the Boltzmann-Fokker-Planck equation, accurately approximate the exact distributions and which minimizes the computational expenses

15. Analysis of transport of collimated radiation in a participating media using the lattice Boltzmann method

Application of the lattice Boltzmann method (LBM) recently proposed by Asinari et al. [Asinari P, Mishra SC, Borchiellini R. A lattice Boltzmann formulation to the analysis of radiative heat transfer problems in a participating medium. Numer Heat Transfer B 2010; 57:126–146] is extended to the analysis of transport of collimated radiation in a planar participating medium. To deal with azimuthally symmetric radiation in planar medium, a new lattice structure for the LBM is used. The transport of the collimated component in the medium is analysed by two different, viz., flux splitting and direct approaches. For different angles of incidence of the collimated radiation, the LBM formulation is tested for the effects of the extinction coefficient, the anisotropy factor, and the boundary emissivities on heat flux and emissive power distributions. Results are compared with the benchmark results obtained using the finite volume method. Both the approaches in LBM provide accurate results. -- Highlights: ► Transport of collimated radiation in participating media is studied. ► Usage of Lattice Boltzmann method (LBM) is extended in this study. ► In LBM, flux splitting and direct approaches are proposed. ► Effects of various parameters are studied on heat flux and temperature profiles. ► In all cases, LBM provides correct results.

16. 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...

17. 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

18. The relativistic linear Boltzmann transport equation

In this thesis the relativistic linear Boltzmann transport equation is applied to an experiment in pion production by 740 MeV protons incident on a variety of nuclei. This equation is solved by the Monte Carlo method of generating a single particle intranuclear cascade. The transport equation is derived starting with the N-body equation of motion for quantum mechanics in phase in order to determine under what conditions it is a valid approximation. It is shown that it should be a valid semi-classical approximation provided that: (1) The kinetic energy of the transport particle is much greater than its energy of interaction with the mean nuclear potential field. (2) The two-body collision interactions which make up the single particle intranuclear cascade take place over space and time intervals which are small relative to the internucleon space and time intervals for interactions within the nucleus and also compared to the space and time scales over which the probability distribution undergoes variation. In the pion production calculation condition (2) is only approximately met but reasonable agreement with the experimental data is obtained similar to that obtained in other theoretical calculations compared to this experiment

19. 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

20. Lattice Boltzmann Modeling of Micro-fluidic Devices

Clague, D S

2002-01-28

The results to date do indeed show that the lattice Boltzmann method accurately solves relevant, non-trivial flow problems. The parallelization of both the fluid and the mobile species in flow has enhanced this capability such that it is useful for solving relevant problems in a timely fashion. The initial studies of stationary or capture species revealed evidence of hydrodynamic screening between upstream and downstream particles. Numerical studies reveal that the critical length for which the test particle is hydrodynamically decoupled from upstream and downstream particles is on the order of 30 sphere radii. For mobile species, the LB capability was shown to be naturally suited for predicting the hydrodynamic lift phenomenon (inertial lift). A conversion factor was developed based on scaling arguments to include relevant forces generated by external fields. Using this conversion, an analytic solution for the Dielectrophoretic force was included into the LB capability which enabled the study of Dielectrophoretic particle capture. The Non-Newtonian enhancements have expanded the applicability of the LB capability to more physical systems. Specifically, with the bead-n-spring representation of macromolecules researchers will be able to study chain dynamics in micro-, physiological and Bio-MEMS environments. Furthermore, the ability to capture the shear thinning behavior, without any increase in computational time, positions this capability to be applied to a whole host of new problems involving biofluids.

1. 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

2. Lattice Boltzmann Modeling of Micro-fluidic Devices

The results to date do indeed show that the lattice Boltzmann method accurately solves relevant, non-trivial flow problems. The parallelization of both the fluid and the mobile species in flow has enhanced this capability such that it is useful for solving relevant problems in a timely fashion. The initial studies of stationary or capture species revealed evidence of hydrodynamic screening between upstream and downstream particles. Numerical studies reveal that the critical length for which the test particle is hydrodynamically decoupled from upstream and downstream particles is on the order of 30 sphere radii. For mobile species, the LB capability was shown to be naturally suited for predicting the hydrodynamic lift phenomenon (inertial lift). A conversion factor was developed based on scaling arguments to include relevant forces generated by external fields. Using this conversion, an analytic solution for the Dielectrophoretic force was included into the LB capability which enabled the study of Dielectrophoretic particle capture. The Non-Newtonian enhancements have expanded the applicability of the LB capability to more physical systems. Specifically, with the bead-n-spring representation of macromolecules researchers will be able to study chain dynamics in micro-, physiological and Bio-MEMS environments. Furthermore, the ability to capture the shear thinning behavior, without any increase in computational time, positions this capability to be applied to a whole host of new problems involving biofluids

3. 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.

4. Lattice Boltzmann method and its applications in engineering thermophysics

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.

5. The Boltzmann equation theory of charged particle transport

It is shown how a formally exact Kubo-like response theory equivalent to the Boltzmann equation theory of charged particle transport can be constructed. The response theory gives the general wavevector and time-dependent velocity distribution at any time in terms of an initial distribution function, to which is added the response induced by a generalized perturbation over the intervening time. The usual Kubo linear response result for the distribution function is recovered by choosing the initial velocity distribution to be Maxwellian. For completeness the response theory introduces an exponential convergence function into the response time integral. This is equivalent to using a modified Boltzmann equation but the general form of the transport theory is not changed. The modified transport theory can be used to advantage where possible convergence difficulties occur in numerical solutions of the Boltzmann equation. This paper gives a systematic development of the modified transport theory and shows how the response theory fits into the broader scheme of solving the Boltzmann equation. The discussion extends both the work of Kumar et al. (1980), where the distribution function is expanded out in terms of tensor functions, and the propagator description where the non-hydrodynamic time development of the distribution function is related to the wavevector dependent Green function of the Boltzmann equation

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.)

7. 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...

8. Lattice Boltzmann Model for Compressible Fluid on a Square Lattice

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

9. 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...

10. 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.

11. Axisymmetric multiphase Lattice Boltzmann method for generic equations of state

Reijers, Sten A; Toschi, Federico

2015-01-01

We present an axisymmetric lattice Boltzmann model based on the Kupershtokh et al. multiphase model that is capable of solving liquid-gas density ratios up to $10^3$. Appropriate source terms are added to the lattice Boltzmann evolution equation to fully recover the axisymmetric multiphase conservation equations. We validate the model by showing that a stationary droplet obeys the Young-Laplace law, comparing the second oscillation mode of a droplet with respect to an analytical solution and showing correct mass conservation of a propagating density wave.

12. 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.

13. 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.

14. Jet propagation within a Linearized Boltzmann Transport Model

Luo, Tan; Wang, Xin-Nian; Zhu, Yan

2015-01-01

A Linear Boltzmann Transport (LBT) model has been developed for the study of jet propagation inside a quark-gluon plasma. Both leading and thermal recoiled partons are transported according to the Boltzmann equations to account for jet-induced medium excitations. In this talk, we present our study within the LBT model in which we implement the complete set of elastic parton scattering processes. We investigate elastic parton energy loss and their energy and length dependence. We further investigate elastic energy loss and transverse shape of reconstructed jets. Contributions from the recoiled thermal partons are found to have significant influences on the jet energy loss and transverse profile.

15. Lattice gas cellular automata and lattice Boltzmann models an introduction

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.

16. Boltzmann learning of parameters in cellular neural networks

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 ...... 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 by...

17. 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

18. The Boltzmann-Hamel Equations for Optimal Control

Maruskin, Jared M.; Bloch, Anthony M.

2007-01-01

We extend the Boltzmann-Hamel equations to the optimal control setting, producing a set of equations for both kinematic and dynamic nonholonomic optimal control problems. In particular, we will show the dynamic optimal control problem can be written as a minimal set of 4n-2m first order differential equations of motion.

19. Metamaterial characterization using Boltzmann's kinetic equation for electrons

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...

20. Thermal creep problems by the discrete Boltzmann equation

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.

1. Lattice Boltzmann simulations of attenuation-driven acoustic streaming

We show that lattice Boltzmann simulations can be used to model the attenuation-driven acoustic streaming produced by a travelling wave. Comparisons are made to analytical results and to the streaming pattern produced by an imposed body force approximating the Reynolds stresses. We predict the streaming patterns around a porous material in an attenuating acoustic field

2. Nonequilibrium phenomena in QCD and BEC. Boltzmann and beyond

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.)

3. A path-integral approach to the collisional Boltzmann gas

Chen, C Y

2000-01-01

Collisional effects are included in the path-integral formulation that was proposed in one of our previous paper for the collisionless Boltzmann gas. In calculating the number of molecules entering a six-dimensional phase volume element due to collisions, both the colliding molecules and the scattered molecules are allowed to have distributions; thus the calculation is done smoothly and no singularities arise.

4. On the linearized relativistic Boltzmann equation. II. Existence of hydrodynamics

Solutions are analyzed of the linearized relativistic Boltzmann equation for initial data from L2(r, p) in long-time and/or small-mean-free-path limits. In both limits solutions of this equation converge to approximate ones constructed with solutions of the set of differential equations called the equations of relativistic hydrodynamics

5. Existence of the scattering operator for the linear Boltzmann equation

Existence theorems are proven in a study of the scattering problem for the linear Boltzmann equation (transport equation), describing the motion of a cloud of nonself-interacting particles (neutrons) in phase space. Also Simon's weak coupling result is discussed, and a meaningful wave operator in the presence of trapped particles is defined and its existence proven. 7 references

6. 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…

7. Variably saturated flow described with the anisotropic Lattice Boltzmann methods

Ginzburg, I.

2006-01-01

This paper addresses the numerical solution of highly nonlinear parabolic equations with Lattice Boltzmann techniques. They are first developed for generic advection and anisotropic dispersion equations (AADE). Collision configurations handle the anisotropic diffusion forms by using either anisotropic eigenvalue sets or anisotropic equilibrium functions. The coordinate transformation from the orthorhombic (rectangular) discretization grid to the cuboid computational grid is equivalen...

8. Solving the Homogeneous Boltzmann Equation with Arbitrary Scattering Kernel

Hohenegger, A

2008-01-01

With applications in astroparticle physics in mind, we generalize a method for the solution of the nonlinear, space homogeneous Boltzmann equation with isotropic distribution function to arbitrary matrix elements. The method is based on the expansion of the matrix element in terms of two cosines of the "scattering angles". The scattering functions used by previous authors in particle physics for matrix elements in Fermi-approximation are retrieved as lowest order results in this expansion. The method is designed for the unified treatment of reactive mixtures of particles obeying different scattering laws, including the quantum statistical terms for blocking or stimulated emission, in possibly large networks of Boltzmann equations. Although our notation is the relativistic one, as it is used in astroparticle physics, the results can also be applied in the classical case.

9. Solving the homogeneous Boltzmann equation with arbitrary scattering kernel

With applications in astroparticle physics in mind, we generalize a method for the solution of the nonlinear, space-homogeneous Boltzmann equation with an isotropic distribution function to arbitrary matrix elements. The method is based on the expansion of the scattering kernel in terms of two cosines of the 'scattering angles'. The scattering functions used by previous authors in particle physics for matrix elements in the Fermi approximation are retrieved as lowest order results in this expansion. The method is designed for the unified treatment of reactive mixtures of particles obeying different scattering laws, including the quantum statistical terms for blocking or stimulated emission, in possibly large networks of Boltzmann equations. Although our notation is the relativistic one, as it is used in astroparticle physics, the results can also be applied in the classical case.

10. Boltzmann Samplers, P\\'olya Theory, and Cycle Pointing

Bodirsky, Manuel; Kang, Mihyun; Vigerske, Stefan

2010-01-01

We introduce a general method to count unlabeled combinatorial structures and to efficiently generate them at random. The approach is based on pointing unlabeled structures in an "unbiased" way that a structure of size n gives rise to n pointed structures. We extend Polya theory to the corresponding pointing operator, and present a random sampling framework based on both the principles of Boltzmann sampling and on P\\'olya operators. All previously known unlabeled construction principles for Boltzmann samplers are special cases of our new results. Our method is illustrated on several examples: in each case, we provide enumerative results and efficient random samplers. The approach applies to unlabeled families of plane and nonplane unrooted trees, and tree-like structures in general, but also to families of graphs (such as cacti graphs and outerplanar graphs) and families of planar maps.

11. 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.

12. 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...

13. Simulation of ship airwakes using a lattice Boltzmann method

The first step in analysing the fully-coupled helicopter/ship-airwake environment is to study the airwake of a frigate or frigate-like shape in isolation. This will give an understanding of the flow field that a helicopter pilot will encounter when operating from a maritime platform. The flow around a simplified frigate shape is simulated using a lattice Boltzmann algorithm and the results compared to experimentally obtained surface data. The mean flow field of the unsteady simulations captures all the flow features of the three ship yaw angles studied. The strength of these features as predicted by the numerical simulations decreases slightly in accuracy as the approaching wind angle is increased. The results presented demonstrate the ability of the lattice Boltzmann method to predict accurately the mean airwake of a frigate-like geometry. (author)

14. 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.

15. 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...

16. Modified Lattice Boltzmann method for compressible fluid simulations

A modified lattice Boltzmann algorithm is shown to have much better stability to growing temperature perturbations, when compared with the standard lattice Boltzmann algorithm. The damping rates of long-wavelength waves, which determine stability, are derived using a collisional equilibrium distribution function which has the property that the Euler equations are obtained exactly in the limit of zero time step. Using this equilibrium distribution function, we show that our algorithm has inherent positive hyperviscosity and hyperdiffusivity, for very small values of viscosity and thermal diffusivity, which are lacking in the standard algorithm. Short-wavelength modes are shown to be stable for temperatures greater than a lower limit. Results from a computer code are used to compare these algorithms, and to confirm the damping rate predictions made analytically. Finite amplitude sound waves in the simulated fluid steepen, as expected from gas dynamic theory

17. Modified lattice Boltzmann method for compressible fluid simulations.

Hinton, F L; Rosenbluth, M N; Wong, S K; Lin-Liu, Y R; Miller, R L

2001-06-01

A modified lattice Boltzmann algorithm is shown to have much better stability to growing temperature perturbations, when compared with the standard lattice Boltzmann algorithm. The damping rates of long-wavelength waves, which determine stability, are derived using a collisional equilibrium distribution function which has the property that the Euler equations are obtained exactly in the limit of zero time step. Using this equilibrium distribution function, we show that our algorithm has inherent positive hyperviscosity and hyperdiffusivity, for very small values of viscosity and thermal diffusivity, which are lacking in the standard algorithm. Short-wavelength modes are shown to be stable for temperatures greater than a lower limit. Results from a computer code are used to compare these algorithms, and to confirm the damping rate predictions made analytically. Finite amplitude sound waves in the simulated fluid steepen, as expected from gas dynamic theory. PMID:11415085

18. Dynamics of annihilation. I. Linearized Boltzmann equation and hydrodynamics.

García de Soria, María Isabel; Maynar, Pablo; Schehr, Grégory; Barrat, Alain; Trizac, Emmanuel

2008-05-01

We study the nonequilibrium statistical mechanics of a system of freely moving particles, in which binary encounters lead either to an elastic collision or to the disappearance of the pair. Such a system of ballistic annihilation therefore constantly loses particles. The dynamics of perturbations around the free decay regime is investigated using the spectral properties of the linearized Boltzmann operator, which characterize linear excitations on all time scales. The linearized Boltzmann equation is solved in the hydrodynamic limit by a projection technique, which yields the evolution equations for the relevant coarse-grained fields and expressions for the transport coefficients. We finally present the results of molecular dynamics simulations that validate the theoretical predictions. PMID:18643046

19. Pointwise Description for the Linearized Fokker-Planck-Boltzmann Model

Wu, Kung-Chien

2015-09-01

In this paper, we study the pointwise (in the space variable) behavior of the linearized Fokker-Planck-Boltzmann model for nonsmooth initial perturbations. The result reveals both the fluid and kinetic aspects of this model. The fluid-like waves are constructed as the long-wave expansion in the spectrum of the Fourier modes for the space variable, and it has polynomial time decay rate. We design a Picard-type iteration for constructing the increasingly regular kinetic-like waves, which are carried by the transport equations and have exponential time decay rate. The Mixture Lemma plays an important role in constructing the kinetic-like waves, this lemma was originally introduced by Liu-Yu (Commun Pure Appl Math 57:1543-1608, 2004) for Boltzmann equation, but the Fokker-Planck term in this paper creates some technical difficulties.

20. Higher-order Boltzmann machines and entropy bounds

Apolloni, Bruno; Battistini, Egidio; de Falco, Diego

1999-07-01

We examine some aspects of the interface area between mathematical statistics and statistical physics relevant to the study of Boltzmann machines. The Boltzmann machine learning algorithm is based on a variational principle (Gibbs' lemma for relative entropy). This fact suggests the possibility of a scheme of successive approximations: here we consider successive approximations parametrized by the order of many-body interactions among individual units. We prove bounds on the gain in relative entropy in the crucial step of adding, and estimating by Hebb's rule, a new parameter. We address the problem of providing, on the basis of local observations, upper and lower bounds on the entropy. While upper bounds are easily obtained by subadditivity, lower bounds involve localization of Hirschman bounds on a dual quantum system.

1. 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.

2. Solving the Homogeneous Boltzmann Equation with Arbitrary Scattering Kernel

Hohenegger, A.

2008-01-01

With applications in astroparticle physics in mind, we generalize a method for the solution of the nonlinear, space homogeneous Boltzmann equation with isotropic distribution function to arbitrary matrix elements. The method is based on the expansion of the matrix element in terms of two cosines of the "scattering angles". The scattering functions used by previous authors in particle physics for matrix elements in Fermi-approximation are retrieved as lowest order results in this expansion. Th...

3. Topological interactions in a Boltzmann-type framework

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...

4. Coupling Lattice Boltzmann and Molecular Dynamics models for dense fluids

Dupuis, A.; Kotsalis, E. M.; Koumoutsakos, P.

2006-01-01

We propose a hybrid model, coupling Lattice Boltzmann and Molecular Dynamics models, for the simulation of dense fluids. Time and length scales are decoupled by using an iterative Schwarz domain decomposition algorithm. The MD and LB formulations communicate via the exchange of velocities and velocity gradients at the interface. We validate the present LB-MD model in simulations of flows of liquid argon past and through a carbon nanotube. Comparisons with existing hybrid algorithms and with r...

5. Weighted particle method for solving the Boltzmann equation

We propose a new, deterministic, method of solution of the nuclear Boltzmann equation. In this Weighted Particle Method two-body collisions are treated by a Master equation for an occupation probability of each numerical particle. We apply the method to the quadrupole motion of 12C. A comparison with usual stochastic methods is made. Advantages and disadvantages of the Weighted Particle Method are discussed

6. 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...

7. 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.

8. 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...

9. 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...

10. 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...

11. 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 ...

12. Dynamics of Annihilation I : Linearized Boltzmann Equation and Hydrodynamics

de Soria, M. I. Garcia; Maynar, P.; Schehr, G.; Barrat, A.; Trizac, E.

2008-01-01

We study the non-equilibrium statistical mechanics of a system of freely moving particles, in which binary encounters lead either to an elastic collision or to the disappearance of the pair. Such a system of {\\em ballistic annihilation} therefore constantly looses particles. The dynamics of perturbations around the free decay regime is investigated from the spectral properties of the linearized Boltzmann operator, that characterize linear excitations on all time scales. The linearized Boltzma...

13. 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...

14. 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 ...

15. Simulation of particle saltation using the lattice Boltzmann method

Dolanský, Jindřich; Chára, Zdeněk; Vlasák, Pavel; Kysela, Bohuš

Delft : Delft University of Technology, 2015. ISBN 978-83-927084-8-3. ISSN 0867-7964. [17th International Conference on Transport and Sedimentation of Solid Particles. Delft (NL), 22.09.2015-25.09.2015] R&D Projects: GA ČR GA15-18870S Institutional support: RVO:67985874 Keywords : lattice Boltzmann method * entropic LBM * particle-laden turbulent flow * particle-fluid interaction * PIV Subject RIV: BK - Fluid Dynamics

16. Simulation of particle saltation using the lattice Boltzmann method

Dolanský, Jindřich; Chára, Zdeněk; Vlasák, Pavel; Kysela, Bohuš

Delft : Delft University of Technology, 2015, s. 53-60. ISBN 978-83-927084-8-3. ISSN 0867-7964. [17th International Conference on Transport and Sedimentation of Solid Particles. Delft (NL), 22.09.2015-25.09.2015] R&D Projects: GA ČR(CZ) GA15-18870S Institutional support: RVO:67985874 Keywords : lattice Boltzmann method * entropic LBM * particle-laden turbulent flow * particle-fluid interaction * PIV Subject RIV: BK - Fluid Dynamics

17. 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...

18. 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.

19. 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.

20. Non-linear effects in the Boltzmann equation

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.)

1. On half-space problems for the discrete Boltzmann equation

We study typical half-space problems of rarefied gas dynamics, including the problems of Milne and Kramer, for the discrete Boltzmann equation (a general discrete velocity model, DVM, with an arbitrary finite number of velocities). Then the discrete Boltzmann equation reduces to a system of Odes. The data for the outgoing particles at the boundary are assigned, possibly linearly depending on the data for the incoming particles. A classification of well-posed half-space problems for the homogeneous, as well as the inhomogeneous, linearized discrete Boltzmann equation is made. In the non-linear case the solutions are assumed to tend to an assigned Maxwellian at infinity. The conditions on the data at the boundary needed for the existence of a unique (in a neighborhood of the assigned Maxwellian) solution of the problem are investigated. In the non-degenerate case (corresponding, in the continuous case, to the case when the Mach number at the Maxwellian at infinity is different of (1, 0 and 1) implicit conditions are found. Furthermore, under certain assumptions explicit conditions are found, both in the non-degenerate and degenerate cases. An application to axially symmetric models is also studied.

2. Cauchy Annealing Schedule: An Annealing Schedule for Boltzmann Selection Scheme in Evolutionary Algorithms

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...

3. Numerical Simulation of Two Phase Flow in Reconstructed Pore Network Based on Lattice Boltzmann Method

Song Rui

2013-01-01

Full Text Available Accurate prediction and understanding of the disorder microstructures in the porous media contribute to acquiring the macroscopic physical properties such as conductivity, permeability, formation factor, elastic moduli etc. Based on the rock serial sectioning images of Berea sandstone acquired by the core scanning system developed by our research group, the reconstructed rock model is established in the Mimics software and the extracted pore network of the porous rock is accomplished by the self-programming software in C++ programming language based on the revised Medial axis based algorithm and the Maximal ball algorithm. Using a lattice Boltzmann method, the single and two C phase flow are accomplished. Both of the pore-scale networks and the seepage mechanism of the single- and two Cphase flow are identical with the benchmark experimental data.

4. Nanoscale simulation of shale transport properties using the lattice Boltzmann method: permeability and diffusivity

Chen, Li; Kang, Qinjun; Yao, Jun; Tao, Wenquan

2014-01-01

Porous structures of shales are reconstructed based on scanning electron microscopy (SEM) images of shale samples from Sichuan Basin, China. Characterization analyzes of the nanoscale reconstructed shales are performed, including porosity, pore size distribution, specific surface area and pore connectivity. The multiple-relaxation-time (MRT) lattice Boltzmann method (LBM) fluid flow model and single-relaxation-time (SRT) LBM diffusion model are adopted to simulate the fluid flow and Knudsen diffusion process within the reconstructed shales, respectively. Tortuosity, intrinsic permeability and effective Knudsen diffusivity are numerically predicted. The tortuosity is much higher than that commonly employed in Bruggeman equation. Correction of the intrinsic permeability by taking into consideration the contribution of Knudsen diffusion, which leads to the apparent permeability, is performed. The correction factor under different Knudsen number and pressure are estimated and compared with existing corrections re...

5. 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

6. 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.

7. El suicidio de Ludwig Boltzmann: el misterio persiste

Sierra Cuartas, Carlos Eduardo de Jesus

1994-01-01

En este año de 1994 se cumple 150 años del nacimiento del muy ilustre, y malogrado según veremos, físico austriaco Ludwig Boltzmann. Tal circunstancia ha motivado la publicación de escritos diversos acerca de su vida y obra, máxime si se tiene en cuenta su indiscutible aporte al inicio del esclarecimiento del misterio de la fecha del tiempo, merced a su trabajo sobre entropía, una medida del cambio que siempre aumenta con el tiempo en un sistema termodinámico aislado, en tanto se alcanza el ...

8. LATTICE BOLTZMANN EQUATION MODEL IN THE CORIOLIS FIELD

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.

9. Hydrodynamic limit with geometric correction of stationary Boltzmann equation

Wu, Lei

2016-05-01

We consider the hydrodynamic limit of a stationary Boltzmann equation in a unit plate with in-flow boundary. The classical theory claims that the solution can be approximated by the sum of interior solution which satisfies steady incompressible Navier-Stokes-Fourier system, and boundary layer derived from Milne problem. In this paper, we construct counterexamples to disprove such formulation in L∞ both for its proof and result. Also, we show the hydrodynamic limit with a different boundary layer expansion with geometric correction.

10. 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

11. 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...

12. Lattice Boltzmann method and its application in engineering

Guo, Zhaoli

2013-01-01

Lattice Boltzmann method (LBM) is a relatively new simulation technique for the modeling of complex fluid systems and has attracted interest from researchers in computational physics. Unlike the traditional CFD methods, which solve the conservation equations of macroscopic properties (i.e., mass, momentum, and energy) numerically, LBM models the fluid consisting of fictive particles, and such particles perform consecutive propagation and collision processes over a discrete lattice mesh.This book will cover the fundamental and practical application of LBM. The first part of the book consists of

13. 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.

14. 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.

15. Existence of the scattering matrix for the linearized Boltzmann equation

Following Hejtmanek, we consider neutrons in infinite space obeying a linearized Boltzmann equation describing their interaction with matter in some compact set D. We prove existence of the S-matrix and subcriticality of the dynamics in the (weak-coupling) case where the mean free path is larger than the diameter of D uniform in the velocity. We prove existence of the S-matrix also for the case where D is convex and filled with uniformly absorbent material. In an appendix, we present an explicit example where the dynamics is not invertible on L+1, the cone of positive elements in L1. (orig.)

16. Pointwise Behavior of the Linearized Boltzmann Equation on Torus

Wu, Kung-Chien

2013-01-01

We study the pointwise behavior of the linearized Boltzmann equation on torus for non-smooth initial perturbation. The result reveals both the fluid and kinetic aspects of this model. The fluid-like waves are constructed as part of the long-wave expansion in the spectrum of the Fourier mode for the space variable, the time decay rate of the fluid-like waves depends on the size of the domain. We design a Picard-type iteration for constructing the increasingly regular kinetic-like waves, which ...

17. Jet propagation within a Linearized Boltzmann Transport model

A Linearized Boltzmann Transport (LBT) model has been developed for the study of parton propagation inside quark–gluon plasma. Both leading and thermal recoiled partons are tracked in order to include the effect of jet-induced medium excitation. In this talk, we present a study within the LBT model in which we implement the complete set of elastic parton scattering processes. We investigate elastic parton energy loss and their energy and length dependence. We further investigate energy loss and transverse shape of reconstructed jets. Contributions from the recoiled thermal partons and jet-induced medium excitations are found to have significant influences on the jet energy loss and transverse profile

18. Jet propagation within a Linearized Boltzmann Transport model

Luo, Tan; He, Yayun [Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079 (China); Wang, Xin-Nian [Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079 (China); Nuclear Science Division, Mailstop 70R0319, Lawrence Berkeley National Laboratory, Berkeley, CA 94740 (United States); Zhu, Yan [Departamento de Física de Partículas and IGFAE, Universidade de Santiago de Compostela, E-15706 Santiago de Compostela, Galicia (Spain)

2014-12-15

A Linearized Boltzmann Transport (LBT) model has been developed for the study of parton propagation inside quark–gluon plasma. Both leading and thermal recoiled partons are tracked in order to include the effect of jet-induced medium excitation. In this talk, we present a study within the LBT model in which we implement the complete set of elastic parton scattering processes. We investigate elastic parton energy loss and their energy and length dependence. We further investigate energy loss and transverse shape of reconstructed jets. Contributions from the recoiled thermal partons and jet-induced medium excitations are found to have significant influences on the jet energy loss and transverse profile.

19. 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.

20. An implicit Lagrangian lattice Boltzmann method for the compressible flows

Yan, Guangwu; Dong, Yinfeng; Liu, Yanhong

2006-08-01

In this paper, we propose a new Lagrangian lattice Boltzmann method (LBM) for simulating the compressible flows. The new scheme simulates fluid flows based on the displacement distribution functions. The compressible flows, such as shock waves and contact discontinuities are modelled by using Lagrangian LBM. In this model, we select the element in the Lagrangian coordinate to satisfy the basic fluid laws. This model is a simpler version than the corresponding Eulerian coordinates, because the convection term of the Euler equations disappears. The numerical simulations conform to classical results.

1. Relativistic Rotating Boltzmann Gas Using the Tetrad Formalism

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.

2. Linearized Boltzmann collision integral with the correct cutoff

Chang, Yongbin; White, R. D.

2014-07-01

In the calculation of the linearized Boltzmann collision operator for an inverse-square force law interaction (Coulomb interaction) F(r)=κ /r2, we found the widely used scattering angle cutoff θ ≥θmin is a wrong practise since the divergence still exists after the cutoff has been made. When the correct velocity change cutoff |v '-v|≥δmin is employed, the scattering angle can be integrated. A unified linearized Boltzmann collision operator for both inverse-square force law and rigid-sphere interactions is obtained. Like many other unified quantities such as transition moments, Fokker-Planck expansion coefficients and energy exchange rates obtained recently [Y. B. Chang and L. A. Viehland, AIP Adv. 1, 032128 (2011)], the difference between the two kinds of interactions is characterized by a parameter, γ, which is 1 for rigid-sphere interactions and -3 for inverse-square force law interactions. When the cutoff is removed by setting δmin=0, Hilbert's well known kernel for rigid-sphere interactions is recovered for γ = 1.

3. Linearized Boltzmann collision integral with the correct cutoff

In the calculation of the linearized Boltzmann collision operator for an inverse-square force law interaction (Coulomb interaction) F(r)=κ/r2, we found the widely used scattering angle cutoff θ≥θmin is a wrong practise since the divergence still exists after the cutoff has been made. When the correct velocity change cutoff |v′−v|≥δmin is employed, the scattering angle can be integrated. A unified linearized Boltzmann collision operator for both inverse-square force law and rigid-sphere interactions is obtained. Like many other unified quantities such as transition moments, Fokker-Planck expansion coefficients and energy exchange rates obtained recently [Y. B. Chang and L. A. Viehland, AIP Adv. 1, 032128 (2011)], the difference between the two kinds of interactions is characterized by a parameter, γ, which is 1 for rigid-sphere interactions and −3 for inverse-square force law interactions. When the cutoff is removed by setting δmin=0, Hilbert's well known kernel for rigid-sphere interactions is recovered for γ = 1

4. Lattice Boltzmann simulation of flow around a confined circular cyclinder

A two dimensional lattice Boltzmann model (LBM) based on a single time relaxation BGK model has been developed. Several benchmark problems including the Poiseuille flow, the lid driven cavity flow and the flow around a circular cylinder have been performed employing a d2q9 lattice. The laminar flow around a circular cylinder within a channel has been extensively investigated using the present lattice Boltzmann model. Both symmetric and asymmetric placement configurations of the circular cylinder within the channel have been considered. A new treatment for the outlet velocity and pressure (density) boundary conditions has been proposed and validated. The present LBM results are in excellent agreement with those of the other existing CFD results. Careful examination of the LBM results and an appropriate calculation of the lift coefficient based on the rectangular lattice representation of the circular cylinder reveals that the periodic oscillation of the lift coefficient has a second harmonic when the cylinder is placed asymmetrically within the channel. The second harmonic could be associated with an asymmetrical shedding pattern of the vortices behind the cylinder from the upper and lower sides of the cylinder. (author)

5. Reciprocal relations based on the non-stationary Boltzmann equation

Sharipov, Felix

2012-03-01

The reciprocal relations for open gaseous systems are obtained on the basis of main properties of the non-stationary Boltzmann equation and gas-surface interaction law. It is shown that the main principles to derive the kinetic coefficients satisfying the reciprocal relations remain the same as those used for time-independent gaseous systems [F. Sharipov, Onsager-Casimir reciprocal relations based on the Boltzmann equation and gas-surface interaction law single gas, Phys. Rev. 73 (2006) 026110]. First, the kinetic coefficients are obtained from the entropy production expression; then it is proved that the coefficient matrix calculated for time reversed source functions is symmetric. The proof is based on the reversibility of the gas-gas and gas-surface interactions. Three examples of applications of the present theory are given. None of these examples can be treated in the frame of the classical Onsager-Casimir reciprocal relations, which are valid only in a particular case, when the kinetic coefficients are odd or even with respect to the time reversion. The approach is generalized for gaseous mixtures.

6. Fault diagnosis via neural networks: The Boltzmann machine

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

7. 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.

8. Avoiding Boltzmann Brain domination in holographic dark energy models

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.

9. 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.

10. A hybrid method for the solution of linear Boltzmann equation

Highlights: • The paper presents a novel method for the solution of linear Boltzmann equation. • The hybrid method, based on multiple collisions, combines transport with diffusion. • The physical basis of the method is discussed together with the mathematical model. • Results show its performance in terms of accuracy and computational time. • The extension of the method to more general configurations is discussed. - Abstract: This paper presents a novel approach devised to solve the transport of neutral particles in scattering and absorbing media. The solution to the linear Boltzmann equation is sought starting from a multi-collision approach of the integro-differential equation which is combined with an approximate model for the description of the residue after truncation of the Neumann series. In the paper, the theoretical basis of such hybrid method is discussed together with the physical intuition at the basis of the methodology. Results for both steady-state and transient problems are presented and an extension to general multi-dimensional, anisotropic problem is reported

11. 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 ...

12. Lattice Boltzmann Method for 3-D Flows with Curved Boundary

Mei, Renwei; Shyy, Wei; Yu, Dazhi; Luo, Li-Shi

2002-01-01

In this work, we investigate two issues that are important to computational efficiency and reliability in fluid dynamics applications of the lattice, Boltzmann equation (LBE): (1) Computational stability and accuracy of different lattice Boltzmann models and (2) the treatment of the boundary conditions on curved solid boundaries and their 3-D implementations. Three athermal 3-D LBE models (D3QI5, D3Ql9, and D3Q27) are studied and compared in terms of efficiency, accuracy, and robustness. The boundary treatment recently developed by Filippova and Hanel and Met et al. in 2-D is extended to and implemented for 3-D. The convergence, stability, and computational efficiency of the 3-D LBE models with the boundary treatment for curved boundaries were tested in simulations of four 3-D flows: (1) Fully developed flows in a square duct, (2) flow in a 3-D lid-driven cavity, (3) fully developed flows in a circular pipe, and (4) a uniform flow over a sphere. We found that while the fifteen-velocity 3-D (D3Ql5) model is more prone to numerical instability and the D3Q27 is more computationally intensive, the 63Q19 model provides a balance between computational reliability and efficiency. Through numerical simulations, we demonstrated that the boundary treatment for 3-D arbitrary curved geometry has second-order accuracy and possesses satisfactory stability characteristics.

13. 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.

14. Lattice Boltzmann Simulation for Complex Flow in a Solar Wall

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.

15. L2-stability of the Vlasov-Maxwell-Boltzmann system near global Maxwellians

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

16. Boltzmann-Gibbs Entropy Versus Tsallis Entropy: Recent Contributions to Resolving the Argument of Einstein Concerning "Neither Herr Boltzmann nor Herr Planck has given a definition of W"?

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.

17. Boltzmann and Einstein: Statistics and dynamics –An unsolved problem

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.

18. Lattice-Boltzmann scheme for computer simulation of two-phase flows; Gitter-Boltzmann-Verfahren zur Simulation von Zweiphasenstroemungen

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.)

19. Gauss Quadratures - the Keystone of Lattice Boltzmann Models

Piaud, Benjamin; Blanco, Stéphane; Fournier, Richard; Ambruş, Victor Eugen; Sofonea, Victor

2014-01-01

In this paper, we compare two families of Lattice Boltzmann (LB) models derived by means of Gauss quadratures in the momentum space. The first one is the HLB(N;Qx,Qy,Qz) family, derived by using the Cartesian coordinate system and the Gauss-Hermite quadrature. The second one is the SLB(N;K,L,M) family, derived by using the spherical coordinate system and the Gauss-Laguerre, as well as the Gauss-Legendre quadratures. These models order themselves according to the maximum order N of the moments of the equilibrium distribution function that are exactly recovered. Microfluidics effects (slip velocity, temperature jump, as well as the longitudinal heat flux that is not driven by a temperature gradient) are accurately captured during the simulation of Couette flow for Knudsen number (kn) up to 0.25.

20. 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

1. 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.

2. 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.

3. Lattice Boltzmann simulations of a viscoelastic shear-thinning fluid.

Papenkort, S; Voigtmann, Th

2015-07-28

We present a hybrid lattice Boltzmann algorithm for the simulation of flow glass-forming fluids, characterized by slow structural relaxation, at the level of the Navier-Stokes equation. The fluid is described in terms of a nonlinear integral constitutive equation, relating the stress tensor locally to the history of flow. As an application, we present results for an integral nonlinear Maxwell model that combines the effects of (linear) viscoelasticity and (nonlinear) shear thinning. We discuss the transient dynamics of velocities, shear stresses, and normal stress differences in planar pressure-driven channel flow, after switching on (startup) and off (cessation) of the driving pressure. This transient dynamics depends nontrivially on the channel width due to an interplay between hydrodynamic momentum diffusion and slow structural relaxation. PMID:26233150

4. 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...

5. 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.

6. 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.

7. Distribution Learning in Evolutionary Strategies and Restricted Boltzmann Machines

Krause, Oswin

The thesis is concerned with learning distributions in the two settings of Evolutionary Strategies (ESs) and Restricted Boltzmann Machines (RBMs). In both cases, the distributions are learned from samples, albeit with different goals. Evolutionary Strategies are concerned with finding an optimum of...... an objective function for which the gradient is not available. The algorithm samples function values from a search distribution and adapts the parameters of the distribution during the optimization process. In the thesis, new update schemes for the covariance matrix used by the CMA-ES are...... second part of the thesis is concerned with RBMs that are fitted to a dataset using maximum log-likelihood. As the computation of the distribution's normalization constant is intractable, Markov Chain Monte Carlo methods are required to estimate and follow the log-likelihood gradient. The thesis...

8. Electric Conductivity from the solution of the Relativistic Boltzmann Equation

Puglisi, A; Greco, V

2014-01-01

We present numerical results of electric conductivity $\\sigma_{el}$ of a fluid obtained solving the Relativistic Transport Boltzmann equation in a box with periodic boundary conditions. We compute $\\sigma_{el}$ using two methods: the definition itself, i.e. applying an external electric field, and the evaluation of the Green-Kubo relation based on the time evolution of the current-current correlator. We find a very good agreement between the two methods. We also compare numerical results with analytic formulas in Relaxation Time Approximation (RTA) where the relaxation time for $\\sigma_{el}$ is determined by the transport cross section $\\sigma_{tr}$, i.e. the differential cross section weighted with the collisional momentum transfer. We investigate the electric conductivity dependence on the microscopic details of the 2-body scatterings: isotropic and anisotropic cross-section, and massless and massive particles. We find that the RTA underestimates considerably $\\sigma_{el}$; for example at screening masses $... 9. Lattice Boltzmann simulations of a viscoelastic shear-thinning fluid Papenkort, S.; Voigtmann, Th. 2015-07-01 We present a hybrid lattice Boltzmann algorithm for the simulation of flow glass-forming fluids, characterized by slow structural relaxation, at the level of the Navier-Stokes equation. The fluid is described in terms of a nonlinear integral constitutive equation, relating the stress tensor locally to the history of flow. As an application, we present results for an integral nonlinear Maxwell model that combines the effects of (linear) viscoelasticity and (nonlinear) shear thinning. We discuss the transient dynamics of velocities, shear stresses, and normal stress differences in planar pressure-driven channel flow, after switching on (startup) and off (cessation) of the driving pressure. This transient dynamics depends nontrivially on the channel width due to an interplay between hydrodynamic momentum diffusion and slow structural relaxation. 10. 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. 11. 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... 12. 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. 13. Lattice Boltzmann Equation On a 2D Rectangular Grid Bouzidi, MHamed; DHumieres, Dominique; Lallemand, Pierre; Luo, Li-Shi; Bushnell, Dennis M. (Technical Monitor) 2002-01-01 We construct a multi-relaxation lattice Boltzmann model on a two-dimensional rectangular grid. The model is partly inspired by a previous work of Koelman to construct a lattice BGK model on a two-dimensional rectangular grid. The linearized dispersion equation is analyzed to obtain the constraints on the isotropy of the transport coefficients and Galilean invariance for various wave propagations in the model. The linear stability of the model is also studied. The model is numerically tested for three cases: (a) a vortex moving with a constant velocity on a mesh periodic boundary conditions; (b) Poiseuille flow with an arbitrasy inclined angle with respect to the lattice orientation: and (c) a cylinder &symmetrically placed in a channel. The numerical results of these tests are compared with either analytic solutions or the results obtained by other methods. Satisfactory results are obtained for the numerical simulations. 14. Exact results for the Boltzmann equation and Smoluchowski's coagulation equation 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.) 15. Lattice Boltzmann model for melting with natural convection 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. 16. Ab initio molecular dynamics on the electronic Boltzmann equilibrium distribution Alonso, J L; Echenique, P [Departamento de Fisica Teorica, Universidad de Zaragoza, Pedro Cerbuna 12, E-50009 Zaragoza (Spain); Castro, A; Polo, V [Institute for Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza, Mariano Esquillor s/n, E-50018 Zaragoza (Spain); Rubio, A [Nano-Bio Spectroscopy group and ETSF Scientific Development Centre, Departamento de Fisica de Materiales, Universidad del PaIs Vasco, Centro de Fisica de Materiales, CSIC-UPV/EHU-MPC and DIPC, E-20018 San Sebastian (Spain); Zueco, D, E-mail: dzueco@unizar.e [Instituto de Ciencia de Materiales de Aragon and Departamento de Fisica de la Materia Condensada, CSIC-Universidad de Zaragoza, E-50009 Zaragoza (Spain) 2010-08-15 We prove that for a combined system of classical and quantum particles, it is possible to describe a dynamics for the classical particles that incorporates in a natural way the Boltzmann equilibrium population for the quantum subsystem. In addition, these molecular dynamics (MD) do not need to assume that the electrons immediately follow the nuclear motion (in contrast to any adiabatic approach) and do not present problems in the presence of crossing points between different potential energy surfaces (conical intersections or spin-crossings). A practical application of this MD to the study of the effect of temperature on molecular systems presenting (nearly) degenerate states-such as the avoided crossing in the ring-closure process of ozone-is presented. 17. Heat conduction in multifunctional nanotrusses studied using Boltzmann transport equation 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 18. 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 19. 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. 20. Thermal Correction to the Molar Polarizability of a Boltzmann Gas Jentschura, U D; Mohr, P J 2013-01-01 Metrology in atomic physics has been crucial for a number of advanced determinations of fundamental constants. In addition to very precise frequency measurements, the molar polarizability of an atomic gas has recently also been measured very accurately. Part of the motivation for the measurements is due to ongoing efforts to redefine the International System of Units (SI) for which an accurate value of the Boltzmann constant is needed. Here, we calculate the dominant shift of the molar polarizability in an atomic gas due to thermal effects. It is given by the relativistic correction to the dipole interaction, which emerges when the probing electric field is Lorenz transformed into the rest frame of the atoms that undergo thermal motion. While this effect is small when compared to currently available experimental accuracy, the relativistic correction to the dipole interaction is much larger than the thermal shift of the polarizability induced by blackbody radiation. 1. Thermal correction to the molar polarizability of a Boltzmann gas Jentschura, U. D.; Puchalski, M.; Mohr, P. J. 2011-12-01 Metrology in atomic physics has been crucial for a number of advanced determinations of fundamental constants. In addition to very precise frequency measurements, the molar polarizability of an atomic gas has recently also been measured very accurately. Part of the motivation for the measurements is due to ongoing efforts to redefine the International System of Units (SI), for which an accurate value of the Boltzmann constant is needed. Here we calculate the dominant shift of the molar polarizability in an atomic gas due to thermal effects. It is given by the relativistic correction to the dipole interaction, which emerges when the probing electric field is Lorentz transformed into the rest frame of the atoms that undergo thermal motion. While this effect is small when compared to currently available experimental accuracy, the relativistic correction to the dipole interaction is much larger than the thermal shift of the polarizability induced by blackbody radiation. 2. 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.

3. 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...

4. Towards Full Aircraft Airframe Noise Prediction: Lattice Boltzmann Simulations

Khorrami, Mehdi R.; Fares, Ehab; Casalino, Damiano

2014-01-01

Computational results for an 18%-scale, semi-span Gulfstream aircraft model are presented. Exa Corporation's lattice Boltzmann PowerFLOW(trademark) solver was used to perform time-dependent simulations of the flow field associated with this high-fidelity aircraft model. The simulations were obtained for free-air at a Mach number of 0.2 with the flap deflected at 39 deg (landing configuration). We focused on accurately predicting the prominent noise sources at the flap tips and main landing gear for the two baseline configurations, namely, landing flap setting without and with gear deployed. Capitalizing on the inherently transient nature of the lattice Boltzmann formulation, the complex time-dependent flow features associated with the flap were resolved very accurately and efficiently. To properly simulate the noise sources over a broad frequency range, the tailored grid was very dense near the flap inboard and outboard tips. Extensive comparison of the computed time-averaged and unsteady surface pressures with wind tunnel measurements showed excellent agreement for the global aerodynamic characteristics and the local flow field at the flap inboard and outboard tips and the main landing gear. In particular, the computed fluctuating surface pressure field for the flap agreed well with the measurements in both amplitude and frequency content, indicating that the prominent airframe noise sources at the tips were captured successfully. Gear-flap interaction effects were remarkably well predicted and were shown to affect only the inboard flap tip, altering the steady and unsteady pressure fields in that region. The simulated farfield noise spectra for both baseline configurations, obtained using a Ffowcs-Williams and Hawkings acoustic analogy approach, were shown to be in close agreement with measured values.

5. Generalizing the Boltzmann equation in complex phase space.

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

6. Kapitza conductance, temperature gradients, and solutions to the Boltzmann equation

In the belief that the study of heat transport requires the study of the transport equation, we present an approach to the problem of the Kapitza conductance h/subK/ between two materials which involves the solutions of the Boltzmann equation. One of our purposes is to investigate the origin of the apparent temperature discontinuity ΔT that is associated with this phenomenon. The hydrodynamic solutions of the Boltzmann equation, which (by definition) are describable in terms of local thermohydrodynamic variables, can transfer heat but are not at all responsible for ΔT; whereas the nonhydrodynamic solutions are completely responsible for ΔT but do not transfer heat. An effective temperature T tilde is defined which approaches the thermodynamic temperature T far from the interface, and which is assumed to be continuous across the interface. With this assumption, formal expressions for ΔT and h/subK/ are derived. In the limit as the properties of the two materials become identical, R/subK/ (=h/subK//sup -1/) approaches zero, as should be the case. Further, this approach has a natural generalization to finite frequencies and includes lifetime effects. It is pointed out that thermometers do not measure T tilde but rather T/subR/ which reflects, in a complicated fashion, the presence of the nonhydrodynamic modes, whose amplitudes fall off exponentially as one moves from the interface. In He II, determination of the exponential damping lengths (as a function of temperature and pressure) would provide information about phonon dispersion and phonon interactions which is at least as detailed as could be obtained by other means

7. 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.

8. A simple lattice Boltzmann scheme for low Mach number reactive flows

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.

9. Lattice Boltzmann simulation of the gas-solid adsorption process in reconstructed random porous media

Zhou, L.; Qu, Z. G.; Ding, T.; Miao, J. Y.

2016-04-01

The gas-solid adsorption process in reconstructed random porous media is numerically studied with the lattice Boltzmann (LB) method at the pore scale with consideration of interparticle, interfacial, and intraparticle mass transfer performances. Adsorbent structures are reconstructed in two dimensions by employing the quartet structure generation set approach. To implement boundary conditions accurately, all the porous interfacial nodes are recognized and classified into 14 types using a proposed universal program called the boundary recognition and classification program. The multiple-relaxation-time LB model and single-relaxation-time LB model are adopted to simulate flow and mass transport, respectively. The interparticle, interfacial, and intraparticle mass transfer capacities are evaluated with the permeability factor and interparticle transfer coefficient, Langmuir adsorption kinetics, and the solid diffusion model, respectively. Adsorption processes are performed in two groups of adsorbent media with different porosities and particle sizes. External and internal mass transfer resistances govern the adsorption system. A large porosity leads to an early time for adsorption equilibrium because of the controlling factor of external resistance. External and internal resistances are dominant at small and large particle sizes, respectively. Particle size, under which the total resistance is minimum, ranges from 3 to 7 μm with the preset parameters. Pore-scale simulation clearly explains the effect of both external and internal mass transfer resistances. The present paper provides both theoretical and practical guidance for the design and optimization of adsorption systems.

10. Factoring

Lenstra, Arjen K.

1994-01-01

Factoring, finding a non-trivial factorization of a composite positive integer, is believed to be a hard problem. How hard we think it is, however, changes almost on a daily basis. Predicting how hard factoring will be in the future, an important issue for cryptographic applications of composite numbers, is therefore a challenging task. The author presents a brief survey of general purpose integer factoring algorithms and their implementations