Journal of Statistical Physics

, Volume 168, Issue 4, pp 772–793 | Cite as

A Rigourous Demonstration of the Validity of Boltzmann’s Scenario for the Spatial Homogenization of a Freely Expanding Gas and the Equilibration of the Kac Ring

  • S. De BièvreEmail author
  • P. E. Parris


Boltzmann provided a scenario to explain why individual macroscopic systems composed of a large number N of microscopic constituents are inevitably (i.e., with overwhelming probability) observed to approach a unique macroscopic state of thermodynamic equilibrium, and why after having done so, they are then observed to remain in that state, apparently forever. We provide here rigourous new results that mathematically prove the basic features of Boltzmann’s scenario for two classical models: a simple boundary-free model for the spatial homogenization of a non-interacting gas of point particles, and the well-known Kac ring model. Our results, based on concentration inequalities that go back to Hoeffding, and which focus on the typical behavior of individual macroscopic systems, improve upon previous results by providing estimates, exponential in N, of probabilities and time scales involved.


Approach to thermodynamic equilibrium Expanding gas Kac ring model 



S.D.B. is supported by the Labex CEMPI (ANR-11-LABX-0007-01) and by the Nord-Pas de Calais Regional Council and the Fonds Européen de Développement Économique Régional (Grant CPER Photonics for Society). P.E.P. thanks the University of Lille and the Labex CEMPI, where part of this work was performed, for their hospitality. The authors thank H. Spohn for bringing the work of J. Beck to their attention, and the latter for communicating his recent unpublished work to them.


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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Université Lille, CNRS, UMR 8524 - Laboratoire Paul PainlevéLilleFrance
  2. 2.Equipe-Projet MEPHYSTO, Centre de Recherche INRIA Futurs, Parc Scientifique de la Haute BorneVilleneuve d’Ascq CedexFrance
  3. 3.Department of PhysicsMissouri University of Science & TechnologyRollaUSA

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