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.

Figure 1

The diffusion coefficient (top) and the transport efficiency (bottom) as a function of the bias force (main panel) and the parameter $\theta$ for $F=4$ (inset). Black dots show the respective quantity for the displacement, while the orange squares correspond to the fluctuations of the displacement around its local mean value. The data were obtained using Langevin simulations in a sine potential $U\left(x\right)=U_{0}sin(2\pi x/L)$ with $U_0=1, L=1$. The temperature and mobility were set to $T=0.2$ and $\mu =1$.