Mapping current fluctuations of stochastic pumps to nonequilibrium steady states

We show that current fluctuations in a stochastic pump can be robustly mapped to fluctuations in a corresponding time-independent nonequilibrium steady state. We thus refine a recently proposed mapping so that it ensures equivalence of not only the averages, but also optimal representation of fluctuations in currents and density. Our mapping leads to a natural decomposition of the entropy production in stochastic pumps similar to the “housekeeping” heat. As a consequence of the decomposition of entropy production, the current fluctuations in weakly perturbed stochastic pumps are shown to satisfy a universal bound determined by the steady state entropy production.

Figure 1

(a) A schematic of the stochastic pump under consideration. Symmetric barriers $B_{ij}$ and energy levels $E_i$ parametrize Arrhenius rates and are varied periodically in time to generate a current. The corresponding nonequilibrium steady state representation of the pump has no time dependence, but rather rates that break detailed balance. (b) The time-periodic steady state probabilities for each site on the graph are shown over an entire period $\tau$. The solid lines show the time-dependent occupation of the pump. The dashed lines show the average occupancy per period, a property matched by the corresponding steady state.

Figure 2

(a) The large deviation rate function for the current around the upper cycle (see Fig. 1) $1\to 2\to 3\to 1$ is shown for the stochastic pump (blue, solid gray), the nonequilibrium steady state with the same average entropy production along each edge as the pump (light green, dot-dashed), and the nonequilibrium steady state with the same average flow along each edge as the pump (red dots). While the nonequilibrium steady state with ${\mathcal{S}}_{\langle \sigma \rangle }$ has the same average current, the character and extent of its fluctuations are extremely different. Choosing ${\mathcal{S}}_{\langle q\rangle }$ preserves even very rare fluctuations in current. (b) The large deviation rate functions for entropy production reveal that the steady state that recapitulates the current fluctuations has a smaller average entropy production. Furthermore, the extent of entropy production fluctuations in the corresponding steady state is much less pronounced (red innermost curve). Thus pump rate function for entropy production is shown in blue (solid) and enhanced fluctuations relative to ${\mathcal{S}}_{\langle \sigma \rangle }$, which is shown in light green (dot-dashed).