Structure of the stationary state of the asymmetric target process

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 ${\rho }_c$.

arXiv comment30 pages, 16 figures

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• Particle Condensation in Pair Exclusion Process
• Zero-range process with finite compartments: Gentile's statistics and glassiness
• Condensation in randomly perturbed zero-range processes
• Condensation and Clustering in the Driven Pair Exclusion Process
• Condensation transition in a conserved generalized interacting zero-range process