Dynamic fluctuations of current and mass in nonequilibrium mass transport processes

Abstract

We study steady-state dynamic fluctuations of current and mass, as well as the corresponding power spectra, in conserved-mass transport processes on a ring of $L$ sites; these processes violate detailed balance, have nontrivial spatial structures, and their steady states are not described by the Boltzmann-Gibbs distribution. We exactly calculate, for all times $T$, the fluctuations $\langle \mathcal{Q}_i^2(T) \rangle$ and $\langle \mathcal{Q}_{sub}^2(l, T) \rangle$ of the cumulative currents upto time $T$ across $i$th bond and across a subsystem of size $l$ (summed over bonds in the subsystem), respectively; we also calculate the (two-point) dynamic correlation function for subsystem mass. In particular, we show that, for large $L \gg 1$, the bond-current fluctuation grows linearly for $T \sim {\cal O}(1)$, subdiffusively for $T \ll L^2$ and then again linearly for $T \gg L^2$. The scaled subsystem current fluctuation $\lim_{l \rightarrow \infty, T \rightarrow \infty} \langle \mathcal{Q}^2_{sub}(l, T) \rangle/2lT$ converges to the density-dependent particle mobility $χ$ when the large subsystem size limit is taken first, followed by the large time limit. Remarkably, the scaled current fluctuation $D \langle \mathcal{Q}_i^2(T)\rangle/2 χL \equiv {\cal W}(y)$ as a function of scaled time $y=DT/L^2$ is expressed in terms of a universal scaling function ${\cal W}(y)$, where $D$ is the bulk-diffusion coefficient. Similarly, the power spectra for current and mass time series are characterized by the respective universal scaling functions, which are calculated exactly. We provide a microscopic derivation of equilibrium-like Green-Kubo and Einstein relations, that connect the steady-state current fluctuations to the response to an external force and to mass fluctuation, respectively.