Stochastic control and non-equilibrium thermodynamics: fundamental limits

TL;DR

This paper links stochastic thermodynamics and optimal control by quantifying the minimal work gap in finite-time transitions between equilibrium states using Wasserstein-2 distance, revealing fundamental thermodynamic limits.

Contribution

It introduces a control-theoretic framework to analyze thermodynamic transitions, connecting optimal transport, gradient flows, and entropy in a multivariable stochastic setting.

Findings

Minimal work gap relates to Wasserstein-2 distance squared divided by transition time.

Optimal control protocols correspond to gradient flows of entropy functionals.

Fundamental thermodynamic limits are derived from control and transport theory.

Abstract

We consider damped stochastic systems in a controlled (time-varying) quadratic potential and study their transition between specified Gibbs-equilibria states in finite time. By the second law of thermodynamics, the minimum amount of work needed to transition from one equilibrium state to another is the difference between the Helmholtz free energy of the two states and can only be achieved by a reversible (infinitely slow) process. The minimal gap between the work needed in a finite-time transition and the work during a reversible one, turns out to equal the square of the optimal mass transport (Wasserstein-2) distance between the two end-point distributions times the inverse of the duration needed for the transition. This result, in fact, relates non-equilibrium optimal control strategies (protocols) to gradient flows of entropy functionals via and the Jordan-Kinderlehrer-Otto scheme. The purpose of this paper is to introduce ideas and results from the emerging field of stochastic thermodynamics in the setting of classical regulator theory, and to draw connections and derive such fundamental relations from a control perspective in a multivariable setting.