A generalization of the thermodynamic uncertainty relation to periodically driven systems

TL;DR

This paper extends the thermodynamic uncertainty relation to systems with periodic driving, broadening its applicability to time-dependent processes like heat engines and molecular machines, and demonstrates its validity with a two-level heat engine model.

Contribution

It introduces a generalized thermodynamic uncertainty relation for periodically driven systems and time-dependent observables, unifying steady-state and fast-driving limits.

Findings

Generalized relation applies to time-dependent current observables.

Recovers original relation in steady-state and fast-driving limits.

Validated with a two-level heat engine model.

Abstract

The thermodynamic uncertainty relation expresses a universal trade-off between precision and entropy production, which applies in its original formulation to current observables in steady-state systems. We generalize this relation to periodically time-dependent systems and, relatedly, to a larger class of inherently time-dependent current observables. In the context of heat engines or molecular machines, our generalization applies not only to the work performed by constant driving forces, but also to the work performed while changing energy levels. The entropic term entering the generalized uncertainty relation is the sum of local rates of entropy production, which are modified by a factor that refers to an effective time-independent probability distribution. The conventional form of the thermodynamic uncertainty relation is recovered for a time-independently driven steady state and, additionally, in the limit of fast driving. We illustrate our results for a simple model of a heat engine with two energy levels.