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.
Highlights
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
In this Letter, we show that, the thermodynamic uncertainty relation (TUR) is the consequence of the symmetry under a different type of time-reversal operation
The continuous time-reversal operation can be thought of as adiabatically changing the direction of the irreversible flows in the system: First, we reduce the magnitude of the flows while keeping the steady state fixed; at θ = 0, the flows vanish and the system is in equilibrium
Summary
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. The TUR, which applies to steady states of irreversible Markovian dynamics, is an inequality between the average and fluctuations of an observable time-integrated current Jτ [see Eq (13)] and the entropy production Sτirr, ( Jτ )2 Var(Jτ ) The continuous time-reversal operation can be thought of as adiabatically changing the direction of the irreversible flows in the system: First, we reduce the magnitude of the flows while keeping the steady state fixed; at θ = 0, the flows vanish and the system is in equilibrium.
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