Numerical study of phase transition in an exclusion model with parallel dynamics

Abstract

A numerical method based on Matrix Product Formalism is proposed to study the phase transitions and shock formation in the Asymmetric Simple Exclusion Process with open boundaries and parallel dynamics. By working in a canonical ensemble, where the total number of the particles is being fixed, we find that the model has a rather non-trivial phase diagram consisting of three different phases which are separated by second-order phase transition. Shocks may evolve in the system for special values of the reaction parameters. During the last decade the study of one-dimensional driven lattice gases has been of increasing interest because besides their application in different areas of non-equilibrium physics they show a variety of fascinating properties such as non-equilibrium phase transitions and spontaneous symmetry breaking [1]. They have also let us study the one-dimensional shocks i.e. discontinuities in the density of particles on the lattice over a microscopic region. Depending on the dynamics these models can be divided into two different classes: models with sequential (continuous time evolution) and parallel (discrete time evolution) dynamics. These models can also have open boundaries (where the particles can enter or leave the lattice) or periodic boundary conditions (with conservation of the number of particle). Different approaches have been used to study the one-dimensional out-ofequilibrium models. Among these approaches the Matrix Product Formalism (MPF) is the one which allows us to study these models under sequential or parallel dynamics with different boundary conditions [2]. According to the Email address: farhad@sadaf.basu.ac.ir (Farhad H Jafarpour).

Exact shock profile for the ASEP with sublattice-parallel update

“…However, the case of fixed particle number has not been studied yet. Our recent numerical investigations show that in the case of fixed particle number the phase diagram of the model highly depends on the total density of the particles on the lattice ρ, α and β [13]. For ρ < 1 2 the phase diagram contains a low-density and a shock phase while for ρ > 1 2 it has a high-density phase and also a shock phase.…”

Multi-shocks in reaction-diffusion models

“…Recently, shocks in one-dimensional reaction-diffusion models have absorbed much interest [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. There are some exact results on shocks in one-dimensional reaction-diffusion models as well as simulations, numeric results [6] and also mean field results [2].…”

“…Recently, shocks in one-dimensional reaction-diffusion models have absorbed much interest [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. There are some exact results on shocks in one-dimensional reaction-diffusion models as well as simulations, numeric results [6] and also mean field results [2]. Formation of localized shocks in one-dimensional driven diffusive systems with spacially homogeneous creation and annihilation of particles has been studied in [12].…”

Phase transitions in systems possessing shock solutions

“…Shocks in one-dimensional reaction-diffusion models have been absorbed many interest recently [2][3][4][5][6][7][8][9][10][11][12]. There are some exact results about shocks in one-dimensional reaction-diffusion models together with simulations, numeric results [7] and also mean field results [3].…”

“…Shocks in one-dimensional reaction-diffusion models have been absorbed many interest recently [2][3][4][5][6][7][8][9][10][11][12]. There are some exact results about shocks in one-dimensional reaction-diffusion models together with simulations, numeric results [7] and also mean field results [3]. Formation of localized shocks in one-dimensional driven diffusive systems with spacially homogeneous creation and annihilation of particles has been studied in [13].…”