Why does entropy drive evolution equations?

TL;DR

This paper unifies various evolution equations driven by entropy by showing they all stem from the principle that entropy characterizes the invariant measure of an underlying stochastic process, explaining their common structure.

Contribution

It introduces a unified framework linking entropy to invariant measures in stochastic processes, clarifying its role in diverse evolution equations.

Findings

Entropy characterizes invariant measures in stochastic processes.

Different forms of entropy in evolution equations are explained by a common principle.

Examples include stochastic processes, gradient flows, and GENERIC systems.

Abstract

`Entropy' appears as driving force in many different evolution equations, both deterministic and stochastic, and in these equations this `entropy' also takes different forms. We show how all these examples can be understood as different instances of a common principle: Entropy drives evolutions because it characterizes the invariant measure of an underlying stochastic process. This interpretation explains the appearance of entropy, the different forms that entropy takes in these equations, and how entropy `drives' these evolution equations. We illustrate this common structure with examples from stochastic processes, gradient flows, and GENERIC systems.