Linear diffusive limit of deterministic systems of hard spheres (Q2450276)
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scientific article; zbMATH DE number 6550064
- The Brownian motion as the limit of a deterministic system of hard-spheres
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| English | Linear diffusive limit of deterministic systems of hard spheres |
scientific article; zbMATH DE number 6550064 |
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Linear diffusive limit of deterministic systems of hard spheres (English)
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The Brownian motion as the limit of a deterministic system of hard-spheres (English)
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19 May 2014
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4 March 2016
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Hilbert's sixth problem
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Brownian motion
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hydrodynamic limit
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hard spheres
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Boltzmann's kinetic theory
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The purpose of this paper is to describe the macroscopic behaviour of a gas consisting of \(N\) interacting particles, with positions and velocities, by a linear Boltzmann equation for tagged particles, with initial data and hard-sphere cross-section, in a low density Lorentz gas, where the obstacles obey a Poisson distribution, and have no dynamics. The working method is to improve \textit{O. E. Lanford~III}'s results [Lect. Notes Phys. 38, 1--111 (1975; Zbl 0329.70011)]; locally the \(N\)-particle distribution should therefore look like a conditioned tensorized Maxwellian. The proof use the Liouville equation in phase space, Gibbs measure, Green's formula, Duhamel formula, Cauchy-Schwarz inequalities, Markov chains, and heat equation. Several extensions of the main theorems are considered.
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0.9547938704490662
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0.7852357029914856
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