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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

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.

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. BisiJ. 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. BodineauI. 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. BodineauI. 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. BoldrighiniL. 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. BisiJ. 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. BodineauI. 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. BodineauI. 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. BoldrighiniL. 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 $\nu$
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)
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