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

Corentin Le Bihan

doi:10.3934/dcds.2021177

Article Contents

2022, Volume 42, Issue 4: 1903-1932. Doi: 10.3934/dcds.2021177

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

Corentin Le Bihan

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

Received: April 2021

Revised: September 2021

Early access: November 2021

Published: April 2022

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 82C40; Secondary: 35Q20, 70L10, 82D05.

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Figure 1. Reflection of particle $ i $ at time $ \tau $

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Figure 2. Evolution of $ \bar{\omega}_i $ when particle $ i $ has a reflection at time $ \tau $

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Figure 3. Pseudotrajectory associated with collision parameters $ ((1, -)_3, (2, +)_4, (1, +)_5, (4, -)_6) $

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Figure 4. A shift of particle $ j $ in $ \mathcal{P}^{i, j, i'}_1 $

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Figure 5. Virtual particle

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