Symmetries of cyclic work distributions for an isolated harmonic oscillator

TL;DR

This paper derives a symmetry relation for the work distribution in a driven isolated harmonic oscillator, valid in both classical and quantum regimes, revealing differences from the Crooks relation and highlighting quantum fluctuations.

Contribution

It introduces a new symmetry relation for cyclic work distributions in isolated harmonic oscillators, extending understanding beyond existing nonequilibrium thermodynamics relations.

Findings

The work distribution satisfies $P(W)= ext{exp}(eta W)P(-W)$ for cyclic protocols.

The symmetry holds in both classical and quantum dynamics.

Quantum uncertainty increases work fluctuations compared to classical case.

Abstract

We have calculated the distribution of work $W$ done on a 1-d harmonic oscillator that is initially in canonical equilibrium at temperature $T$, then thermally isolated and driven by an arbitrary time-dependent cyclic spring constant $\kappa(t)$, and demonstrated that it satisfies $P(W)=\exp(\beta W)P(-W)$, where $\beta=1/k_{B}T$, in both classical and quantum dynamics. This differs from the celebrated Crooks relation of nonequilibrium thermodynamics, since the latter relates distributions for forward and backward protocols of driving. We show that it is a special case of a symmetry that holds for non-cyclic work processes on the isolated oscillator, and that consideration of time reversal invariance shows it to be consistent with the Crooks relation. We have verified that the symmetry holds in both classical and quantum treatments of the dynamics, but that inherent uncertainty in the latter case leads to greater fluctuations in work performed for a given process.