Dynamic fluctuations of current and mass in nonequilibrium mass transport processes

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, ⟨ Q i 2 ( T ) ⟩ c and ⟨ Q sub 2 ( l , T ) ⟩ c , of cumulative (time-integrated) currents up to time T across the 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 of the subsystem mass. In particular, we show that, for large L ≫ 1 , the second cumulant ⟨ Q i 2 ( T ) ⟩ c of the cumulative current up to time T across the i th bond grows linearly as ⟨ Q i 2 ⟩ c ∼ T for initial times T ∼ O ( 1 ) , subdiffusively as ⟨ Q i 2 ⟩ c ∼ T 1 / 2 for intermediate times 1 ≪ T ≪ L 2 , and then again linearly as ⟨ Q i 2 ⟩ c ∼ T for long times T ≫ L 2 . The scaled cumulant lim l → ∞ , T → ∞ ⟨ Q sub 2 ( l , T ) ⟩ c / 2 l T 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 ⟨ Q i 2 ( T ) ⟩ c / 2 χ L ≡ W ( y ) as a function of scaled time y = D T / L 2 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.