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 $⟨{\mathcal{Q}}_i^2(T)⟩$ and $⟨{\mathcal{Q}}_{sub}^2(l,T)⟩$ 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 $T\sim \mathcal{O}(1)$, subdiffusively for $T\ll L^2$ and then again linearly for $T\gg L^2$. The scaled subsystem current fluctuation ${\lim }_{l\to \infty ,T\to \infty }⟨{\mathcal{Q}}_{sub}^2(l,T)⟩/2lT$ converges to the density-dependent particle mobility $\chi$ when the large subsystem size limit is taken first, followed by the large time limit. Remarkably, the scaled current fluctuation $D⟨{\mathcal{Q}}_i^2(T)⟩/2\chi L\equiv \mathcal{W}(y)$ as a function of scaled time $y=DT/L^2$ is expressed in terms of a universal scaling function $\mathcal{W}(y)$, 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.

24 pages, 13 figures

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