Optimal stochastic transport in inhomogeneous thermal environments

TL;DR

This paper investigates how to optimize entropy production in systems with spatially varying temperatures, revealing that the optimal process involves a finite duration and can be understood through a geometric framework involving curved manifolds.

Contribution

It introduces a novel geometric approach to optimize entropy production in inhomogeneous thermal environments, linking anomalous entropy contributions to a potential on a curved manifold.

Findings

Optimal entropy production is achieved by a finite-time process, not quasi-static.

The problem reduces to a deterministic motion on a curved manifold with a potential.

Explicit example with linearly space-dependent diffusion coefficient demonstrates the theory.

Abstract

We consider optimization of the average entropy production in inhomogeneous temperature environments within the framework of stochastic thermodynamics. For systems modeled by Langevin equations (e.g. a colloidal particle in a heat bath) it has been recently shown that a space dependent temperature breaks the time reversal symmetry of the fast velocity degrees of freedom resulting in an anomalous contribution to the entropy production of the overdamped dynamics. We show that optimization of entropy production is determined by an auxiliary deterministic problem describing motion on a curved manifold in a potential. The "anomalous contribution" to entropy plays the role of the potential and the inverse of the diffusion tensor is the metric. We also find that entropy production is not minimized by adiabatically slow, quasi-static protocols but there is a finite optimal duration for the transport process. As an example we discuss the case of a linearly space dependent diffusion coefficient.