De Newton \`a Boltzmann et Einstein: validation des mod\`eles cin\'etiques et de diffusion

TL;DR

This paper reviews the historical and recent mathematical validation of kinetic and diffusion models, especially the Boltzmann equation, from Newtonian mechanics, highlighting key proofs and their implications for statistical physics.

Contribution

It summarizes recent rigorous proofs connecting Newtonian particle systems to the Boltzmann and diffusion equations, extending Lanford's work and validating the linear Boltzmann and Brownian motion limits.

Findings

Lanford's proof extended to short-range potentials

Validation of the linear Boltzmann equation over large times

Derivation of Brownian motion as a limit of particle dynamics

Abstract

The kinetic theory of Maxwell and Boltzmann has been the subject of major scientific controversies. The alleged incompatibility between the reversible nature of the equations of classical mechanics and the increase of entropy, which, in the kinetic theory of gases, is a mathematical property of the Boltzmann equation known as the H Theorem, was one of the most basic arguments against the validity of kinetic theory. About a century later, in 1974, O. Lanford proposed a strategy to prove that the Boltzmann equation describes a particular asymptotic limit of Newton's equations of classical mechanics for a system made of a large number N of spherical particles interacting by elastic collisions. A recent work by I. Gallagher, L. Saint-Raymond and B. Texier completes Lanford's proof and extends it to the case where the interaction between particles is given by a short range potential. In a later article, T. Bodineau, I. Gallagher and L. Saint-Raymond study the dynamics of a tagged particle among N other identical particles in the same asymptotic regime, and prove the validity of the linear Boltzmann equation on arbitrary large periods of time as N tends to infinity. Using classical results on the asymptotic theory of the linear Boltzmann equation, the same authors prove that the stochastic process known as the "Brownian motion" describes a particular limit of the deterministic dynamics of interacting particles.