  Condensation phenomena in non-equilibrium systems have been modeled by the
zero-range process, which is a model of particles hopping between boxes with
Markovian dynamics. 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.
