Continuous time-reversal and equality in the thermodynamic uncertainty\n relation

Abstract

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