Journal
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
Volume 42, Issue 4, Pages 1903-1932Publisher
AMER INST MATHEMATICAL SCIENCES-AIMS
DOI: 10.3934/dcds.2021177
Keywords
Low density; Boltzmann-Grad limit; hard-sphere; stochastic boundary condition; kinetic theory
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This paper rigorously derives the Boltzmann equation in a compact domain with isotropic boundary conditions, showing that the dynamics of a system of hard spheres converge to the solution of the Boltzmann equation at a specific scale.
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] x (R/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 -> 0 to the solution of the Boltzmann equation, with the same short time restriction of Lanford's classical theorem.
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