Optimal protocols for minimal work processes in underdamped stochastic thermodynamics

TL;DR

This paper investigates optimal control protocols for minimal work in underdamped stochastic thermodynamics, revealing that boundary jumps in control parameters are optimal, which can enhance free energy calculation methods.

Contribution

It extends the understanding of optimal protocols from overdamped to underdamped systems, showing boundary delta peaks are optimal for minimizing work.

Findings

Boundary jumps are optimal in underdamped systems.

Including inertia allows for delta peak boundary control.

Results can improve free energy estimation techniques.

Abstract

For systems in an externally controllable time-dependent potential, the optimal protocol minimizes the mean work spent in a finite-time transition between two given equilibrium states. For overdamped dynamics which ignores inertia effects, the optimal protocol has been found to involve jumps of the control parameter at the beginning and end of the process. Including the inertia term, we show that this feature not only persists but that even delta peak-like changes of the control parameter at both boundaries make the process optimal. These results are obtained by analyzing two simple paradigmatic cases: First, a Brownian particle dragged by a harmonic optical trap through a viscous fluid and, second, a Brownian particle subject to an optical trap with time-dependent stiffness. These insights could be used to improve free energy calculations via either thermodynamic integration or "fast growth" methods using Jarzynski's equality.