Geometrical aspects of entropy production in stochastic thermodynamics based on Wasserstein distance

TL;DR

This paper links optimal transport geometry with stochastic thermodynamics, revealing that entropy production bounds relate to Wasserstein distance, leading to new thermodynamic speed limits and optimal protocols.

Contribution

It introduces a geometric interpretation of entropy production using Wasserstein distance, deriving bounds and trade-offs in stochastic thermodynamics.

Findings

Entropy production lower bound equals the Wasserstein path length.

Derived a thermodynamic speed limit relating transition time and entropy production.

Provided a geometric framework for optimal protocols minimizing entropy production.

Abstract

We study a relationship between optimal transport theory and stochastic thermodynamics for the Fokker-Planck equation. We show that the lower bound on the entropy production is the action measured by the path length of the $L^2$-Wasserstein distance. Because the $L^2$-Wasserstein distance is a geometric measure of optimal transport theory, our result implies a geometric interpretation of the entropy production. Based on this interpretation, we obtain a thermodynamic trade-off relation between transition time and the entropy production. This thermodynamic trade-off relation is regarded as a thermodynamic speed limit which gives a tighter bound of the entropy production. We also discuss stochastic thermodynamics for the subsystem and derive a lower bound on the partial entropy production as a generalization of the second law of information thermodynamics. Our formalism also provides a geometric picture of the optimal protocol to minimize the entropy production. We illustrate these results by the optimal stochastic heat engine and show a geometrical bound of the efficiency.