Fluctuation-optimization theorem

TL;DR

This paper introduces a fluctuation theorem linking work to its optimal average, enabling numerical calculation of optimal work from work distributions without requiring the explicit optimal protocol, validated through a Brownian motion example.

Contribution

It presents a new fluctuation-optimization theorem expressed as an Euler-Lagrange equation, allowing the calculation of optimal work using only work distributions.

Findings

The theorem relates work fluctuations to optimal average work.

The relation depends on switching time, driving strength, and protocol.

Numerical methods can compute optimal work without knowing the protocol.

Abstract

A fluctuation theorem relating the work to its optimal average work is presented. The function mediating the relation is increasing and convex, and depends on the switching time $\tau$, driving strength $\delta\lambda/\lambda_0$, and protocol $g(t)$. The result is corroborated by an example of an overdamped white noise Brownian motion subjected to a moving laser harmonic trap. Observing also that the fluctuation-optimization theorem is an Euler-Lagrange equation, I conclude that the function minimizing $\langle h(-\beta W)\rangle$ obeys the relation proposed. The optimal work can now be calculated with numerical methods without knowing the optimal protocol, using only a work distribution of an arbitrary protocol.