Dynamic fluctuations of current and mass in nonequilibrium mass transport processes

Abstract

We study steady-state dynamic fluctuations of current and mass in several variants of random average processes on a ring of L sites. These processes violate detailed balance in the bulk and have nontrivial spatial structures: their steady states are not described by the Boltzmann–Gibbs distribution and can have nonzero spatial correlations. Using a microscopic approach, we exactly calculate the second cumulants, or the variance, ⟨Qi2(T)⟩c and ⟨Qsub2(l,T)⟩c , of cumulative (time-integrated) currents up to time T across the ith bond and across a subsystem of size l (summed over bonds in the subsystem), respectively. We also calculate the (two-point) dynamic correlation function of the subsystem mass. In particular, we show that, for large L≫1 , the second cumulant ⟨Qi2(T)⟩c of the cumulative current up to time T across the ith bond grows linearly as ⟨Qi2⟩c∼T for initial times T∼O(1) , subdiffusively as ⟨Qi2⟩c∼T1/2 for intermediate times 1≪T≪L2 , and then again linearly as ⟨Qi2⟩c∼T for long times T≫L2 . The scaled cumulant liml→∞,T→∞⟨Qsub2(l,T)⟩c/2lT of current across the subsystem of size l and up to time T converges to the density-dependent particle mobility χ(ρ) when the large subsystem-size limit is taken first, followed by the large-time limit; when the limits are reversed, it simply vanishes. Remarkably, regardless of the dynamical rules, the scaled current cumulant D⟨Qi2(T)⟩c/2χL≡W(y) as a function of scaled time y=DT/L2 can be expressed in terms of a universal scaling function W(y) , where D is the bulk-diffusion coefficient; interestingly, the intermediate-time subdiffusive and long-time diffusive growths can be connected through a single scaling function W(y) . The power spectra for current and mass are also exactly characterized by the respective scaling functions. Furthermore, we provide a microscopic derivation of equilibrium-like Green–Kubo and Einstein relations that connect the steady-state current fluctuations to an ‘operational’ mobility (i.e. the response to an external force field) and mass fluctuation, respectively.