Continuous time reversal and equality in the thermodynamic uncertainty relation

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.

Automated summary

In this work, a continuous time-reversal operation is introduced to connect the time-forward and time-reversed trajectories in the steady state of an irreversible Markovian dynamics through a continuous family of stochastic dynamics. This enables a tighter version of the thermodynamic uncertainty relation (TUR) involving observables evaluated relative to their local mean value. Furthermore, a family of dynamics realizing the continuous time reversal contains an equilibrium dynamics midway between the time-forward and time-reversed dynamics. By leveraging this equilibrium dynamics and an appropriate choice of observable, the inequality in the TUR can be turned into an equality, as demonstrated in the example of a particle diffusing in a tilted periodic potential.