Optimal protocols and optimal transport in stochastic thermodynamics

TL;DR

This paper demonstrates that optimal control problems in small system thermodynamics can be solved using optimal transport theory, linking thermodynamic optimization to well-established mathematical methods and equations like Burgers' equation.

Contribution

It establishes a connection between thermodynamic control optimization and optimal transport, extending solvable problems in small system thermodynamics with efficient numerical methods.

Findings

Optimal control problems are solved by deterministic optimal transport.

Minimizing heat or work relates to Burgers' equation in cosmology.

Extends the range of solvable thermodynamic optimization problems.

Abstract

Thermodynamics of small systems has become an important field of statistical physics. They are driven out of equilibrium by a control, and the question is naturally posed how such a control can be optimized. We show that optimization problems in small system thermodynamics are solved by (deterministic) optimal transport, for which very efficient numerical methods have been developed, and of which there are applications in Cosmology, fluid mechanics, logistics, and many other fields. We show, in particular, that minimizing expected heat released or work done during a non-equilibrium transition in finite time is solved by Burgers equation of Cosmology and mass transport by the Burgers velocity field. Our contribution hence considerably extends the range of solvable optimization problems in small system thermodynamics.