Structure of the stationary state of the asymmetric target process

TL;DR

This paper introduces the target process, a dual to the zero-range process, and studies its stationary states, revealing dimension-dependent behaviors including homogeneous states and phase transitions with condensate formation.

Contribution

It presents the first analysis of the asymmetric target process's stationary states, highlighting its duality with ZRP and the impact of asymmetry and dimensionality on phase transitions.

Findings

In 1D, the system remains homogeneous at all densities.

In higher dimensions, a phase transition leads to condensate formation.

The critical density for condensation is independent of dimension.

Abstract

We introduce a novel migration process, the target process. This process is dual to the zero-range process (ZRP) in the sense that, while for the ZRP the rate of transfer of a particle only depends on the occupation of the departure site, it only depends on the occupation of the arrival site for the target process. More precisely, duality associates to a given ZRP a unique target process, and vice-versa. If the dynamics is symmetric, i.e., in the absence of a bias, both processes have the same stationary-state product measure. In this work we focus our interest on the situation where the latter measure exhibits a continuous condensation transition at some finite critical density $\rho_c$, irrespective of the dimensionality. The novelty comes from the case of asymmetric dynamics, where the target process has a nontrivial fluctuating stationary state, whose characteristics depend on the dimensionality. In one dimension, the system remains homogeneous at any finite density. An alternating scenario however prevails in the high-density regime: typical configurations consist of long alternating sequences of highly occupied and less occupied sites. The local density of the latter is equal to $\rho_c$ and their occupation distribution is critical. In dimension two and above, the asymmetric target process exhibits a phase transition at a threshold density $\rho_0$ much larger than $\rho_c$. The system is homogeneous at any density below $\rho_0$, whereas for higher densities it exhibits an extended condensate elongated along the direction of the mean current, on top of a critical background with density $\rho_c$.