Degenerate optimal paths in thermally isolated systems

TL;DR

This paper identifies finite-time switching protocols in thermally isolated systems that achieve work equal to the quasistatic limit, revealing a connection between optimal work paths and adiabatic invariants.

Contribution

It introduces a class of optimal switching protocols within linear response that minimize work in isolated systems, linking work optimization to adiabatic invariants.

Findings

Existence of finite-time protocols with work equal to quasistatic value.

Optimal protocols consist of a linear part plus a time-reversal odd function.

Analytical solution for harmonic oscillator and numerical results for anharmonic systems.

Abstract

We present an analysis of the work performed on a system of interest that is kept thermally isolated during the switching of a control parameter. We show that there exists, for a certain class of systems, a finite-time family of switching protocols for which the work is equal to the quasistatic value. These optimal paths are obtained within linear response for systems initially prepared in a canonical distribution. According to our approach, such protocols are composed of a linear part plus a function which is odd with respect to time reversal. For systems with one degree of freedom, we claim that these optimal paths may also lead to the conservation of the corresponding adiabatic invariant. This points to an interesting connection between work and the conservation of the volume enclosed by the energy shell. To illustrate our findings, we solve analytically the harmonic oscillator and present numerical results for certain anharmonic examples.