Minimal Entropy Production in the Presence of Anisotropic Fluctuations

Abstract

Anisotropy in temperature, chemical potential, or ion concentration, provides the fuel that feeds dynamical processes that sustain life. At the same time, anisotropy is a root cause of incurred losses manifested as entropy production. In this work we study how to minimize such entropic losses using the framework of stochastic optimal control. Specifically, we consider a rudimentary model of an overdamped stochastic thermodynamic system that is in contact with heat baths of different temperatures simultaneously, and seek the entropy-minimizing control that steers the system between thermodynamic states in finite time. It is known that when the system is in contact with a single heat bath, the entropy production is due to dissipation and can be expressed in terms of a quadratic functional in the control effort—the square of the $W_{2}$ optimal mass transport (OMT) distance traversed by the thermodynamic state. Anisotropy on the other hand complicates substantially the mechanism of entropy production since, besides dissipation, seepage of energy between the heat sources by way of the system dynamics is often a major additional contributing factor. A key result of this article is to show that in the presence of anisotropy, minimization of entropy production can once again be expressed via a modified OMT problem. However, in contrast to the isotropic situation that leads to classical OMT and the so-called Wasserstein $W_{2}$ metric, entropy production may not vanish when the thermodynamic state remains unchanged; this is due to the fact that maintaining a nonequilibrium steady-state (NESS) incurs an intrinsic entropic cost that can be traced back to the seepage of heat between heat baths. NESSs represent hallmarks of life, since living matter by necessity operates far from equilibrium. Therefore, the question studied herein, to characterize minimal entropy production in anisotropic environments, appears of central importance in biological processes and on how such processes may have evolved to optimize for available usage of resources.