The partially asymmetric zero range process with quenched disorder

TL;DR

This paper studies a one-dimensional zero range process with quenched disorder, revealing condensation phenomena, exact dynamical exponents, and detailed steady-state and coarsening behaviors.

Contribution

It introduces an exact analysis of the steady state and dynamics of the disordered zero range process, including the calculation of the dynamical exponent and coarsening laws.

Findings

Current vanishes as J ~ L^{-z} with exactly calculated z.

Transport mechanisms differ for z<1 and z>1, involving active particles and anomalous diffusion.

Condensate growth follows n_L ~ t^{1/(1+z)} for large times.

Abstract

We consider the one-dimensional partially asymmetric zero range process where the hopping rates as well as the easy direction of hopping are random variables. For this type of disorder there is a condensation phenomena in the thermodynamic limit: the particles typically occupy one single site and the fraction of particles outside the condensate is vanishing. We use extreme value statistics and an asymptotically exact strong disorder renormalization group method to explore the properties of the steady state. In a finite system of $L$ sites the current vanishes as $J \sim L^{-z}$, where the dynamical exponent, $z$, is exactly calculated. For $0<z<1$ the transport is realized by $N_a \sim L^{1-z}$ active particles, which move with a constant velocity, whereas for $z>1$ the transport is due to the anomalous diffusion of a single Brownian particle. Inactive particles are localized at a second special site and their number in rare realizations is macroscopic. The average density profile of inactive particles has a width of, $\xi \sim \delta^{-2}$, in terms of the asymmetry parameter, $\delta$. In addition to this, we have investigated the approach to the steady state of the system through a coarsening process and found that the size of the condensate grows as $n_L \sim t^{1/(1+z)}$ for large times. For the unbiased model $z$ is formally infinite and the coarsening is logarithmically slow.