Computing the optimal protocol for finite-time processes in stochastic thermodynamics

TL;DR

This paper develops methods to find optimal control protocols that minimize work in finite-time thermodynamic processes, providing analytical solutions for linear systems and numerical solutions for nonlinear systems, revealing multiple optimal protocols with jumps.

Contribution

It introduces analytical solutions for linear systems and numerical methods for nonlinear systems to determine multiple optimal protocols with jumps in finite-time stochastic thermodynamics.

Findings

Analytical solutions for linear systems' optimal protocols.

Numerical methods for nonlinear systems' optimal protocols.

Existence of multiple distinct optimal protocols with jumps.

Abstract

Asking for the optimal protocol of an external control parameter that minimizes the mean work required to drive a nano-scale system from one equilibrium state to another in finite time, Schmiedl and Seifert ({\it Phys. Rev. Lett.} {\bf 98}, 108301 (2007)) found the Euler-Lagrange equation to be a non-local integro-differential equation of correlation functions. For two linear examples, we show how this integro-differential equation can be solved analytically. For non-linear physical systems we show how the optimal protocol can be found numerically and demonstrate that there may exist several distinct optimal protocols simultaneously, and we present optimal protocols that have one, two, and three jumps, respectively.