Universal bounds on current fluctuations

For current fluctuations in nonequilibrium steady states of Markovian processes, we derive four different universal bounds valid beyond the Gaussian regime. Different variants of these bounds apply to either the entropy change or any individual current, e.g., the rate of substrate consumption in a chemical reaction or the electron current in an electronic device. The bounds vary with respect to their degree of universality and tightness. A universal parabolic bound on the generating function of an arbitrary current depends solely on the average entropy production. A second, stronger bound requires knowledge both of the thermodynamic forces that drive the system and of the topology of the network of states. These two bounds are conjectures based on extensive numerics. An exponential bound that depends only on the average entropy production and the average number of transitions per time is rigorously proved. This bound has no obvious relation to the parabolic bound but it is typically tighter further away from equilibrium. An asymptotic bound that depends on the specific transition rates and becomes tight for large fluctuations is also derived. This bound allows for the prediction of the asymptotic growth of the generating function. Even though our results are restricted to networks with a finite number of states, we show that the parabolic bound is also valid for three paradigmatic examples of driven diffusive systems for which the generating function can be calculated using the additivity principle. Our bounds provide a general class of constraints for nonequilibrium systems.

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Figure 1

(a) Unicyclic network with affinity $\mathcal{A}$ and five states. (b) Multicyclic network with two fundamental cycles, one with three states and affinity ${\mathcal{A}}_1$ and the other with four states and affinity ${\mathcal{A}}_2$. The red dashed lines indicate a cycle with affinity ${\mathcal{A}}_1+{\mathcal{A}}_2$ and five states. (c) Multicyclic network with three fundamental cycles with three states each. The affinities of these cycles are ${\mathcal{A}}_1,{\mathcal{A}}_2$, and ${\mathcal{A}}_3$. The red dashed lines indicate a cycle with affinity ${\mathcal{A}}_1+{\mathcal{A}}_2$ and four states.

Figure 2

Illustration of our two main results for the generating function of an individual cycle current (thick black curve). The parabolic bound that depends on the entropy production is shown as a green dotted line and the exponential bound that depends on the entropy production and the activity is shown as a blue dashed line. The generating function refers to the cycle $\alpha =1$ of the “house-shaped” network shown in Fig. 1. The affinities are ${\mathcal{A}}_1=8$ and ${\mathcal{A}}_2=6$, all transition rates were set to 1 except for $k_{15}=exp({\mathcal{A}}_1/2),k_{51}=exp(-{\mathcal{A}}_1/2),k_{41}=exp({\mathcal{A}}_2/2)$, and $k_{14}=exp(-{\mathcal{A}}_2/2)$. The quantities relevant for the bounds are the average current $J_{\alpha }\simeq 0.36$, the entropy production $\sigma \simeq 3.6$, and the activity $R\simeq 2.13$.

Figure 3

The generating function $\lambda \left(z\right)$ and the rate function $h\left(\xi \right)$ of the asymmetric random walk for selected affinities $\mathcal{A}$ ($2,5,10,50$) and $N=1$. Black arrows indicate the direction of increasing $\mathcal{A}$. The parabolic bound for the generating function (40) and for the rate function (41) are shown as black dashed curves.

Figure 4

Generating functions ${\lambda }_{\alpha }\left(z\right)$ for an individual current in fully connected networks with random transition rates. The black curves in the upper and lower panels correspond to networks with $N=4$ and $N=6$ vertices, respectively. The parabolic bound (43) is shown as a dashed curve.

Figure 5

Generating function $\lambda \left(z\right)$ scaled by the steady state current $J$ in unicyclic networks with $N=10$ states and fixed affinity $\mathcal{A}=100$. The gray-scale (color-scale) encodes the standard deviation (SD) used for sampling the transition rates (see Appendix pp2). Dark gray (blue) corresponds to a nearly uniform distribution of transition rates and light gray (yellow) to a broad distribution of transition rates. The lower and upper bounds (50) and (51), respectively, are shown as (red) dashed lines.

Figure 6

Generating function $\lambda \left(z\right)$ scaled by the steady state current $J$ in unicyclic networks with $N=10$ for three families of distinct affinities $\mathcal{A}$. For each family, transition rates were sampled according to the procedure described in Appendix pp2. The black curves refer to the lower and upper bound from Eqs. (50) and (51).

Figure 7

Generating functions for entropy change (a)–(c) and randomly selected individual currents $X_{\alpha }$ (d) for the network of Fig. 1. The affinities are $\mathcal{A}=(1,2,3)$ in (a), $\mathcal{A}=(11,12,13)$ in (b) and (d), and $\mathcal{A}=(-11,12,13)$ in (c). The black curves correspond to the parabolic bound and the red dashed curves correspond to the hyperbolic cosine bound. The generating functions were generated as explained in Appendix pp2. In panel (a) ${\mathcal{A}}^{*}/n^{*}=1/3$, in panels (b) and (d) ${\mathcal{A}}^{*}/n^{*}=11/3$, and in panel (c) ${\mathcal{A}}^{*}/n^{*}=1/4$. For small values of ${\mathcal{A}}^{*}/n^{*}$, as in panels (a) and (c), the parabolic and hyperbolic cosine bound are closer to each other.

Figure 8

Generating function (left) and rate function (right) for a five-state unicyclic network with rates $ln{k}_i^{+}=(3,3,2,3,2)$ and $ln{k}_i^{-}=(-1,0,-1,0,0)$, affinity $\mathcal{A}=15$, current $J\simeq 2.25$, and activity $R\simeq 12.9$. The functions are shown as solid lines and the exponential bound (65) as dashed lines. Analytic continuations of the piecewise defined functions are shown as dotted curves.

Figure 9

Asymptotic bound for the house-shaped network with two fundamental cycles shown in Fig. 1. The color code represents the ratio (77) between the generating function and the bound. Black dashed lines indicate the borders between sectors with constant relevant cycles $\widehat{C}\left(\mathit{z}\right)$. For each sector, the relevant cycle $\widehat{\mathcal{C}}\left(\mathit{z}\right)$ is shown in white. The affinity of the three-cycle is ${\mathcal{A}}_1=8$ and the affinity of the four-cycle is ${\mathcal{A}}_2=6$.

Figure 10

Schematic illustrations of the WASEP (a) and the KMP (b) models. For the WASEP, in the bulk the particles jump with rates $p\equiv 1/2+\nu /\left(2L\right)$ to the right and $q\equiv 1/2-\nu /\left(2L\right)$ to the left, where the SSEP corresponds to $\nu =0$. At the boundaries particles are exchanged with the reservoirs. The model also has the exclusion principle, i.e., the maximum number of particles in a site is one. For the KMP model energy flows from a hot reservoir at temperature $T_L$ to a cold reservoir at temperature $T_R$. In the bulk a randomly chosen pair of sites exchange energy, which is a continuous variable, in such a way that the total energy is conserved. At the boundaries energy is exchanged with the reservoirs. The precise rules of these models can be found in [29] for the WASEP and [28] for the KMP model.

Figure 11

Comparison of the parabolic bound (42) with generating functions for driven diffusive systems. For the SSEP the densities of the left and right reservoirs were chosen as ${\varrho }_{\mathrm{L}}=0.99$ and ${\varrho }_{\mathrm{R}}=0.01$. For the WASEP the parameters are $\nu =10,{\varrho }_{\mathrm{L}}=4/7$, and ${\varrho }_{\mathrm{R}}=5/18$, as in Ref. [29]. For the KMP model the parameters are $T_{\mathrm{L}}=2$ and $T_{\mathrm{R}}=1$, as in Ref. [28].

Figure 12

Summary of the four bounds for a unicyclic network with four states. Transition rates are $ln{k}_i^{+}=(3,4,5,4)$ and $ln{k}_i^{+}=(0,-1,1,1)$, leading to the affinity $\mathcal{A}=15$ and the current $J\simeq 10.307$.

Figure 13

The polynomial $P\left(x\right)$ for a generic unicyclic network with $N=6$ states. The tangent at $x=0$ and the values $y=y_1$ and $x=x_1$ are shown as dashed lines.