June  2022, 15(3): 517-534. doi: 10.3934/krm.2022001

The Boltzmann-Grad limit for the Lorentz gas with a Poisson distribution of obstacles

CMLS, École polytechnique, 91128 Palaiseau Cedex, France

In memory of Prof. Robert T. Glassey (1946-2020)

Received  September 2021 Revised  November 2021 Published  June 2022 Early access  January 2022

In this note, we propose a slightly different proof of Gallavotti's theorem ["Statistical Mechanics: A Short Treatise", Springer, 1999, pp. 48-55] on the derivation of the linear Boltzmann equation for the Lorentz gas with a Poisson distribution of obstacles in the Boltzmann-Grad limit.

Citation: 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
References:
[1]

V. Agoshkov, Boundary Value Problems for Transport Equations, Birkhäuser, Boston, 1998. doi: 10.1007/978-1-4612-1994-1.

[2]

C. Bardos, Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d'approximation; application à l'équation de transport, Ann. Scient. École Normale Sup., 3 (1970), 185-233.  doi: 10.24033/asens.1190.

[3]

C. BoldrighiniL. A. Bunimovich and Ya. G. Sinai, On the Boltzmann equation for the Lorentz gas, J. Stat. Phys., 32 (1983), 477-501.  doi: 10.1007/BF01008951.

[4]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer Science+Business Media, New York, 1994. doi: 10.1007/978-1-4419-8524-8.

[5]

D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes. Vol. I: Elementary Theory and Methods, 2$^nd$ edition, Springer-Verlag, New York, Berlin, Heidelberg, 2003.

[6]

I. Gallagher, L. Saint-Raymond and B. Texier, From Newton to Boltzmann: Hard Spheres and Short-Range Potentials, Zurich Lectures in Advanced Mathematics. European Math. Soc., 2013.

[7]

G. Gallavotti, Divergences and the approach to equilibrium in the Lorentz and the wind-tree models, Phys. Rev., 185 (1969), 308-322.  doi: 10.1103/PhysRev.185.308.

[8]

G. Gallavotti, Rigorous theory of the Boltzmann equation in the Lorentz gas, Nota Interna No. 358, Istituto di Fisica, Università di Roma, 1972.

[9]

G. Gallavotti, Statistical Mechanics: A Short Treatise, Appendix 1.A.2, pp. 48–55, Springer, Berlin, Heidelberg, 1999. doi: 10.1007/978-3-662-03952-6.

[10]

R. T. Glassey, The Cauchy Problem in Kinetic Theory, SIAM, Philadelphia 1996. doi: 10.1137/1.9781611971477.

[11]

R. T. Glassey and W. A. Strauss, Singularity formation in a collisionless plasma could occur only at high velocities, Arch. Rational Mech. Anal., 92 (1986), 59-90.  doi: 10.1007/BF00250732.

[12]

H. Grad, Principles of the Kinetic theory of Gases, in Handbuch der Physik Band XII. Thermodynamik der Gase (ed. S. Flügge), Springer-Verlag, 1958,205–294.

[13] J. F. C. Kingman, Poisson Processes, Oxford University Press, 1993. 
[14]

O. E. Lanford, Time evolution of large classical systems, in Dynamical Systems, Theory And Applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974), Lect. Notes in Physics 38, (ed. J. Moser) Springer Verlag, 1975, 1–111.

[15] G. Last and M. Penrose, Lectures on the Poisson Process, Cambridge Univ. Press, 2018. 
[16]

H. Lorentz, Le mouvement des électrons dans les métaux, Arch. Neerl., 10 (1905), 336-371. 

[17]

G. C. Papanicolaou, Asymptotic analysis of transport processes, Bull. Amer. Math. Soc., 81 (1975), 330-392.  doi: 10.1090/S0002-9904-1975-13744-X.

[18]

Y. Sone, Molecular Gas Dynamics, Theory, Techniques and Applications, Birkhäuser, Boston, 2007. doi: 10.1007/978-0-8176-4573-1.

[19]

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

show all references

References:
[1]

V. Agoshkov, Boundary Value Problems for Transport Equations, Birkhäuser, Boston, 1998. doi: 10.1007/978-1-4612-1994-1.

[2]

C. Bardos, Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d'approximation; application à l'équation de transport, Ann. Scient. École Normale Sup., 3 (1970), 185-233.  doi: 10.24033/asens.1190.

[3]

C. BoldrighiniL. A. Bunimovich and Ya. G. Sinai, On the Boltzmann equation for the Lorentz gas, J. Stat. Phys., 32 (1983), 477-501.  doi: 10.1007/BF01008951.

[4]

C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer Science+Business Media, New York, 1994. doi: 10.1007/978-1-4419-8524-8.

[5]

D. J. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes. Vol. I: Elementary Theory and Methods, 2$^nd$ edition, Springer-Verlag, New York, Berlin, Heidelberg, 2003.

[6]

I. Gallagher, L. Saint-Raymond and B. Texier, From Newton to Boltzmann: Hard Spheres and Short-Range Potentials, Zurich Lectures in Advanced Mathematics. European Math. Soc., 2013.

[7]

G. Gallavotti, Divergences and the approach to equilibrium in the Lorentz and the wind-tree models, Phys. Rev., 185 (1969), 308-322.  doi: 10.1103/PhysRev.185.308.

[8]

G. Gallavotti, Rigorous theory of the Boltzmann equation in the Lorentz gas, Nota Interna No. 358, Istituto di Fisica, Università di Roma, 1972.

[9]

G. Gallavotti, Statistical Mechanics: A Short Treatise, Appendix 1.A.2, pp. 48–55, Springer, Berlin, Heidelberg, 1999. doi: 10.1007/978-3-662-03952-6.

[10]

R. T. Glassey, The Cauchy Problem in Kinetic Theory, SIAM, Philadelphia 1996. doi: 10.1137/1.9781611971477.

[11]

R. T. Glassey and W. A. Strauss, Singularity formation in a collisionless plasma could occur only at high velocities, Arch. Rational Mech. Anal., 92 (1986), 59-90.  doi: 10.1007/BF00250732.

[12]

H. Grad, Principles of the Kinetic theory of Gases, in Handbuch der Physik Band XII. Thermodynamik der Gase (ed. S. Flügge), Springer-Verlag, 1958,205–294.

[13] J. F. C. Kingman, Poisson Processes, Oxford University Press, 1993. 
[14]

O. E. Lanford, Time evolution of large classical systems, in Dynamical Systems, Theory And Applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974), Lect. Notes in Physics 38, (ed. J. Moser) Springer Verlag, 1975, 1–111.

[15] G. Last and M. Penrose, Lectures on the Poisson Process, Cambridge Univ. Press, 2018. 
[16]

H. Lorentz, Le mouvement des électrons dans les métaux, Arch. Neerl., 10 (1905), 336-371. 

[17]

G. C. Papanicolaou, Asymptotic analysis of transport processes, Bull. Amer. Math. Soc., 81 (1975), 330-392.  doi: 10.1090/S0002-9904-1975-13744-X.

[18]

Y. Sone, Molecular Gas Dynamics, Theory, Techniques and Applications, Birkhäuser, Boston, 2007. doi: 10.1007/978-0-8176-4573-1.

[19]

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

[1]

Corentin Le Bihan. Boltzmann-Grad limit of a hard sphere system in a box with isotropic boundary conditions. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1903-1932. doi: 10.3934/dcds.2021177

[2]

Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361

[3]

Hai-Liang Li, Tong Yang, Mingying Zhong. Diffusion limit of the Vlasov-Poisson-Boltzmann system. Kinetic and Related Models, 2021, 14 (2) : 211-255. doi: 10.3934/krm.2021003

[4]

Yong-Kum Cho. On the Boltzmann equation with the symmetric stable Lévy process. Kinetic and Related Models, 2015, 8 (1) : 53-77. doi: 10.3934/krm.2015.8.53

[5]

Marco A. Fontelos, Lucía B. Gamboa. On the structure of double layers in Poisson-Boltzmann equation. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1939-1967. doi: 10.3934/dcdsb.2012.17.1939

[6]

Juhi Jang, Ning Jiang. Acoustic limit of the Boltzmann equation: Classical solutions. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 869-882. doi: 10.3934/dcds.2009.25.869

[7]

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

[8]

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

[9]

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

[10]

Robert M. Strain. Coordinates in the relativistic Boltzmann theory. Kinetic and Related Models, 2011, 4 (1) : 345-359. doi: 10.3934/krm.2011.4.345

[11]

Renjun Duan, Tong Yang, Changjiang Zhu. Boltzmann equation with external force and Vlasov-Poisson-Boltzmann system in infinite vacuum. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 253-277. doi: 10.3934/dcds.2006.16.253

[12]

Laurent Bernis, Laurent Desvillettes. Propagation of singularities for classical solutions of the Vlasov-Poisson-Boltzmann equation. Discrete and Continuous Dynamical Systems, 2009, 24 (1) : 13-33. doi: 10.3934/dcds.2009.24.13

[13]

Marc Briant. Perturbative theory for the Boltzmann equation in bounded domains with different boundary conditions. Kinetic and Related Models, 2017, 10 (2) : 329-371. doi: 10.3934/krm.2017014

[14]

Stefan Possanner, Claudia Negulescu. Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport. Kinetic and Related Models, 2011, 4 (4) : 1159-1191. doi: 10.3934/krm.2011.4.1159

[15]

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

[16]

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

[17]

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

[18]

Hongxu Chen. Cercignani-Lampis boundary in the Boltzmann theory. Kinetic and Related Models, 2020, 13 (3) : 549-597. doi: 10.3934/krm.2020019

[19]

Seung-Yeal Ha, Ho Lee, Seok Bae Yun. Uniform $L^p$-stability theory for the space-inhomogeneous Boltzmann equation with external forces. Discrete and Continuous Dynamical Systems, 2009, 24 (1) : 115-143. doi: 10.3934/dcds.2009.24.115

[20]

Xuwen Chen, Yan Guo. On the weak coupling limit of quantum many-body dynamics and the quantum Boltzmann equation. Kinetic and Related Models, 2015, 8 (3) : 443-465. doi: 10.3934/krm.2015.8.443

2020 Impact Factor: 1.432

Metrics

  • PDF downloads (176)
  • HTML views (104)
  • Cited by (0)

Other articles
by authors

[Back to Top]