Temporally correlated zero-range process with open boundaries: Steady state and fluctuations

Abstract

We study an open-boundary version of the on-off zero-range process introduced in Hirschberg et al. [Phys. Rev. Lett. 103, 090602 (2009)]. This model includes temporal correlations which can promote the condensation of particles, a situation observed in real-world dynamics. We derive the exact solution for the steady state of the one-site system, as well as a mean-field approximation for larger one-dimensional lattices, and also explore the large deviation properties of the particle current. Analytical and numerical calculations show that, although the particle distribution is well described by an effective Markovian solution, the probability of rare currents differs from the memoryless case. In particular, we find evidence for a memory-induced dynamical phase transition.

Fluctuations in interacting particle systems with memory

“…where, for our model with α(j) = α 0 + aj, the fixed point j * is given by (15). Now, as argued in section 3, we expect long-time diffusive behaviour with modified diffusion coefficient D * /(1 − 2A * ) when A * of (16) is less than 1/2.…”

“…Much less is known about non-Markovian many-particle systems but some aspects of the stationary-state properties (e.g., mean current as a function of density, conditions for a condensation transition) have been investigated for models with internal states or non-exponential waiting times [12,13,14]. Going beyond the typical behaviour, the current fluctuations in a temporally-correlated zero-range process have also recently been explored (and compared to the equivalent memoryless model) although exact analytical calculations proved possible only for a single site [15].…”

Trapping in bottlenecks: Interplay between microscopic dynamics and large scale effects

“…We also mention that, in the recent literature, ZRP with modified blockage rules have also been studied in different frameworks, e.g. non-Markovian processes and traffic models [20,21,22,23]. This model can be thought of as a basic model for pedestrians on a lane with a width shrinkage due to a corridor or a door.…”

Blockage-induced condensation controlled by a local reaction

“…Related results have been also proved in [7]. It is also worth mentioning that in the recent literature, in different framework as non-Markovian process and traffic models, ZRP with modified blockage rules have been considered [8][9][10][11].…”