Fluctuation-dissipation relation, Maxwell-Boltzmann statistics, equipartition theorem, and stochastic calculus

TL;DR

This paper investigates the fundamental relations of statistical mechanics using stochastic calculus, deriving the fluctuation-dissipation relation and analyzing the physical relevance of solutions in stochastic differential equations.

Contribution

It introduces two stochastic analytical approaches to connect classical statistical mechanics results and clarifies the physical significance of solutions in stochastic differential equations.

Findings

Backward stochastic differential equations elucidate classical relations.

Itô calculus yields the physically relevant unique solution.

Stratonovich calculus admits multiple solutions, not all physically meaningful.

Abstract

We derive the fluctuation-dissipation relation and explore its connection with the equipartition theorem and Maxwell-Boltzmann statistics through the use of different stochastic analytical techniques. Our first approach is the theory of backward stochastic differential equations, which arises naturally in this context, and facilitates the understanding of the interplay between these classical results of statistical mechanics. The second approach consists in deriving forward stochastic differential equations for the energy of an electric system according to both It\^o and Stratonovich stochastic calculus rules. While the It\^o equation possesses a unique solution, which is the physically relevant one, the Stratonovich equation admits this solution along with infinitely many more, none of which has a physical nature. Despite of this fact, some, but not all of them, obey the fluctuation-dissipation relation.