Figure 1.
The collision parameter $\nu$
-
Previous Article
Global regularity for a model of Navier-Stokes equations with logarithmic sub-dissipation
- KRM Home
- This Issue
-
Next Article
Mathematical modeling of a discontinuous solution of the generalized Poisson-Nernst-Planck problem in a two-phase medium
February
2018, 11(1): 137-177.
doi: 10.3934/krm.2018008
The derivation of the linear Boltzmann equation from a Rayleigh gas particle model
| 1. | Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom |
| 2. | Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom |
A linear Boltzmann equation is derived in the Boltzmann-Grad scaling for the deterministic dynamics of many interacting particles with random initial data. We study a Rayleigh gas where a tagged particle is undergoing hard-sphere collisions with background particles, which do not interact among each other. In the Boltzmann-Grad scaling, we derive the validity of a linear Boltzmann equation for arbitrary long times under moderate assumptions on higher moments of the initial distributions of the tagged particle and the possibly non-equilibrium distribution of the background. The convergence of the empiric dynamics to the Boltzmann dynamics is shown using Kolmogorov equations for associated probability measures on collision histories.
Mathematics Subject Classification:
Primary: 82C40; Secondary: 35Q20, 37L05, 60K35, 76P05, 82C22.
Citation:
Karsten Matthies, George Stone, Florian Theil. The derivation of the linear Boltzmann equation from a Rayleigh gas particle model. Kinetic and Related Models,
2018, 11
(1)
: 137-177.
doi: 10.3934/krm.2018008
![]()
References:
| [1] |
W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander,
Vector-valued Laplace Transforms and Cauchy Problems volume 96 of Monographs in Mathematics. Birkhäuser/Springer Basel AG, Basel, second edition, 2011.
doi: 10.1007/978-3-0348-0087-7. |
| [2] |
L. Arlotti and B. Lods,
Integral representation of the linear Boltzmann operator for granular gas dynamics with applications, Journal of Statistical Physics, 129 (2007), 517-536.
doi: 10.1007/s10955-007-9402-1. |
| [3] |
J. Banasiak and L. Arlotti,
Perturbations of Positive Semigroups with Applications Springer Monographs in Mathematics. Springer-Verlag London, Ltd. , London, 2006. |
| [4] |
M. Bisi, J. A. Cañizo and B. Lods,
Entropy dissipation estimates for the linear Boltzmann operator, Journal of Functional Analysis, 269 (2015), 1028-1069.
doi: 10.1016/j.jfa.2015.05.002. |
| [5] |
T. Bodineau, I. Gallagher and L. Saint-Raymond,
The Brownian motion as the limit of a deterministic system of hard-spheres, Inventiones Mathematicae, 203 (2016), 493-553.
doi: 10.1007/s00222-015-0593-9. |
| [6] |
T. Bodineau, I. Gallagher and L. Saint-Raymond,
From hard spheres dynamics to the Stokes-Fourier equations: An L$^2$ analysis of the Boltzmann-Grad limit, Comptes Rendus Mathematique, 353 (2015), 623-627.
doi: 10.1016/j.crma.2015.04.013. |
| [7] |
C. Boldrighini, L. A. Bunimovich and Y. G. Sinaǐ,
On the Boltzmann equation for the Lorentz gas, J. Statist. Phys., 32 (1983), 477-501.
doi: 10.1007/BF01008951. |
| [8] |
M. Born and H. S. Green,
A general kinetic theory of liquids. i. the molecular distribution functions, In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, The Royal Society, 188 (1946), 10-18.
doi: 10.1098/rspa.1946.0093. |
| [9] |
T. Carleman,
Problemes Mathématiques Dans la Théorie Cinétique de Gaz volume 2. Almqvist & Wiksells boktr, 1957. |
| [10] |
C. Cercignani,
The Boltzmann Equation and Its Applications volume 67 of Applied Mathematical Sciences Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4612-1039-9. |
| [11] |
C. Cercignani, R. Illner and M. Pulvirenti,
The Mathematical Theory of Dilute Gases volume 106 of Applied Mathematical Sciences Springer-Verlag, 1994.
doi: 10.1007/978-1-4419-8524-8. |
| [12] |
D. L. Cohn,
Measure Theory Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser/Springer, New York, second edition, 2013.
doi: 10.1007/978-1-4614-6956-8. |
| [13] |
L. Desvillettes and V. Ricci,
The Boltzmann-Grad limit of a stochastic Lorentz gas in a force field, Bull. Inst. Math. Acad. Sin. (N.S.), 2 (2007), 637-648.
|
| [14] |
I. Gallagher, L. Saint-Raymond and B. Texier,
From Newton to Boltzmann: Hard Spheres and Short-Range Potentials Zurich Lectures in Advanced Mathematics. European Mathematical Society, 2013. |
| [15] |
G. Gallavotti,
Statistical Mechanics. A Short Treatise Theoretical and Mathematical Physics. Springer-Verlag Berlin Heidelberg, 1999.
doi: 10.1007/978-3-662-03952-6. |
| [16] |
F. Golse,
On the periodic Lorentz gas and the Lorentz kinetic equation, Ann. Fac. Sci. Toulouse Math. (6), 17 (2008), 735-749.
doi: 10.5802/afst.1200. |
| [17] |
O. E. Lanford,
Dynamical systems, theory and applications: Battelle seattle 1974 rencontres, Chapter Time Evolution of Large Classical Systems, Springer Berlin Heidelberg, Berlin, Heidelberg, 38 (1975), 1-111.
|
| [18] |
J. L. Lebowitz and H. Spohn,
Transport properties of the Lorentz gas: Fourier's law, J. Statist. Phys., 19 (1978), 633-654.
doi: 10.1007/BF01011774. |
| [19] |
J. L. Lebowitz and H. Spohn,
Microscopic basis for Fick's law for self-diffusion, J. Statist. Phys., 28 (1982), 539-556.
doi: 10.1007/BF01008323. |
| [20] |
J. L. Lebowitz and H. Spohn,
Steady state self-diffusion at low density, J. Statist. Phys., 29 (1982), 39-55.
doi: 10.1007/BF01008247. |
| [21] |
H. Lorentz,
The motion of electrons in metallic bodies i, Koninklijke Nederlandsche Akademie van Wetenschappen Proceedings, 7 (1905), 438-453.
|
| [22] |
A. Pazy, J. Marklof, Kinetic transport in crystals, In XVIth International Congress on Mathematical Physics, pages 162-179. World Sci. Publ., Hackensack, NJ, 2010.
doi: 10.1142/9789814304634_0009. |
| [23] |
J. Marklof and A. Strömbergsson,
The Boltzmann-Grad limit of the periodic Lorentz gas, Ann. of Math. (2), 174 (2011), 225-298.
doi: 10.4007/annals.2011.174.1.7. |
| [24] |
K. Matthies and F. Theil,
Validity and non-validity of propagation of chaos, Analysis and Stochastics of growth processes and Interface Models, (2008), 101-119.
doi: 10.1093/acprof:oso/9780199239252.003.0005. |
| [25] |
K. Matthies and F. Theil,
Validity and failure of the Boltzmann approximation of kinetic annihilation, Journal of Nonlinear Science, 20 (2010), 1-46.
doi: 10.1007/s00332-009-9049-y. |
| [26] |
K. Matthies and F. Theil,
A semigroup approach to the justification of kinetic theory, SIAM Journal on Mathematical Analysis, 44 (2012), 4345-4379.
doi: 10.1137/120865598. |
| [27] |
F. A. Molinet,
Existence, uniqueness and properties of the solutions of the Boltzmann kinetic equation for a weakly ionized gas. i, Journal of Mathematical Physics, 18 (1977), 984-996.
doi: 10.1063/1.523380. |
| [28] |
A. Pazy,
Semigroups of Linear Operators and Applications to Partial Differential Equations volume 44 of Applied Mathematical Sciences Springer-Verlag, 1983.
doi: 10.1007/978-1-4612-5561-1. |
| [29] |
M. Pulvirenti, C. Saffirio and S. Simonella, On the validity of the Boltzmann equation for short range potentials Reviews in Mathematical Physics 26 (2014), 1450001, 64 pp.
doi: 10.1142/S0129055X14500019. |
| [30] |
H. Spohn,
The Lorentz process converges to a random flight process, Comm. Math. Phys., 60 (1978), 277-290.
doi: 10.1007/BF01612893. |
| [31] |
A. Pazy, H. Spohn, Kinetic equations from Hamiltonian dynamics: The Markovian approximations, In Kinetic theory and gas dynamics, volume 293 of CISM Courses and Lect., pages 183-211. Springer, Vienna, 1988.
doi: 10.1007/978-3-7091-2762-9_6. |
| [32] |
H. Spohn,
Large Scale Dynamics of Interacting Particles Texts and Monographs in Physics. Springer Berlin Heidelberg, 1991.
doi: 10.1007/978-3-642-84371-6. |
| [33] |
K. Uchiyama,
Derivation of the Boltzmann equation from particle dynamics, Hiroshima Math. J., 18 (1988), 245-297.
|
| [34] |
H. van Beijeren, O. E. Lanford, Ⅲ, J. L. Lebowitz and H. Spohn, Equilibrium time correlation functions in the low-density limit, J. Statist. Phys., 22 (1980), 237-257.
doi: 10.1007/BF01008050. |
show all references
References:
| [1] |
W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander,
Vector-valued Laplace Transforms and Cauchy Problems volume 96 of Monographs in Mathematics. Birkhäuser/Springer Basel AG, Basel, second edition, 2011.
doi: 10.1007/978-3-0348-0087-7. |
| [2] |
L. Arlotti and B. Lods,
Integral representation of the linear Boltzmann operator for granular gas dynamics with applications, Journal of Statistical Physics, 129 (2007), 517-536.
doi: 10.1007/s10955-007-9402-1. |
| [3] |
J. Banasiak and L. Arlotti,
Perturbations of Positive Semigroups with Applications Springer Monographs in Mathematics. Springer-Verlag London, Ltd. , London, 2006. |
| [4] |
M. Bisi, J. A. Cañizo and B. Lods,
Entropy dissipation estimates for the linear Boltzmann operator, Journal of Functional Analysis, 269 (2015), 1028-1069.
doi: 10.1016/j.jfa.2015.05.002. |
| [5] |
T. Bodineau, I. Gallagher and L. Saint-Raymond,
The Brownian motion as the limit of a deterministic system of hard-spheres, Inventiones Mathematicae, 203 (2016), 493-553.
doi: 10.1007/s00222-015-0593-9. |
| [6] |
T. Bodineau, I. Gallagher and L. Saint-Raymond,
From hard spheres dynamics to the Stokes-Fourier equations: An L$^2$ analysis of the Boltzmann-Grad limit, Comptes Rendus Mathematique, 353 (2015), 623-627.
doi: 10.1016/j.crma.2015.04.013. |
| [7] |
C. Boldrighini, L. A. Bunimovich and Y. G. Sinaǐ,
On the Boltzmann equation for the Lorentz gas, J. Statist. Phys., 32 (1983), 477-501.
doi: 10.1007/BF01008951. |
| [8] |
M. Born and H. S. Green,
A general kinetic theory of liquids. i. the molecular distribution functions, In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, The Royal Society, 188 (1946), 10-18.
doi: 10.1098/rspa.1946.0093. |
| [9] |
T. Carleman,
Problemes Mathématiques Dans la Théorie Cinétique de Gaz volume 2. Almqvist & Wiksells boktr, 1957. |
| [10] |
C. Cercignani,
The Boltzmann Equation and Its Applications volume 67 of Applied Mathematical Sciences Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4612-1039-9. |
| [11] |
C. Cercignani, R. Illner and M. Pulvirenti,
The Mathematical Theory of Dilute Gases volume 106 of Applied Mathematical Sciences Springer-Verlag, 1994.
doi: 10.1007/978-1-4419-8524-8. |
| [12] |
D. L. Cohn,
Measure Theory Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser/Springer, New York, second edition, 2013.
doi: 10.1007/978-1-4614-6956-8. |
| [13] |
L. Desvillettes and V. Ricci,
The Boltzmann-Grad limit of a stochastic Lorentz gas in a force field, Bull. Inst. Math. Acad. Sin. (N.S.), 2 (2007), 637-648.
|
| [14] |
I. Gallagher, L. Saint-Raymond and B. Texier,
From Newton to Boltzmann: Hard Spheres and Short-Range Potentials Zurich Lectures in Advanced Mathematics. European Mathematical Society, 2013. |
| [15] |
G. Gallavotti,
Statistical Mechanics. A Short Treatise Theoretical and Mathematical Physics. Springer-Verlag Berlin Heidelberg, 1999.
doi: 10.1007/978-3-662-03952-6. |
| [16] |
F. Golse,
On the periodic Lorentz gas and the Lorentz kinetic equation, Ann. Fac. Sci. Toulouse Math. (6), 17 (2008), 735-749.
doi: 10.5802/afst.1200. |
| [17] |
O. E. Lanford,
Dynamical systems, theory and applications: Battelle seattle 1974 rencontres, Chapter Time Evolution of Large Classical Systems, Springer Berlin Heidelberg, Berlin, Heidelberg, 38 (1975), 1-111.
|
| [18] |
J. L. Lebowitz and H. Spohn,
Transport properties of the Lorentz gas: Fourier's law, J. Statist. Phys., 19 (1978), 633-654.
doi: 10.1007/BF01011774. |
| [19] |
J. L. Lebowitz and H. Spohn,
Microscopic basis for Fick's law for self-diffusion, J. Statist. Phys., 28 (1982), 539-556.
doi: 10.1007/BF01008323. |
| [20] |
J. L. Lebowitz and H. Spohn,
Steady state self-diffusion at low density, J. Statist. Phys., 29 (1982), 39-55.
doi: 10.1007/BF01008247. |
| [21] |
H. Lorentz,
The motion of electrons in metallic bodies i, Koninklijke Nederlandsche Akademie van Wetenschappen Proceedings, 7 (1905), 438-453.
|
| [22] |
A. Pazy, J. Marklof, Kinetic transport in crystals, In XVIth International Congress on Mathematical Physics, pages 162-179. World Sci. Publ., Hackensack, NJ, 2010.
doi: 10.1142/9789814304634_0009. |
| [23] |
J. Marklof and A. Strömbergsson,
The Boltzmann-Grad limit of the periodic Lorentz gas, Ann. of Math. (2), 174 (2011), 225-298.
doi: 10.4007/annals.2011.174.1.7. |
| [24] |
K. Matthies and F. Theil,
Validity and non-validity of propagation of chaos, Analysis and Stochastics of growth processes and Interface Models, (2008), 101-119.
doi: 10.1093/acprof:oso/9780199239252.003.0005. |
| [25] |
K. Matthies and F. Theil,
Validity and failure of the Boltzmann approximation of kinetic annihilation, Journal of Nonlinear Science, 20 (2010), 1-46.
doi: 10.1007/s00332-009-9049-y. |
| [26] |
K. Matthies and F. Theil,
A semigroup approach to the justification of kinetic theory, SIAM Journal on Mathematical Analysis, 44 (2012), 4345-4379.
doi: 10.1137/120865598. |
| [27] |
F. A. Molinet,
Existence, uniqueness and properties of the solutions of the Boltzmann kinetic equation for a weakly ionized gas. i, Journal of Mathematical Physics, 18 (1977), 984-996.
doi: 10.1063/1.523380. |
| [28] |
A. Pazy,
Semigroups of Linear Operators and Applications to Partial Differential Equations volume 44 of Applied Mathematical Sciences Springer-Verlag, 1983.
doi: 10.1007/978-1-4612-5561-1. |
| [29] |
M. Pulvirenti, C. Saffirio and S. Simonella, On the validity of the Boltzmann equation for short range potentials Reviews in Mathematical Physics 26 (2014), 1450001, 64 pp.
doi: 10.1142/S0129055X14500019. |
| [30] |
H. Spohn,
The Lorentz process converges to a random flight process, Comm. Math. Phys., 60 (1978), 277-290.
doi: 10.1007/BF01612893. |
| [31] |
A. Pazy, H. Spohn, Kinetic equations from Hamiltonian dynamics: The Markovian approximations, In Kinetic theory and gas dynamics, volume 293 of CISM Courses and Lect., pages 183-211. Springer, Vienna, 1988.
doi: 10.1007/978-3-7091-2762-9_6. |
| [32] |
H. Spohn,
Large Scale Dynamics of Interacting Particles Texts and Monographs in Physics. Springer Berlin Heidelberg, 1991.
doi: 10.1007/978-3-642-84371-6. |
| [33] |
K. Uchiyama,
Derivation of the Boltzmann equation from particle dynamics, Hiroshima Math. J., 18 (1988), 245-297.
|
| [34] |
H. van Beijeren, O. E. Lanford, Ⅲ, J. L. Lebowitz and H. Spohn, Equilibrium time correlation functions in the low-density limit, J. Statist. Phys., 22 (1980), 237-257.
doi: 10.1007/BF01008050. |
Figure 2.
Two example trees
Figure 3.
In the case $v'=0$ we are calculating the volume of $\Delta$ , since we know the background particle cannot start in $\Delta$ . For $v'\neq 0$ the cylinders get shifted but the principle is the same. (Diagram not to scale)
| [1] |
Karsten Matthies, George Stone. Derivation of a non-autonomous linear Boltzmann equation from a heterogeneous Rayleigh gas. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3299-3355. doi: 10.3934/dcds.2018143 |
| [2] |
Stéphane Brull, Pierre Charrier, Luc Mieussens. Gas-surface interaction and boundary conditions for the Boltzmann equation. Kinetic and Related Models, 2014, 7 (2) : 219-251. doi: 10.3934/krm.2014.7.219 |
| [3] |
Raffaele Esposito, Mario Pulvirenti. Rigorous validity of the Boltzmann equation for a thin layer of a rarefied gas. Kinetic and Related Models, 2010, 3 (2) : 281-297. doi: 10.3934/krm.2010.3.281 |
| [4] |
Darya V. Verveyko, Andrey Yu. Verisokin. Application of He's method to the modified Rayleigh equation. Conference Publications, 2011, 2011 (Special) : 1423-1431. doi: 10.3934/proc.2011.2011.1423 |
| [5] |
François Golse. The Boltzmann-Grad limit for the Lorentz gas with a Poisson distribution of obstacles. Kinetic and Related Models, 2022, 15 (3) : 517-534. doi: 10.3934/krm.2022001 |
| [6] |
Thibaut Allemand. Derivation of a two-fluids model for a Bose gas from a quantum kinetic system. Kinetic and Related Models, 2009, 2 (2) : 379-402. doi: 10.3934/krm.2009.2.379 |
| [7] |
Etienne Bernard, Laurent Desvillettes, Franç cois Golse, Valeria Ricci. A derivation of the Vlasov-Stokes system for aerosol flows from the kinetic theory of binary gas mixtures. Kinetic and Related Models, 2018, 11 (1) : 43-69. doi: 10.3934/krm.2018003 |
| [8] |
Kay Kirkpatrick. Rigorous derivation of the Landau equation in the weak coupling limit. Communications on Pure and Applied Analysis, 2009, 8 (6) : 1895-1916. doi: 10.3934/cpaa.2009.8.1895 |
| [9] |
Tai-Ping Liu, Shih-Hsien Yu. Boltzmann equation, boundary effects. Discrete and Continuous Dynamical Systems, 2009, 24 (1) : 145-157. doi: 10.3934/dcds.2009.24.145 |
| [10] |
Leif Arkeryd, Anne Nouri. On a Boltzmann equation for Haldane statistics. Kinetic and Related Models, 2019, 12 (2) : 323-346. doi: 10.3934/krm.2019014 |
| [11] |
Claude Bardos, François Golse, Ivan Moyano. Linear Boltzmann equation and fractional diffusion. Kinetic and Related Models, 2018, 11 (4) : 1011-1036. doi: 10.3934/krm.2018039 |
| [12] |
Yan Guo, Juhi Jang, Ning Jiang. Local Hilbert expansion for the Boltzmann equation. Kinetic and Related Models, 2009, 2 (1) : 205-214. doi: 10.3934/krm.2009.2.205 |
| [13] |
Raffaele Esposito, Yan Guo, Rossana Marra. Validity of the Boltzmann equation with an external force. Kinetic and Related Models, 2011, 4 (2) : 499-515. doi: 10.3934/krm.2011.4.499 |
| [14] |
El Miloud Zaoui, Marc Laforest. Stability and modeling error for the Boltzmann equation. Kinetic and Related Models, 2014, 7 (2) : 401-414. doi: 10.3934/krm.2014.7.401 |
| [15] |
Alexander Bobylev, Åsa Windfäll. Boltzmann equation and hydrodynamics at the Burnett level. Kinetic and Related Models, 2012, 5 (2) : 237-260. doi: 10.3934/krm.2012.5.237 |
| [16] |
Radjesvarane Alexandre. A review of Boltzmann equation with singular kernels. Kinetic and Related Models, 2009, 2 (4) : 551-646. doi: 10.3934/krm.2009.2.551 |
| [17] |
Carlos Jerez-Hanckes, Irina Pettersson, Volodymyr Rybalko. Derivation of cable equation by multiscale analysis for a model of myelinated axons. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 815-839. doi: 10.3934/dcdsb.2019191 |
| [18] |
Jacek Banasiak, Wilson Lamb. The discrete fragmentation equation: Semigroups, compactness and asynchronous exponential growth. Kinetic and Related Models, 2012, 5 (2) : 223-236. doi: 10.3934/krm.2012.5.223 |
| [19] |
Robert Denk, Yoshihiro Shibata. Generation of semigroups for the thermoelastic plate equation with free boundary conditions. Evolution Equations and Control Theory, 2019, 8 (2) : 301-313. doi: 10.3934/eect.2019016 |
| [20] |
Yan Yong, Weiyuan Zou. Macroscopic regularity for the relativistic Boltzmann equation with initial singularities. Kinetic and Related Models, 2019, 12 (5) : 945-967. doi: 10.3934/krm.2019036 |
2020 Impact Factor: 1.432
Tools
Metrics
Other articles
by authors
[Back to Top]