Continuous time-reversal and equality in the thermodynamic uncertainty relation

TL;DR

This paper introduces a continuous time-reversal operation in Markovian dynamics, leading to a tighter thermodynamic uncertainty relation and conditions under which it becomes an equality, demonstrated with a particle in a tilted potential.

Contribution

It develops a continuous time-reversal framework connecting forward and reversed dynamics, deriving a refined TUR and identifying conditions for equality.

Findings

Derived a tighter TUR involving local mean values.

Identified an equilibrium dynamics that makes TUR an equality.

Validated results with a particle in a tilted periodic potential.

Abstract

We introduce a continuous time-reversal operation which connects the time-forward and time-reversed trajectories in the steady state of an irreversible Markovian dynamics via a continuous family of stochastic dynamics. This continuous time-reversal allows us to derive a tighter version of the thermodynamic uncertainty relation (TUR) involving observables evaluated relative to their local mean value. Moreover, the family of dynamics realizing the continuous time-reversal contains an equilibrium dynamics halfway between the time-forward and time-reversed dynamics. We show that this equilibrium dynamics, together with an appropriate choice of the observable, turns the inequality in the TUR into an equality. We demonstrate our findings for the example of a particle diffusing in a tilted periodic potential.