Large deviations and the Boltzmann entropy formula

TL;DR

This paper explores the connection between large deviations theory and Boltzmann's entropy formula, highlighting historical insights and its role in understanding non-equilibrium statistical mechanics.

Contribution

It provides a historical and conceptual analysis linking large deviations to Boltzmann's entropy, emphasizing Einstein's interpretation and its influence on modern non-equilibrium theory.

Findings

Einstein interpreted Boltzmann's entropy as a large deviation formula.

Historical reconstruction shows the deep connection between large deviations and thermodynamic entropy.

The interpretation has influenced recent developments in non-equilibrium statistical mechanics.

Abstract

In the last decades the theory of large deviations has become a main tool in statistical mechanics especially in the study of non--equilibrium. In a rational reconstruction of the story one must recognize the ideal connection and debt of some recent work, to discussions taking place at the beginning of the twentieth century. The famous equation $S=k\ln W$ usually attributed to Boltzmann, actually written in this final form by Planck on his route to the quantum hypothesis, was interpreted by Einstein as a large deviation formula. This interpretation, on which he based his theory of thermodynamic equilibrium fluctuations, has been a source of inspiration in recent developments of non--equilibrium statistical mechanics. In this paper we briefly illustrate this aspect.