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arXiv:1708.09153 · Physical review. E

Matrix product solution of a left-permeable two-species asymmetric exclusion process

Arvind Ayyer, Caley Finn, D. Roy

Published Jan 31, 2018• 9 citations
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Abstract

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.

DOI
10.1103/physreve.97.012151

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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.
In recent years, one-dimensional driven diffusion models consisting of particles on a lattice have been studied. The most famous topics of these models are solvability and phase transitions. For example, the exact stationary-state solution of the asymmetric exclusion process (ASEP) with an open boundary condition can be obtained as a matrix product form and its physical quantities exhibit boundary-induced phase transitions in the thermodynamic limit. There are three regimes, namely, maximal-current (MC) phase, low-density (LD) phase and high-density (HD) phase in the phase diagram. The two-species models or three-state models are also interesting. The stationary-state solutions of the two-species totally asymmetric exclusion process (TASEP) with a periodic boundary condition and an open boundary condition can be obtained as matrix product forms. The model in ref. 6 exhibits two phases in which symmetry breaks spontaneously. Furthermore, the model in ref. 7 exhibits phase separation. In this note, the two-species TASEP with a new open boundary condition different from the condition in ref. 6 is introduced and we obtain the stationary-state solution exactly as a matrix product form and calculate the physical quantities in the thermodynamic limit. Let us consider a stochastic process on an L-site chain with two-species particles. We note j 1⁄4 0 if site j is empty, j 1⁄4 1 if site j is occupied by the first-class particle and j 1⁄4 2 if site j is occupied by the second-class particle. The exchange rule in the bulk of the chain is expressed as 10 ! 01 with a rate 1; ð1Þ 20 ! 02 with a rate 1; ð2Þ 12 ! 21 with a rate 1: ð3Þ
We find the exact solution for the stationary state measure of the partially asymmetric exclusion process on a ring with multiple species of particles. The solution is in the form of a matrix product representation where the matrices for a system of N species are defined recursively in terms of the matrices for a system of N − 1 species. A complete proof is given, based on the quadratic relations verified by these matrices. This matrix product construction is interpreted in terms of the action of a transfer matrix.
Abstract We present an explicit representation for the matrix product ansatz for some two-species totally asymmetric exclusion process with open boundary conditions. The construction relies on the integrability of the models, a property that constrains the possible rates at the boundaries. The realisation is built on a tensor product of copies of the DEHP algebras. Using this explicit construction, we are able to calculate the partition function of the models. The densities and currents in the stationary state are also computed and led to the phase diagrams of the models. Depending on the values of the boundary rates, we obtain for each species shock waves, maximal current, or low/high densities phases.
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.