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


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.



Keywords and Phrases

Approach to thermodynamic equilibrium; Expanding gas; Kac ring model

International Standard Serial Number (ISSN)

0022-4715; 1572-9613

Document Type

Article - Journal

Document Version


File Type





© 2018 Springer, All rights reserved.

Publication Date

01 Aug 2017