Thermodynamics Beyond Molecules: Statistical Thermodynamics of Distributions

TL;DR

This paper redefines thermodynamics as a variational calculus of probability distributions, extending its principles beyond molecules to any stochastic process, unifying thermodynamics, information theory, and Bayesian inference.

Contribution

It introduces a generalized, physics-independent framework for statistical thermodynamics based on variational calculus of distributions, connecting it with information theory and Bayesian methods.

Findings

Derives thermodynamic relationships without physical assumptions

Unifies statistical mechanics, information theory, and Bayesian inference

Provides a probabilistic foundation for thermodynamics

Abstract

Statistical thermodynamics has a universal appeal that extends beyond molecular systems, and yet, as its tools are being transplanted to fields outside physics, the fundamental question, \textit{what is thermodynamics?}, has remained unanswered. We answer this question here. Generalized statistical thermodynamics is a variational calculus of probability distributions. It is independent of physical hypotheses but provides the means to incorporate our knowledge, assumptions and physical models about a stochastic processes that gives rise to the probability in question. We derive the familiar calculus of thermodynamics via a probabilistic argument that makes no reference to physics. At the heart of the theory is a space of distributions and a special functional that assigns probabilities to this space. The maximization of this functional generates the entire mathematical network of thermodynamic relationships. We obtain statistical mechanics as a special case and make contact with Information Theory and Bayesian inference.