Dynamical Transitions in a One-Dimensional Katz–Lebowitz–Spohn Model

Alessandro Pelizzola,Marco Pretti,Francesco Puccioni

doi:10.3390/e21111028

Abstract

Dynamical transitions, already found in the high- and low-density phases of the Totally Asymmetric Simple Exclusion Process and a couple of its generalizations, are singularities in the rate of relaxation towards the Non-Equilibrium Stationary State (NESS), which do not correspond to any transition in the NESS itself. We investigate dynamical transitions in the one-dimensional Katz–Lebowitz–Spohn model, a further generalization of the Totally Asymmetric Simple Exclusion Process where the hopping rate depends on the occupation state of the 2 nodes adjacent to the nodes affected by the hop. Following previous work, we choose Glauber rates and bulk-adapted boundary conditions. In particular, we consider a value of the repulsion which parameterizes the Glauber rates such that the fundamental diagram of the model exhibits 2 maxima and a minimum, and the NESS phase diagram is especially rich. We provide evidence, based on pair approximation, domain wall theory and exact finite size results, that dynamical transitions also occur in the one-dimensional Katz–Lebowitz–Spohn model, and discuss 2 new phenomena which are peculiar to this model.

Highlights

The Totally Asymmetric Simple Exclusion Process (TASEP, see References [1,2] for recent reviews), originally inspired by biological traffic problems [3,4], has gained a prominent role in non-equilibrium statistical physics, becoming a paradigmatic minimal model, with a status analogous to that of the Ising model in equilibrium statistical physics
In the portion of the phase diagram corresponding to each phase we report an identifier of the phase (I to VII) and the value of the bulk density
Dynamical transitions are singularities in the relaxation rate, which do not correspond to any transition in the non-equilibrium stationary state (NESS) itself

Summary

Introduction

The Totally Asymmetric Simple Exclusion Process (TASEP, see References [1,2] for recent reviews), originally inspired by biological traffic problems [3,4], has gained a prominent role in non-equilibrium statistical physics, becoming a paradigmatic minimal model, with a status analogous to that of the Ising model in equilibrium statistical physics. This 1D KLS model is more general than the AS model, since here the hopping rate between nodes i and i + 1 depends on the states of both nodes i − 1 and i + 2 (in the AS model there is no dependence on node i − 1) It exhibits a richer NESS phase diagram, due to the appearance, for certain values of the model parameters, of 2 maxima in the bulk current-density relation (the fundamental diagram, as it is called in the traffic literature). The NESS phase diagram of the 1D KLS model with Glauber hopping rates has been determined in References [12,13] using a Markov chain approach to kinetics (MCAK), kinetic Monte Carlo (KMC) and the theory of boundary-induced phase transitions. In the present work we look for dynamical transitions in the 1D KLS model with Glauber hopping rates and bulk-adapted boundary conditions, using three different methods: the pair approximation (PA), a modified domain wall theory (mDWT) and extrapolation of exact finite size results. We found this technique useful in our study of the AS model [19]

Model and Methods

Results

Discussion