Einstein relation and hydrodynamics of nonequilibrium mass transport processes

We derive hydrodynamics of paradigmatic conserved-mass transport processes on a ring. The systems, governed by chipping, diffusion, and coalescence of masses, eventually reach a nonequilibrium steady state, having nontrivial correlations, with steady-state measures in most cases not known. In these processes, we analytically calculate two transport coefficients, bulk-diffusion coefficient and conductivity. Remarkably, the two transport coefficients obey an equilibrium-like Einstein relation even when the microscopic dynamics violates detailed balance and systems are far from equilibrium. Moreover, we show, using a macroscopic fluctuation theory, that the probability of large deviation in density, obtained from the above hydrodynamics, is in complete agreement with the same derived earlier by Das et al. [Phys. Rev. E 93, 062135 (2016)] using an additivity property.

Figure 1

Weakly asymmetric mass transfers: Model I (random sequential update), steady-state probability distribution $P_v\left(m\right)$ is plotted as a function of subsystem mass $m$ for $\lambda =0,0.25$ and 0.5 and subsystem volume $v=10$.

Figure 2

Strongly asymmetric mass transfers: Model I (random sequential update), steady-state probability distribution $P_v\left(m\right)$ is plotted as a function of subsystem mass $m$ for $\lambda =0,0.25$, and 0.5 and subsystem volume $v=10$.