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

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.

arXiv comment16 pages, 6 figures, to appear in Physical Review Research

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• Geometric thermodynamics for the Fokker-Planck equation: Stochastic thermodynamic links between information geometry and optimal transport
• Geometric decomposition of entropy production into excess, housekeeping and coupling parts
• A unified, geometric framework for nonequilibrium protocol optimization
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• Thermal brachistochrone for harmonically confined Brownian particles
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