Dynamic fluctuations of current and mass in nonequilibrium mass transport processes

TL;DR

This paper provides an exact analysis of steady-state fluctuations and power spectra of current and mass in nonequilibrium conserved-mass transport processes on a ring, revealing universal scaling functions and microscopic relations.

Contribution

It introduces exact calculations of current and mass fluctuations, power spectra, and universal scaling functions in nonequilibrium mass transport models, extending understanding beyond equilibrium distributions.

Findings

Bond-current fluctuation grows linearly, then subdiffusively, then linearly again with time.

Scaled subsystem current fluctuation converges to particle mobility in the large limit.

Universal scaling functions describe power spectra and fluctuations, with Green-Kubo and Einstein relations derived.

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 $\chi$ when the large subsystem size limit is taken first, followed by the large time limit. Remarkably, the scaled current fluctuation $D \langle \mathcal{Q}_i^2(T)\rangle/2 \chi L \equiv {\cal W}(y)$ 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.