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

TL;DR

This paper rigorously proves Boltzmann's scenario for the approach to equilibrium in a non-interacting gas and the Kac ring model, using concentration inequalities to estimate probabilities and timescales.

Contribution

It provides new mathematical results confirming Boltzmann's scenario for two classical models, with exponential probability estimates and timescale bounds.

Findings

Proves the approach to equilibrium for a non-interacting gas model.

Establishes the validity of Boltzmann's scenario for the Kac ring model.

Provides exponential estimates for probabilities and timescales involved.

Abstract

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.