Condensation in the Zero Range Process

Stochastic particle systems far from equilibrium are frequently applied in Physics or Biology and show a great variety of collective phenomena already in one dimension. The zero range process (ZRP) is an important model of this class, closely related to driven diffusive systems modeling e.g. biopolymerization or transport across membranes. We study the ZRP on a periodic lattice with rates inducing an effective attraction between particles. If the particle density exceeds a critical value, the system phase separates into a homogeneous background and a condensate, where the excess particles accumulate. We proof this by showing the equivalence of the canonical and the grand-canonical stationary measure. We also show that for large systems the condensed phase typically consists only of a single, randomly located site. Using heuristic arguments supported by Monte Carlo simulations, we study the dynamics of the condensation and its dependence on space dimension and symmetry of the jump rates. The results are extended to condensation in two-component systems, showing a particularly rich critical behaviour.