Stochastic Processes and Statistical Mechanics

TL;DR

This paper demonstrates that the formalism of thermodynamics can be universally applied to any stochastic process, providing a thermodynamic framework for understanding probability distributions and stochastic dynamics.

Contribution

It generalizes thermodynamic principles to stochastic processes, showing they follow the same variational and Legendre structure as in statistical physics.

Findings

Feasible distributions obey thermodynamic laws

The most probable distribution is the canonical distribution

Thermodynamic formalism acts as a universal language for stochastic processes

Abstract

Statistical thermodynamics delivers the probability distribution of the equilibrium state of matter through the constrained maximization of a special functional, entropy. Its elegance and enormous success have led to numerous attempts to decipher its language and make it available to problems outside physics, but a formal generalization has remained elusive. Here we show how the formalism of thermodynamics can be applied to any stochastic process. We view a stochastic process as a random walk on the event space of a random variable that produces a feasible distribution of states. The set of feasible distributions obeys thermodynamics: the most probable distribution is the canonical distribution, it maximizes the functionals of statistical mechanics, and its parameters satisfy the same Legendre relationships. Thus the formalism of thermodynamics -- no new functionals beyond those already encountered in statistical physics -- is shown to be a stochastic calculus, a universal language of probability distributions and stochastic processes.