Speed Limit for Classical Stochastic Processes

We consider the speed limit for classical stochastic Markov processes with and without the local detailed balance condition. We find that, for both cases, a trade-off inequality exists between the speed of the state transformation and the entropy production. The dynamical activity is related to a time scale and plays a crucial role in the inequality. For the dynamics without the local detailed balance condition, we use the Hatano-Sasa entropy production instead of the standard entropy production. Our inequalities consist of the quantities that are commonly used in stochastic thermodynamics and explicitly show underlying physical mechanisms.

Figure 1

(a) Demonstration of Eq. (14) for a single bath. In the bottom plot, the purple dotted line is the linear reference line $y=\tau$, and the green solid curve is $y={\tau }_{\mathrm{I}}$ as a function of $\tau$. These two lines almost agree with each other from the aspect of the smallness of the relative error $(\tau -{\tau }_{\mathrm{I}})/\tau$, which clearly shows the tightness of the bound (14). (b) Demonstration of Eq. (18) for two baths with $\alpha =2/3$ (which means that a finite stationary current exists). The bottom plot shows that Eq. (14) (${\tau }_{\mathrm{I}}\le \tau$: the green solid line) is a poor bound, while Eq. (18) (${\tau }_{\mathrm{II}}\le \tau$: the blue dashed-dotted line) still provides a meaningful bound.