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Journal of Statistical Mechanics Theory and Experiment

Phase transitions in the two-species totally asymmetric exclusion process with open boundaries

Chikashi Arita

Published Dec 7, 2006• 32 citations
Publisher

Abstract

The two-species totally asymmetric exclusion process on a chain with the following boundary conditions is studied: the first-class particles are injected at the left boundary and extracted at the right boundary, while the second-class particles are confined to the chain. We construct the stationary-state solution and calculate physical quantities exactly by means of the matrix product ansatz. In the thermodynamic limit, these quantities exhibit boundary-induced phase transitions.

DOI
10.1088/1742-5468/2006/12/p12008

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The two-species totally asymmetric exclusion process on a chain with the following boundary conditions is studied: the first-class particles are injected at the left boundary and extracted at the right boundary, while the second-class particles are confined to the chain. We construct the stationary-state solution and calculate physical quantities exactly by means of the matrix product ansatz. In the thermodynamic limit, these quantities exhibit boundary-induced phase transitions.
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We study a two-species partially asymmetric exclusion process where the left boundary is permeable for the "slower" species but the right boundary is not. We find a matrix product solution for the stationary state and the exact stationary phase diagram for the densities and currents. By calculating the density of each species at the boundaries, we find further structure in the stationary phases. In particular, we find that the slower species can reach and accumulate at the far boundary, even in phases where the bulk density of these particles approaches zero.
The stationary state of the partially asymmetric simple exclusion process with open boundaries is reconsidered. The so-called matrix product ansatz is employed. This enables us to construct the stationary state in the form of the product of the matrices D and E . Noticing the fact that the matrix C ( = D + E ) for the model is related to certain q -orthogonal polynomials, the model is analysed for a wide range of parameters. The current and the correlation length are evaluated in the thermodynamic limit. It turns out that the phase diagram for the correlation length is richer than that for the totally asymmetric case.
We give a complete classification of integrable Markovian boundary conditions for the asymmetric simple exclusion process with two species (or classes) of particles. Some of these boundary conditions lead to non-vanishing particle currents for each species. We explain how the stationary state of all these models can be expressed in a matrix product form, starting from two key components, the Zamolodchikov–Faddeev and Ghoshal–Zamolodchikov relations. This statement is illustrated by studying in detail a specific example, for which the matrix ansatz (involving nine generators) is explicitly constructed and physical observables (such as currents, densities) calculated.
The totally asymmetric simple exclusion process with generalized update is a version of the discrete time totally asymmetric exclusion process with an additional interparticle interaction that controls the degree of particle clustering. Though the model was shown to be integrable on the ring and on the infinite lattice, on the open chain it was studied mainly numerically, while no analytic results existed even for its phase diagram. In this paper, we introduce boundary conditions associated with the infinite translation invariant stationary states of the model, which allow us to obtain the exact phase diagram analytically. We discuss the phase diagram in detail and confirm the analytic predictions by extensive numerical simulations.