Condensation in non-Markovian zero-range dynamics

Many non-equilibrium systems are known to exhibit real-space condensation in which a high density phase condenses into a small region in space. Examples include phase separation in driven diffusive systems, jamming in traffic flow, gelation in networks resulting in a hub with a macroscopic linking number, or the clustering of compartmentalized shaken granular gasses. Condensation phenomena have been modeled by the zero-range process, which is a model of particles hopping between boxes with Markovian dynamics. However, in many cases, memory effects in the dynamics cannot be neglected. In an attempt to understand the possible impact of temporal correlations on the condensate, we introduce and study a process with non-Markovian zero-range dynamics. We find that memory effects have significant impact on the condensation scenario. Specifically, two main results are found: (1) In mean-field dynamics, the steady state corresponds to that of a Markovian ZRP, but with modified hopping rates which can affect condensation, and (2) for nearest-neighbor hopping in one dimension, the condensate occupies two adjacent sites on the lattice and drifts with a finite velocity. The validity of these results in a more general context is discussed.