Boltzmann-Grad limit of a hard sphere system in a box with isotropic boundary conditions. (English) Zbl 1484.82041
Summary: 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.
MSC:
| 82C40 | Kinetic theory of gases in time-dependent statistical mechanics |
| 35Q20 | Boltzmann equations |
| 82D05 | Statistical mechanics of gases |
Keywords:
low density; Boltzmann-Grad limit; hard-sphere; stochastic boundary condition; kinetic theoryReferences:
| [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; L. Saint-Raymond, A microscopic view of the Fourier law, Comptes Rendus Physique, 20, 402-418 (2019) · 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; M. Pulvirenti, The Boltzmann-Grad limit for a one-dimensional Boltzmann equation in a stationary state, Commun. Math. Phys., 177, 63-81 (1996) · Zbl 0852.76080 |
| [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. · Zbl 0961.76002 |
| [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. · Zbl 0813.76001 |
| [8] | R. Denlinger, The propagation of chaos for a rarefied gas of hard spheres in the whole space, Arch. Ration. Mech. Anal., 229, 885-952 (2018) · Zbl 1397.35164 · 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; R. Marra, Non-isothermal boundary in the Boltzmann theory and Fourier law, Commun. Math. Phys., 323, 177-239 (2013) · Zbl 1280.82009 · 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. · Zbl 1403.35195 |
| [13] | R. Esposito; R. Marra, Stationary non equilibrium states in kinetic theory, J. Stat. Phys., 180, 773-809 (2020) · Zbl 1447.82026 · 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. · Zbl 1315.82001 |
| [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. · Zbl 1405.82026 |
| [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. · Zbl 0493.60100 |
| [17] | H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2, 331-407 (1949) · Zbl 0037.13104 · doi:10.1002/cpa.3160020403 |
| [18] | J.-P. Guiraud, Problème aux limites intérieur pour l’équation de Boltzmann linéaire, J. Méc., Paris, 9, 443-490 (1970) · Zbl 0208.28101 |
| [19] | J.-P. Guiraud, Problème aux limites interieur pour l’équation de Boltzmann en regime stationnaire, faiblement non linéaire, J. Méc., Paris, 11, 183-231 (1972) · Zbl 0245.76061 |
| [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. · Zbl 0850.76600 |
| [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. · Zbl 0329.70011 |
| [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. · Zbl 1296.82051 |
| [25] | M. Pulvirenti; S. Simonella, On the evolution of the empirical measure for the hard-sphere dynamics, Bull. Inst. Math. Acad. Sin. (N.S.), 10, 171-204 (2015) · Zbl 1322.82008 |
| [26] | H. Spohn, Large Scale Dynamics of Interacting Particles, Berlin: Springer-Verlag, 1991. · Zbl 0742.76002 |
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