# American Institute of Mathematical Sciences

April  2022, 42(4): 1903-1932. doi: 10.3934/dcds.2021177

## Boltzmann-Grad limit of a hard sphere system in a box with isotropic boundary conditions

 UMPA (UMR CNRS 5669), École Normale Superieur de Lyon, 46 allée d'Italie, 69364 LYON, France

Received  April 2021 Revised  September 2021 Published  April 2022 Early access  November 2021

Fund Project: The author thanks Sergio Simonella, Raphael Winter and the anonymous referees for their suggestions which helped to improve the paper. We also thank Laure Saint Raymond for suggesting the problem

In this paper we present a rigorous derivation of the Boltzmann equation in a compact domain with {isotropic} boundary conditions. We consider a system of $N$ hard spheres of diameter $\epsilon$ in a box $\Lambda : = [0, 1]\times(\mathbb{R}/\mathbb{Z})^2$. When a particle meets the boundary of the domain, it is instantaneously reinjected into the box with a random direction, {but} conserving kinetic energy. We prove that the first marginal of the process converges in the scaling $N\epsilon^2 = 1$, $\epsilon\rightarrow 0$ to the solution of the Boltzmann equation, with the same short time restriction of Lanford's classical theorem.

Citation: 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
##### References:
 [1] R. K. Alexander, The Infinite Hard-Sphere System, ProQuest LLC, Ann Arbor, MI, 1975, URL http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:7615092, Thesis (Ph.D.)–University of California, Berkeley. [2] T. Bodineau, I. Gallagher and L. Saint-Raymond, A microscopic view of the Fourier law, Comptes Rendus Physique, 20 (2019), 402-418.  doi: 10.1016/j.crhy.2019.08.002. [3] L. Boltzmann, Lectures on Gas Theory, Translated by Stephen G. Brush, University of California Press, Berkeley-Los Angeles, Calif., 1964. [4] S. Caprino and M. Pulvirenti, The Boltzmann-Grad limit for a one-dimensional Boltzmann equation in a stationary state, Commun. Math. Phys., 177 (1996), 63-81. [5] N. Catapano, Stime $L^\infty$ Peril Flusso di Knudsen Concondizioni Diffusive al Bordo, Master's thesis. [6] C. Cercignani, Rarefied Gas Dynamics, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2000, From basic concepts to actual calculations. [7] C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, vol. 106 of Applied Mathematical Sciences, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8. [8] R. Denlinger, The propagation of chaos for a rarefied gas of hard spheres in the whole space, Arch. Ration. Mech. Anal., 229 (2018), 885-952.  doi: 10.1007/s00205-018-1229-1. [9] T. Dolmaire, Etude Mathématique de la Dérivation de L'équation de Boltzmann Dans un Domaine À bord, PhD thesis, Université de Paris, 2019. [10] T. Dolmaire, About Lanford's theorem in the half-space with specular reflection, 2021, arXiv: 2102.05513. [11] R. Esposito, Y. Guo, C. Kim and R. Marra, Non-isothermal boundary in the Boltzmann theory and Fourier law, Commun. Math. Phys., 323 (2013), 177-239.  doi: 10.1007/s00220-013-1766-2. [12] R. Esposito, Y. Guo, C. Kim and R. Marra, Stationary solutions to the Boltzmann equation in the hydrodynamic limit, Ann. PDE, 4 (2018), Paper No. 1,119 pp. doi: 10.1007/s40818-017-0037-5. [13] R. Esposito and R. Marra, Stationary non equilibrium states in kinetic theory, J. Stat. Phys., 180 (2020), 773-809.  doi: 10.1007/s10955-020-02528-w. [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 (EMS), Zürich, 2013. [15] V. Gerasimenko and I. Gapyak, Low-density asymptotic behavior of observables of hard sphere fluids, Adv. Math. Phys., 2018 (2018), No 6252919, 11 pp. doi: 10.1155/2018/6252919. [16] S. Goldstein, J. L. Lebowitz and E. Presutti, Mechanical system with stochastic boundaries, Random Fields. Rigorous Results in Statistical Mechanics and Quantum Field Theory, Esztergom 1979, Colloq. Math. Soc. Janos Bolyai, 27 (1981), 403–419. [17] H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407.  doi: 10.1002/cpa.3160020403. [18] J.-P. Guiraud, Problème aux limites intérieur pour l'équation de Boltzmann linéaire, J. Méc., Paris, 9 (1970), 443-490. [19] J.-P. Guiraud, Problème aux limites interieur pour l'équation de Boltzmann en regime stationnaire, faiblement non linéaire, J. Méc., Paris, 11 (1972), 183-231. [20] R. Illner and M. Pulvirenti, Global validity of the Boltzmann equation for two- and three-dimensional rare gas in vacuum. Erratum and improved result: "Global validity of the Boltzmann equation for a two-dimensional rare gas in vacuum", [Comm. Math. Phys., 105 (1986), 189-203; MR0849204 (88d: 82061)] and "Global validity of the Boltzmann equation for a three-dimensional rare gas in vacuum" [ibid. 113 (1987), 79–85; MR0918406 (89b: 82052)] by Pulvirenti, Comm. Math. Phys., 121 (1989), 143–146, URL http://projecteuclid.org/euclid.cmp/1104178007. [21] O. E. Lanford, III Time evolution of large classical systems, in Dynamical Systems, Theory and Applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974). Lecture Notes in Phys., Vol. 38, 1975, 1–111. [22] C. Le Bihan, Convergence D'un Système de Sphères Dures Vers une Solution de L'équation de Stokes-Fourier Avec Bord, Master's thesis, ENS de Lyon, 2019. [23] C. D. Levermore, Mathematics of Kinetic Theory, 2012, Lecture notes, https://terpconnect.umd.edu/ lvrmr/2012-2013-F/Classes/AMSC698/NOTES/Lec06.pdf. [24] M. Pulvirenti, C. Saffirio and S. Simonella, On the validity of the Boltzmann equation for short range potentials, Rev. Math. Phys., 26 (2014), 1450001, 64 pp. doi: 10.1142/S0129055X14500019. [25] M. Pulvirenti and S. Simonella, On the evolution of the empirical measure for the hard-sphere dynamics, Bull. Inst. Math. Acad. Sin. (N.S.), 10 (2015), 171-204. [26] H. Spohn, Large Scale Dynamics of Interacting Particles, Berlin: Springer-Verlag, 1991. doi: 10.1007/978-3-642-84371-6.

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##### References:
 [1] R. K. Alexander, The Infinite Hard-Sphere System, ProQuest LLC, Ann Arbor, MI, 1975, URL http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:7615092, Thesis (Ph.D.)–University of California, Berkeley. [2] T. Bodineau, I. Gallagher and L. Saint-Raymond, A microscopic view of the Fourier law, Comptes Rendus Physique, 20 (2019), 402-418.  doi: 10.1016/j.crhy.2019.08.002. [3] L. Boltzmann, Lectures on Gas Theory, Translated by Stephen G. Brush, University of California Press, Berkeley-Los Angeles, Calif., 1964. [4] S. Caprino and M. Pulvirenti, The Boltzmann-Grad limit for a one-dimensional Boltzmann equation in a stationary state, Commun. Math. Phys., 177 (1996), 63-81. [5] N. Catapano, Stime $L^\infty$ Peril Flusso di Knudsen Concondizioni Diffusive al Bordo, Master's thesis. [6] C. Cercignani, Rarefied Gas Dynamics, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2000, From basic concepts to actual calculations. [7] C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, vol. 106 of Applied Mathematical Sciences, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8524-8. [8] R. Denlinger, The propagation of chaos for a rarefied gas of hard spheres in the whole space, Arch. Ration. Mech. Anal., 229 (2018), 885-952.  doi: 10.1007/s00205-018-1229-1. [9] T. Dolmaire, Etude Mathématique de la Dérivation de L'équation de Boltzmann Dans un Domaine À bord, PhD thesis, Université de Paris, 2019. [10] T. Dolmaire, About Lanford's theorem in the half-space with specular reflection, 2021, arXiv: 2102.05513. [11] R. Esposito, Y. Guo, C. Kim and R. Marra, Non-isothermal boundary in the Boltzmann theory and Fourier law, Commun. Math. Phys., 323 (2013), 177-239.  doi: 10.1007/s00220-013-1766-2. [12] R. Esposito, Y. Guo, C. Kim and R. Marra, Stationary solutions to the Boltzmann equation in the hydrodynamic limit, Ann. PDE, 4 (2018), Paper No. 1,119 pp. doi: 10.1007/s40818-017-0037-5. [13] R. Esposito and R. Marra, Stationary non equilibrium states in kinetic theory, J. Stat. Phys., 180 (2020), 773-809.  doi: 10.1007/s10955-020-02528-w. [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 (EMS), Zürich, 2013. [15] V. Gerasimenko and I. Gapyak, Low-density asymptotic behavior of observables of hard sphere fluids, Adv. Math. Phys., 2018 (2018), No 6252919, 11 pp. doi: 10.1155/2018/6252919. [16] S. Goldstein, J. L. Lebowitz and E. Presutti, Mechanical system with stochastic boundaries, Random Fields. Rigorous Results in Statistical Mechanics and Quantum Field Theory, Esztergom 1979, Colloq. Math. Soc. Janos Bolyai, 27 (1981), 403–419. [17] H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407.  doi: 10.1002/cpa.3160020403. [18] J.-P. Guiraud, Problème aux limites intérieur pour l'équation de Boltzmann linéaire, J. Méc., Paris, 9 (1970), 443-490. [19] J.-P. Guiraud, Problème aux limites interieur pour l'équation de Boltzmann en regime stationnaire, faiblement non linéaire, J. Méc., Paris, 11 (1972), 183-231. [20] R. Illner and M. Pulvirenti, Global validity of the Boltzmann equation for two- and three-dimensional rare gas in vacuum. Erratum and improved result: "Global validity of the Boltzmann equation for a two-dimensional rare gas in vacuum", [Comm. Math. Phys., 105 (1986), 189-203; MR0849204 (88d: 82061)] and "Global validity of the Boltzmann equation for a three-dimensional rare gas in vacuum" [ibid. 113 (1987), 79–85; MR0918406 (89b: 82052)] by Pulvirenti, Comm. Math. Phys., 121 (1989), 143–146, URL http://projecteuclid.org/euclid.cmp/1104178007. [21] O. E. Lanford, III Time evolution of large classical systems, in Dynamical Systems, Theory and Applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974). Lecture Notes in Phys., Vol. 38, 1975, 1–111. [22] C. Le Bihan, Convergence D'un Système de Sphères Dures Vers une Solution de L'équation de Stokes-Fourier Avec Bord, Master's thesis, ENS de Lyon, 2019. [23] C. D. Levermore, Mathematics of Kinetic Theory, 2012, Lecture notes, https://terpconnect.umd.edu/ lvrmr/2012-2013-F/Classes/AMSC698/NOTES/Lec06.pdf. [24] M. Pulvirenti, C. Saffirio and S. Simonella, On the validity of the Boltzmann equation for short range potentials, Rev. Math. Phys., 26 (2014), 1450001, 64 pp. doi: 10.1142/S0129055X14500019. [25] M. Pulvirenti and S. Simonella, On the evolution of the empirical measure for the hard-sphere dynamics, Bull. Inst. Math. Acad. Sin. (N.S.), 10 (2015), 171-204. [26] H. Spohn, Large Scale Dynamics of Interacting Particles, Berlin: Springer-Verlag, 1991. doi: 10.1007/978-3-642-84371-6.
Reflection of particle $i$ at time $\tau$
Evolution of $\bar{\omega}_i$ when particle $i$ has a reflection at time $\tau$
Pseudotrajectory associated with collision parameters $((1, -)_3, (2, +)_4, (1, +)_5, (4, -)_6)$
A shift of particle $j$ in $\mathcal{P}^{i, j, i'}_1$
Virtual particle
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