Zero-range process with open boundaries

TL;DR

This paper derives the exact stationary distribution for a one-dimensional zero-range process with open boundaries, revealing conditions for correlation-free states, condensation phenomena, and boundary effects.

Contribution

It provides the first exact solution for the stationary distribution of the zero-range process with open boundaries for arbitrary rates, including analysis of condensation and boundary-driven dynamics.

Findings

Stationary distribution is correlation-free and characterized by a space-dependent fugacity.

Strong boundary drive can prevent the existence of a stationary state.

Condensation occurs at boundaries or sites depending on system parameters.

Abstract

We calculate the exact stationary distribution of the one-dimensional zero-range process with open boundaries for arbitrary bulk and boundary hopping rates. When such a distribution exists, the steady state has no correlations between sites and is uniquely characterized by a space-dependent fugacity which is a function of the boundary rates and the hopping asymmetry. For strong boundary drive the system has no stationary distribution. In systems which on a ring geometry allow for a condensation transition, a condensate develops at one or both boundary sites. On all other sites the particle distribution approaches a product measure with the finite critical density \rho_c. In systems which do not support condensation on a ring, strong boundary drive leads to a condensate at the boundary. However, in this case the local particle density in the interior exhibits a complex algebraic growth in time. We calculate the bulk and boundary growth exponents as a function of the system parameters.