Inventiones mathematicae

, Volume 203, Issue 2, pp 493–553

The Brownian motion as the limit of a deterministic system of hard-spheres

  • Thierry Bodineau
  • Isabelle Gallagher
  • Laure Saint-Raymond

DOI: 10.1007/s00222-015-0593-9

Cite this article as:
Bodineau, T., Gallagher, I. & Saint-Raymond, L. Invent. math. (2016) 203: 493. doi:10.1007/s00222-015-0593-9


We provide a rigorous derivation of the Brownian motion as the limit of a deterministic system of hard-spheres as the number of particles \(N\) goes to infinity and their diameter \(\varepsilon \) simultaneously goes to \(0\), in the fast relaxation limit \(\alpha = N\varepsilon ^{d-1}\rightarrow \infty \) (with a suitable diffusive scaling of the observation time). As suggested by Hilbert in his sixth problem, we rely on a kinetic formulation as an intermediate level of description between the microscopic and the fluid descriptions: we use indeed the linear Boltzmann equation to describe one tagged particle in a gas close to global equilibrium. Our proof is based on the fundamental ideas of Lanford. The main novelty here is the detailed study of the branching process, leading to explicit estimates on pathological collision trees.

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Thierry Bodineau
    • 1
  • Isabelle Gallagher
    • 2
  • Laure Saint-Raymond
    • 3
  1. 1.CMAPEcole Polytechnique and CNRSPalaiseauFrance
  2. 2.IMJUniversité Paris DiderotParisFrance
  3. 3.DMAEcole Normale Supérieure and Université Pierre et Marie CurieParisFrance

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