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Journal articles on the topic 'Boltzmann's equation'

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Published: 4 June 2021

Last updated: 18 February 2022

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1

Woods, L. C. "Limitations of Boltzmann's kinetic equation." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 464, no. 2095 (2008): 1923–40. http://dx.doi.org/10.1098/rspa.2008.0074.

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It is often assumed that Boltzmann's kinetic equation (BKE) for the evolution of the velocity distribution function f ( r , w , t ) in a gas is valid regardless of the magnitude of the Knudsen number defined by ϵ ≡ τ d ln ϕ /d t , where ϕ is a macroscopic variable like the fluid velocity v or temperature T , and τ is the collision interval. Almost all accounts of transport theory based on BKE are limited to terms in O ( ϵ )≪1, although there are treatments in which terms in O ( ϵ 2 ) are obtained, classic examples being due to Burnett and Grad. The mathematical limitations that arise are discu

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2

Mendoza, M., S. Succi, and H. J. Herrmann. "High-order kinetic relaxation schemes as high-accuracy Poisson solvers." International Journal of Modern Physics C 26, no. 05 (2015): 1550055. http://dx.doi.org/10.1142/s0129183115500552.

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We present a new approach to find accurate solutions to the Poisson equation, as obtained from the steady-state limit of a diffusion equation with strong source terms. For this purpose, we start from Boltzmann's kinetic theory and investigate the influence of higher-order terms on the resulting macroscopic equations. By performing an appropriate expansion of the equilibrium distribution, we provide a method to remove the unnecessary terms up to a desired order and show that it is possible to find, with high level of accuracy, the steady-state solution of the diffusion equation for sizeable Knu

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3

Di, Yana, Yuwei Fan, Ruo Li, and Lingchao Zheng. "Linear Stability of Hyperbolic Moment Models for Boltzmann Equation." Numerical Mathematics: Theory, Methods and Applications 10, no. 2 (2017): 255–77. http://dx.doi.org/10.4208/nmtma.2017.s04.

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AbstractGrad's moment models for Boltzmann equation were recently regularized to globally hyperbolic systems and thus the regularized models attain local well-posedness for Cauchy data. The hyperbolic regularization is only related to the convection term in Boltzmann equation. We in this paper studied the regularized models with the presentation of collision terms. It is proved that the regularized models are linearly stable at the local equilibrium and satisfy Yong's first stability condition with commonly used approximate collision terms, and particularly with Boltzmann's binary collision mo

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4

Lions, P. L. "Conditions at infinity for boltzmann's equation." Communications in Partial Differential Equations 19, no. 1-2 (1994): 335–67. http://dx.doi.org/10.1080/03605309408821019.

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5

Maass, W. "Stability properties of a class kinetic equations including Boltzmann's equation." Physica A: Statistical Mechanics and its Applications 133, no. 3 (1985): 539–50. http://dx.doi.org/10.1016/0378-4371(85)90148-7.

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6

Jagannathan, Kannan. "Anxiety and the Equation: Understanding Boltzmann's Entropy." American Journal of Physics 87, no. 9 (2019): 765. http://dx.doi.org/10.1119/1.5116583.

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7

Busoni, Giorgio, F. Glose, and B. Perthame. "Stationary Boltzmann's Equation With a Source Term." Communications in Partial Differential Equations 15, no. 12 (1990): 471–81. http://dx.doi.org/10.1080/03605309908820747.

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8

Banggu, Z. "Isolated collisionless uniform spherical solution of Boltzmann's equation." Journal of Physics A: Mathematical and General 20, no. 15 (1987): L959—L963. http://dx.doi.org/10.1088/0305-4470/20/15/005.

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9

Rousseau, M., and J. C. De Jaeger. "2D-Hydrodynamic Energy Model Including Avalanche Breakdown Phenomenon for Power Field Effect Transistors." VLSI Design 13, no. 1-4 (2001): 323–28. http://dx.doi.org/10.1155/2001/69472.

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A 2D-Hydrodynamic model is carried out to predict the breakdown voltage of microwave field effect transistors. The model is based on the conservation equations inferred from Boltzmann's transport equation, coupled with Poisson’s equation. In order to take into account the channel avalanche breakdown, the charge conservation equations for electrons and holes are considered and a generation term is introduced. The set of equations is solved using finite difference and different computational methods have been tested to save computing time. The model allows us to obtain accurate predictions for p

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10

Snyder, Randy W., C. Wade Sheen, and Paul C. Painter. "The Effect of Temperature on the Infrared Spectra of a Polyimide." Applied Spectroscopy 42, no. 3 (1988): 503–8. http://dx.doi.org/10.1366/0003702884427870.

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The effect of high temperature on the infrared spectra of fully cured PMDA/ODA polyimide is discussed. Changes in some bands can be explained by Boltzmann effects on the distribution of excited states. Other bands, in particular the 1780-cm−1 band that is often used for cure measurements, change in ways that cannot be related to Boltzmann's equation. These band position and intensity changes are explained as configurational changes occurring during heating. Determination of correction factors for integrated peak areas from spectra taken at elevated temperature are discussed.

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11

DiPerna, R. J., and P. L. Lions. "Global solutions of Boltzmann's equation and the entropy inequality." Archive for Rational Mechanics and Analysis 114, no. 1 (1991): 47–55. http://dx.doi.org/10.1007/bf00375684.

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12

Robson, RE. "Comment on FTI Method and Transport Coefficient Definitions for Charged Particle Swarms in Gases." Australian Journal of Physics 48, no. 4 (1995): 677. http://dx.doi.org/10.1071/ph950677.

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The kinetic theory of charged particle swarms in gases is based upon solution of the space and time dependent Boltzmann's equation for the phase space distribution function f(r, c, t). Hydrodynamic transport coefficients are defined in connection with a density gradient expansion (DGE) of f(r, c, t) and it is believed that these are the quantities measured in experiment. On the other hand, Ikuta and coworkers start with the spatially independent form of the Boltzmann equation, which they solve iteratively as in path-integral methods, and define transport coefficients in terms of the 'starting

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13

KESSLER, DAVID A., ELAINE S. ORAN, and CAROLYN R. KAPLAN. "Towards the development of a multiscale, multiphysics method for the simulation of rarefied gas flows." Journal of Fluid Mechanics 661 (August 2, 2010): 262–93. http://dx.doi.org/10.1017/s0022112010002934.

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We introduce a coupled multiscale, multiphysics method (CM3) for solving for the behaviour of rarefied gas flows. The approach is to solve the kinetic equation for rarefied gases (the Boltzmann equation) over a very short interval of time in order to obtain accurate estimates of the components of the stress tensor and heat-flux vector. These estimates are used to close the conservation laws for mass, momentum and energy, which are subsequently used to advance continuum-level flow variables forward in time. After a finite time interval, the Boltzmann equation is solved again for the new continu

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14

Winterberg, F. "Quantum Mechanics Derived from Boltzmann's Equation for the Planck Aether." Zeitschrift für Naturforschung A 50, no. 6 (1995): 601–5. http://dx.doi.org/10.1515/zna-1995-0613.

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Abstract In the Planck aether hypothesis it is assumed that space is densely filled with an equal number of locally interacting positive and negative Planck masses obeying a nonrelativistic law of motion. If described by a quantum mechanical two-component nonrelativistic nonlinear operator field equation, this model has a spectrum of particles greatly resembling the particles of the standard model, with Lorentz invariance as a derived dynamical symmetry valid in the limit of energies small com pared to the Planck energy. Here we show that quantum mechanics itself can be derived from the Newto

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15

Zumbrun, Kevin. "L^∞ resolvent bounds for steady Boltzmann's Equation." Kinetic & Related Models 10, no. 4 (2017): 1255–57. http://dx.doi.org/10.3934/krm.2017048.

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16

Wieser, W., and H. Neunzert. "A note on the solvability of Boltzmann's equation via comparison methods." Mathematical Methods in the Applied Sciences 7, no. 1 (1985): 332–39. http://dx.doi.org/10.1002/mma.1670070122.

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17

Palczewski, Andrzej. "Stationary Boltzmann's equation with Maxwell's boundary conditions in a bounded domain." Mathematical Methods in the Applied Sciences 15, no. 6 (1992): 375–93. http://dx.doi.org/10.1002/mma.1670150602.

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18

Ladd, Anthony J. C., and William G. Hoover. "Lorentz gas shear viscosity via nonequilibrium molecular dynamics and Boltzmann's equation." Journal of Statistical Physics 38, no. 5-6 (1985): 973–88. http://dx.doi.org/10.1007/bf01010425.

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19

Diebner, Hans H., and Otto E. Rössler. "A Deterministic Entropy Based on the Instantaneous Phase Space Volume." Zeitschrift für Naturforschung A 53, no. 1-2 (1998): 51–60. http://dx.doi.org/10.1515/zna-1998-1-209.

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Abstract A deterministic entropic measure is derived for the time evolution of Newtonian N-particle systems based on the volume of the instantaneously occupied phase space (IOPS). This measure is found as a natural extension of Boltzmann's entropy. The instantaneous arrangement of the particles is exploited in the form of spatial correlations. The new entropy is a bridge between the time-dependent Boltzmann entropy, formulated on the basis of densities in the one-particle phase space, and the static Gibbs entropy which uses densities in the full phase space. We apply the new concept in a molec

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20

Schnittler, Ch, and S. Scherf. "Alternative Quasi-Classical Transport Equations. II. Boltzmann's Equation and the Electron Distribution Function for MIS Systems." physica status solidi (b) 143, no. 1 (1987): 325–33. http://dx.doi.org/10.1002/pssb.2221430137.

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21

Plastino, A. R., and A. Plastino. "Information theory, approximate time dependent solutions of Boltzmann's equation and Tsallis' entropy." Physics Letters A 193, no. 3 (1994): 251–58. http://dx.doi.org/10.1016/0375-9601(94)90592-4.

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22

Robson, RE, K. Maeda, T. Makabe, and RD White. "Frequency Variation of the Mean Energy of rf Electron Swarms." Australian Journal of Physics 48, no. 3 (1995): 335. http://dx.doi.org/10.1071/ph950335.

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Starting from the general momentum and energy balance equations for charged particle swarms in a gas, as furnished by momentum-transfer theory, we obtain expressions for mean velocity and mean energy of an electron swarm in an r.f. electric field under spatially uniform conditions, in the frequency range WTe > 1, where Te is the energy collisional relaxation time. If Te is a decreasing function of energy, it is shown that the cycle-averaged mean energy reaches a maximum at a certain frequency. Physical arguments are provided to support this result and the prediction is verified for a consta

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23

Esmaeili, Asghar, Mehdi Faraji, and Somayyeh Karimi. "The conduction band non-parabolicity of degenerate AZO semiconductors: k.p method." European Physical Journal Applied Physics 83, no. 3 (2018): 30101. http://dx.doi.org/10.1051/epjap/2018180013.

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We present a discussion regarding the conduction band non-parabolicity and the Fermi energy of Al doped ZnO (AZO) degenerate semiconductors using the higher orders of Fermi–Dirac (F-D) integrals. We find an analytical expression for Fermi energy, based on two-band k.p theory and modified Boltzmann's classical equation. We examine the accuracy of resulted expression using absolute error value.

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24

White, Ron, S. Dujko, K. F. Ness, R. E. Robson, Z. Raspopovic, and Z. Lj Petrovic. "Time dependent multi-term solution of Boltzmann's equation for magnetised low-temperature plasmas." ANZIAM Journal 48 (May 1, 2007): 69. http://dx.doi.org/10.21914/anziamj.v48i0.97.

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25

Winkler, R., G. L. Braglia, and J. Wilhelm. "Electron Kinetics with Attachment and Ionization from Higher Order Solutions of Boltzmann's Equation." Annalen der Physik 501, no. 1 (1989): 21–40. http://dx.doi.org/10.1002/andp.19895010104.

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26

Koushki, A. M., M. Zand, H. Haghighi, and R. Neshati. "Numerical solution of Boltzmann's transport equation to predict characteristics of TEA CO2 lasers." Optik 126, no. 23 (2015): 3601–4. http://dx.doi.org/10.1016/j.ijleo.2015.08.230.

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27

Cuperman, S., and D. Zoler. "An extended analytical solution of the Boltzmann equation for non-homogeneous fusion and astrophysical plasmas." Journal of Plasma Physics 40, no. 3 (1988): 441–53. http://dx.doi.org/10.1017/s0022377800013416.

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The perturbative Chapman-Enskog procedure for solving Boltzmann's equation, holding when f1 ≪ f0 (f = f0 + f1 + …), is replaced by a method that is free of such a limitation. This work represents an extension to the case of strongly anisotropic plasma systems and the spherical geometry of that of Campbell (1984, 1986). The solution obtained here is expressed in terms of prescribed ratios of mean free path for collisions, as well as electric and gravitational fields, to the temperature- and density-gradient lengths. This solution is also used to discuss the limitation of the conduction transpor

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28

Holland, R. "Comparison of FDTD particle pushing and direct differencing of Boltzmann's equation for SGEMP problems." IEEE Transactions on Electromagnetic Compatibility 37, no. 3 (1995): 433–42. http://dx.doi.org/10.1109/15.406532.

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29

Pauls, H. L., and R. A. Burger. "Eigenfunction solution of Boltzmann's equation for the case of a focusing magnetic field with finite helicity." Astrophysical Journal 427 (June 1994): 927. http://dx.doi.org/10.1086/174198.

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30

Yang, Jia, Chen Yi Wang, Li Hua Teng, and Kai Long Zhang. "Quantitative Calculation of Polynuclear Aluminum Content in the Forced Hydrolysis-Polymerization Course of Aluminum (III) Salt Solutions." Advanced Materials Research 1096 (April 2015): 351–55. http://dx.doi.org/10.4028/www.scientific.net/amr.1096.351.

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According to the results of potentiometric titration of aluminum salt solutions under the moderate slow rate of injecting base and the three critical feature points on its titration curves, at the same time, using the self-contained Boltzmann's equation of Origin software fitting the curve, the quantitative formula of poly-aluminum content have been given. This formula can calculate conveniently the polynuclear aluminum content in the forced hydrolysis-polymerization process. The value of pattern calculation coincides with that of the Al-ferron timed spectrophotometry assay, which offers a kin

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31

Sabbane, M., and M. Tij. "Analyse du flux de Poiseuille bidimensionnel via l'équation de Boltzmann." Canadian Journal of Physics 82, no. 3 (2004): 213–25. http://dx.doi.org/10.1139/p04-002.

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The two-dimensional Poiseuille flow induced by an external force is analysed in the framework of BoltzmannMaxwell kinetic theory. In the limit of small Knudsen numbers (Kn [Formula: see text] 0.1), Boltzmann's nonlinear equation, written in terms of moments, is solved using perturbation theory. In our results, the hydrodynamic variable profiles are determined up to the fourth order in the perturbation parameter. Nonetheless, the method of solution remains valid to obtain all physical quantities of a gas undergoing Poiseuille flow. The major conclusion of our analysis has two elements. First,

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32

Chen, Hsien-Pu, Laszlo B. Kish, Claes-Göran Granqvist, and Gabor Schmera. "Do Electromagnetic Waves Exist in a Short Cable at Low Frequencies? What Does Physics Say?" Fluctuation and Noise Letters 13, no. 02 (2014): 1450016. http://dx.doi.org/10.1142/s0219477514500163.

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We refute a physical model, recently proposed by Gunn, Allison and Abbott (GAA) [ http://arxiv.org/pdf/1402.2709v2.pdf ], to utilize electromagnetic waves for eavesdropping on the Kirchhoff-law–Johnson-noise (KLJN) secure key distribution. Their model, and its theoretical underpinnings, is found to be fundamentally flawed because their assumption of electromagnetic waves violates not only the wave equation but also the second law of thermodynamics, the principle of detailed balance, Boltzmann's energy equipartition theorem, and Planck's formula by implying infinitely strong blackbody radiation

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33

Sodha, Mahendra Singh, and Sujeet Kumar Agarwal. "Application of Boltzmann's transfer equation to nonlinear propagation of a plane polarized monochromatic EM wave in ionospheric plasma." Physics of Plasmas 26, no. 10 (2019): 102902. http://dx.doi.org/10.1063/1.5097325.

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34

Roig, Francesc S., and Jacques E. Schoutens. "Remarks on the use of Boltzmann's equation for electrical conduction calculations in metal matrix and in situ composites." Journal of Materials Science 21, no. 8 (1986): 2767–70. http://dx.doi.org/10.1007/bf00551486.

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35

BENNUN, L. "CONSIDERATIONS ABOUT THE NEURAL NETWORK APPROACH FOR ATOMIC AND NUCLEAR SPECTRAL ANALYSIS." Advances in Adaptive Data Analysis 03, no. 03 (2011): 351–61. http://dx.doi.org/10.1142/s1793536911000854.

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Although Neural Networks have proliferated for spectroscopic data interpretation, this paper shows, with simple ideas and well-known examples that they are not well suited for this kind of analysis. This conclusion can be understood after we demonstrate that spectroscopic results are absolutely equivalent, from a statistical point of view, to a sequence of statistically independent events. Stochastic independent events, like a dice throw or a coin tossing, cannot be better described than by a purely statistical method. This conclusion remains unchanged independently if the results are linear o

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36

LOMHOLT, STIG B., PEDER WORNING, LARS ØGENDAL, KARSTEN B. QVIST, DOUGLAS B. HYSLOP, and ROGERT BAUER. "Kinetics of the renneting reaction followed by measurement of turbidity as a function of wavelength." Journal of Dairy Research 65, no. 4 (1998): 545–54. http://dx.doi.org/10.1017/s0022029998003148.

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In order to describe the kinetics of rennet coagulation, measurements of turbidity as a function of wavelength were used to determine the weight-average degree of polymerization, x¯w, during renneting of milk at three different concentrations of enzyme and three concentrations of casein, including the normal casein concentration of milk. The change of x¯w as a function of time was described using Von Smoluchowski's equation, testing a number of expressions for the aggregation rate constant, kij. The best description was achieved when kij was taken as a function of an energy barrier against agg

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37

Saveliev, Andrey. "On the Complex-Valued Distribution Function of Charged Particles in Magnetic Fields." Mathematics 9, no. 19 (2021): 2382. http://dx.doi.org/10.3390/math9192382.

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In this work, we revisit Boltzmann’s distribution function, which, together with the Boltzmann equation, forms the basis for the kinetic theory of gases and solutions to problems in hydrodynamics. We show that magnetic fields may be included as an intrinsic constituent of the distribution function by theoretically motivating, deriving and analyzing its complex-valued version in its most general form. We then validate these considerations by using it to derive the equations of ideal magnetohydrodynamics, thus showing that our method, based on Boltzmann’s formalism, is suitable to describe the d

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38

Mužík, Juraj. "Lattice Boltzmann Method for Two-Dimensional Unsteady Incompressible Flow." Civil and Environmental Engineering 12, no. 2 (2016): 122–27. http://dx.doi.org/10.1515/cee-2016-0017.

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Abstract A Lattice Boltzmann method is used to analyse incompressible fluid flow in a two-dimensional cavity and flow in the channel past cylindrical obstacle. The method solves the Boltzmann’s transport equation using simple computational grid - lattice. With the proper choice of the collision operator, the Boltzmann’s equation can be converted into incompressible Navier-Stokes equation. Lid-driven cavity benchmark case for various Reynolds numbers and flow past cylinder is presented in the article. The method produces stable solutions with results comparable to those in literature and is ver

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39

Grzesik, J. A. "Field-driven ion migration against dead-stop collisional braking." Journal of Plasma Physics 39, no. 1 (1988): 53–60. http://dx.doi.org/10.1017/s0022377800012848.

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The steady-state migration of ions, driven by a uniform electric field against full-stop collisions, is investigated in some detail. The required phase-space distribution is obtained very easily from Boltzmann's equation together with explicit recognition of energy conservation and population balance for the stagnant ion pool. We go on to decompose this aggregate solution into ion tiers classified by the number of background impacts previously endured. Such a decomposition permits us to detect the presence of Poisson statistics (as to collision number) lurking within the composite, thermalized

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40

Badino, Massimiliano. "Eric Johnson, Anxiety and the Equation: Understanding Boltzmann's Entropy. Cambridge, MA: MIT Press, 2018. Pp. ix + 179. ISBN 978-0-2620-3861-4. $22.95/£17.99 (cloth)." British Journal for the History of Science 52, no. 4 (2019): 721–22. http://dx.doi.org/10.1017/s0007087419000773.

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41

QU, KUN, CHANG SHU, and JINSHENG CAI. "DEVELOPING LBM-BASED FLUX SOLVER AND ITS APPLICATIONS IN MULTI-DIMENSION SIMULATIONS." International Journal of Modern Physics: Conference Series 19 (January 2012): 90–99. http://dx.doi.org/10.1142/s2010194512008628.

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In this paper, a new flux solver was developed based on a lattice Boltzmann model. Different from solving discrete velocity Boltzmann equation and lattice Boltzmann equation, Euler/Navier-Stokes (NS) equations were solved in this approach, and the flux at the interface was evaluated with a compressible lattice Boltzmann model. This method combined lattice Boltzmann method with finite volume method to solve Euler/NS equations. The proposed approach was validated by some simulations of one-dimensional and multi-dimensional problems.

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42

Chliamovitch, Gregor, and Yann Thorimbert. "Turbulence through the Spyglass of Bilocal Kinetics." Entropy 20, no. 7 (2018): 539. http://dx.doi.org/10.3390/e20070539.

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In two recent papers we introduced a generalization of Boltzmann’s assumption of molecular chaos based on a criterion of maximum entropy, which allowed setting up a bilocal version of Boltzmann’s kinetic equation. The present paper aims to investigate how the essentially non-local character of turbulent flows can be addressed through this bilocal kinetic description, instead of the more standard approach through the local Euler/Navier–Stokes equation. Balance equations appropriate to this kinetic scheme are derived and closed so as to provide bilocal hydrodynamical equations at the non-viscous

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43

Garrett, A. J. M. "Kinetic theory of cross-modulation in a weakly ionized plasma." Journal of Plasma Physics 46, no. 3 (1991): 365–90. http://dx.doi.org/10.1017/s0022377800016196.

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Cross-modulation in plasma is an electromagnetic wave interaction in which the modulation of one ‘disturbing’ wave is imposed nonlinearly on the transport properties of the medium, and thence onto a second, ‘wanted’ wave propagating linearly through it. This analysis is restricted to weakly ionized plasma with allowance for ambient magnetic field, as in the lower ionosphere. A kinetic description is used, based on the Boltzmann equation for the electrons, with electron-molecule collisions described by Boltzmann's collision integral. Because of the small mass ratio this simplifies to a differen

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44

Ghidaoui, Mohamed S., and Nanzhou Li. "Generalized Boltzmann equation for shallow water flows." Journal of Hydroinformatics 5, no. 1 (2003): 1–10. http://dx.doi.org/10.2166/hydro.2003.0001.

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This invited review paper introduces the Boltzmann-based approach for the numerical modelling of surface water flows to hydroinformaticians. The paper draws upon earlier work by our group as well as others. This review formulates the generalized Boltzmann equation for 1D and 2D shallow water flows and shows that the statistical moments of these generalized equations provide the classical continuity and momentum equations in shallow waters. The connection between the generalized Boltzmann equation and classical shallow water equations provides a framework for formulating new computational appro

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45

Adzhiev, Sergey, Janina Batishcheva, Igor Melikhov, and Victor Vedenyapin. "Kinetic Equations for Particle Clusters Differing in Shape and the H-theorem." Physics 1, no. 2 (2019): 229–53. http://dx.doi.org/10.3390/physics1020019.

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The question of constructing models for the evolution of clusters that differ in shape based on the Boltzmann’s H-theorem is investigated. The first, simplest kinetic equations are proposed and their properties are studied: the conditions for fulfilling the H-theorem (the conditions for detailed and semidetailed balance). These equations are to generalize the classical coagulation–fragmentation type equations for cases when not only mass but also particle shape is taken into account. To construct correct (physically grounded) kinetic models, the fulfillment of the condition of detailed balance

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46

Sakabekov, A., and Y. Auzhani. "Boltzmann’s Six-Moment One-Dimensional Nonlinear System Equations with the Maxwell-Auzhan Boundary Conditions." Journal of Applied Mathematics 2016 (2016): 1–8. http://dx.doi.org/10.1155/2016/5834620.

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We prove existence and uniqueness of the solution of the problem with initial and Maxwell-Auzhan boundary conditions for nonstationary nonlinear one-dimensional Boltzmann’s six-moment system equations in space of functions continuous in time and summable in square by a spatial variable. In order to obtain a priori estimation of the initial and boundary value problem for nonstationary nonlinear one-dimensional Boltzmann’s six-moment system equations we get the integral equality and then use the spherical representation of vector. Then we obtain the initial value problem for Riccati equation. We

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47

Majorana, Armando. "A BGK model for charge transport in graphene." Communications in Applied and Industrial Mathematics 10, no. 1 (2019): 153–61. http://dx.doi.org/10.1515/caim-2019-0018.

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Abstract The classical Boltzmann equation describes well temporal behaviour of a rarefied perfect gas. Modified kinetic equations have been proposed for studying the dynamics of different type of gases. An important example is the transport equation, which describes the charged particles flow, in the semi-classical regime, in electronic devices. In order to reduce the difficulties in solving the Boltzmann equation, simple expressions of a collision operator have been proposed to replace the standard Boltzmann integral term. These new equations are called kinetic models. The most popular and wi

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48

Nawnit, Kumar. "Modified Boltzmann equation and extended Navier–Stokes equations." Physics of Fluids 32, no. 2 (2020): 022001. http://dx.doi.org/10.1063/1.5139501.

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49

Wu, Lei, Jianping Meng, and Yonghao Zhang. "Kinetic modelling of the quantum gases in the normal phase." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 468, no. 2142 (2012): 1799–823. http://dx.doi.org/10.1098/rspa.2011.0673.

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Using the maximum entropy principle, a kinetic model equation is proposed to simplify the intricate collision term in the semi-classical Boltzmann equation for dilute quantum gases in the normal phase. The kinetic model equation keeps the main properties of the Boltzmann equation, including conservation of mass, momentum and energy, the entropy dissipation property, and rotational invariance. It also produces the correct Prandtl numbers for the Fermi gases. To validate the proposed model, the kinetic model equation is numerically solved in the hydrodynamic and kinetic flow regimes using the as

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Simonis, Stephan, Martin Frank, and Mathias J. Krause. "On relaxation systems and their relation to discrete velocity Boltzmann models for scalar advection–diffusion equations." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 378, no. 2175 (2020): 20190400. http://dx.doi.org/10.1098/rsta.2019.0400.

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Abstract:

The connection of relaxation systems and discrete velocity models is essential to the progress of stability as well as convergence results for lattice Boltzmann methods. In the present study we propose a formal perturbation ansatz starting from a scalar one-dimensional target equation, which yields a relaxation system specifically constructed for its equivalence to a discrete velocity Boltzmann model as commonly found in lattice Boltzmann methods. Further, the investigation of stability structures for the discrete velocity Boltzmann equation allows for algebraic characterizations of the equili

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