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February  2018, 11(1): 137-177. doi: 10.3934/krm.2018008

The derivation of the linear Boltzmann equation from a Rayleigh gas particle model

Karsten Matthies 1,, , George Stone 1, and  Florian Theil 2,

1. 

Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom
2. 

Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

* Corresponding author: Karsten Matthies

Received  March 2016 Revised  March 2017 Published  August 2017

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.

Keywords: Derivation, Boltzmann equation, semigroups, Rayleigh gas.
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 & 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. Google Scholar

[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. Google Scholar

[3]

J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications Springer Monographs in Mathematics. Springer-Verlag London, Ltd. , London, 2006. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

T. Carleman, Problemes Mathématiques Dans la Théorie Cinétique de Gaz volume 2. Almqvist & Wiksells boktr, 1957. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[21]

H. Lorentz, The motion of electrons in metallic bodies i, Koninklijke Nederlandsche Akademie van Wetenschappen Proceedings, 7 (1905), 438-453. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[30]

H. Spohn, The Lorentz process converges to a random flight process, Comm. Math. Phys., 60 (1978), 277-290. doi: 10.1007/BF01612893. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

K. Uchiyama, Derivation of the Boltzmann equation from particle dynamics, Hiroshima Math. J., 18 (1988), 245-297. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[3]

J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications Springer Monographs in Mathematics. Springer-Verlag London, Ltd. , London, 2006. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

T. Carleman, Problemes Mathématiques Dans la Théorie Cinétique de Gaz volume 2. Almqvist & Wiksells boktr, 1957. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[21]

H. Lorentz, The motion of electrons in metallic bodies i, Koninklijke Nederlandsche Akademie van Wetenschappen Proceedings, 7 (1905), 438-453. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[30]

H. Spohn, The Lorentz process converges to a random flight process, Comm. Math. Phys., 60 (1978), 277-290. doi: 10.1007/BF01612893. Google Scholar

[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. Google Scholar

[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. Google Scholar

[33]

K. Uchiyama, Derivation of the Boltzmann equation from particle dynamics, Hiroshima Math. J., 18 (1988), 245-297. Google Scholar

[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. Google Scholar

Figure 1.  The collision parameter $\nu$
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Figure 2.  Two example trees
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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)
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