Dissipation Bounds All Steady-State Current Fluctuations

Near equilibrium, small current fluctuations are described by a Gaussian distribution with a linear-response variance regulated by the dissipation. Here, we demonstrate that dissipation still plays a dominant role in structuring large fluctuations arbitrarily far from equilibrium. In particular, we prove a linear-response-like bound on the large deviation function for currents in Markov jump processes. We find that nonequilibrium current fluctuations are always more likely than what is expected from a linear-response analysis. As a small-fluctuations corollary, we derive a recently conjectured uncertainty bound on the variance of current fluctuations.

Figure 1

Current fluctuations illustration: for a four-state model (inset) in a nonequilibrium steady state, the integrated current $J_T$—the net number of hops between pairs of states—along each edge is plotted as a function of time. Each integrated current displays an average rate perturbed by stochastic fluctuations.

Figure 2

Generalized-current fluctuations in the four-state model: generalized currents $j_{\mathit{d}}$ were constructed randomly by choosing $d(y,z)\in [-1,1)$. All the $I(j_{\mathit{d}})$ (colored) and $I(\mathrm{\Sigma })$ (dashed black) fall in the blue-shaded region that satisfies the bound (4), with the differences $\mathrm{\Delta }=I(j_d)-I_{\text{WLR}}(j_d)$ plotted in the inset. Rates: $r(1,2)=3$, $r(1,3)=10$, $r(1,4)=9$, $r(2,1)=10$, $r(2,3)=1$, $r(2,4)=2$, $r(3,1)=6$, $r(3,2)=4$, $r(3,4)=1$, $r(4,1)=7$, $r(4,2)=9$, $r(4,3)=5$.

Figure 3

ASEP current fluctuations: the total current rate function $I(j_{\rho })$ (black dashed) is bounded by $I_{\mathrm{WLR}}(j_{\rho })$ (blue) for the $L=15$ ASEP. Inset: schematic of the $L=15$ ASEP with open boundaries. Particles (shaded circles) jump to neighboring sites with rates $p$, $q$ in the bulk, and with rates given by Greek letters in and out of the boundary reservoirs (shaded semicircles). Blocked transitions are marked by a red “$x$.” Results are shown using the parameters $\alpha =1.25$, $\beta =0.5$, $\gamma =0.5$, $\delta =1.5$, $p=1$, and $q=0.5$, corresponding to a high-density phase [33].