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Matrix product solution to multi-species ASEP with open boundaries

Caley Finn, E. Ragoucy, Matthieu Vanicat

Published Jan 15, 2018• 14 citations
OpenAlex

Abstract

We study a class of multi-species ASEP with open boundaries. The boundariesare chosen in such a way that all species of particles interact non-triviallywith the boundaries and are present in the stationary state. We give the exactexpression of the stationary state in a matrix product form and we compute itsnormalisation. Densities and currents for the different species are thencomputed in term of this normalisation.

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Matrix product solution to multi-species ASEP with open boundaries

Caley Finn, E. Ragoucy, Matthieu Vanicat

Abstract We study a class of multi-species ASEP with open boundaries. The boundaries are chosen in such a way that all species of particles interact non-trivially with the boundaries, and are present in the stationary state. We give the exact expression of the stationary state in a matrix product form, and compute its normalisation. Densities and currents for the different species are then computed in terms of this normalisation.

arXiv:1712.06809•Journal of Statistical Mechanics Theory and Experiment•Apr 1, 2018•11 citations

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

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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.

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Open two-species exclusion processes with integrable boundaries

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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.

arXiv:1412.5939•Journal of Physics A Mathematical and Theoretical•Apr 8, 2015•29 citations

Exact Analysis of Two-Species Totally Asymmetric Exclusion Process with Open Boundary Condition

Chikashi Arita

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

Journal of the Physical Society of Japan•Jun 8, 2006•16 citations

Matrix product solution to a 2-species TASEP with open integrable boundaries

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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.

arXiv:1606.08148•Journal of Physics A Mathematical and Theoretical•Oct 31, 2016•17 citations

Multi-species asymmetric exclusion process in ordered sequential update

M. Ebrahim Fouladvand, Farhad H Jafarpour

A multi-species generalization of the asymmetric simple exclusion process is studied with ordered sequential and sub-lattice parallel updating schemes. In this model, particles hop with their own specific probabilities to their rightmost empty site and fast particles overtake slow ones with a definite probability. Using the matrix product ansatz technique, we obtain the relevant algebra and study the uncorrelated stationary state of the model both for an open system and on a ring. A complete comparison between the physical results in these updates and those of random sequential introduced in Karimipour V 1999 A multi-species ASEP and its relation to traffic flow Phys. Rev. E 59 205 and Karimipour V 1998 A multi-species ASEP, steady state and correlation functions on a periodic lattice Preprint cond-mat/9809193, is made.

arXiv:cond-mat/9901007•Journal of Physics A Mathematical and General•Jul 29, 1999•9 citations
We study a class of multi-species ASEP with open boundaries. The boundariesare chosen in such a way that all species of particles interact non-triviallywith the boundaries and are present in the stationary state. We give the exactexpression of the stationary state in a matrix product form and we compute itsnormalisation. Densities and currents for the different species are thencomputed in term of this normalisation.
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.
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.
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
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.
A multi-species generalization of the asymmetric simple exclusion process is studied with ordered sequential and sub-lattice parallel updating schemes. In this model, particles hop with their own specific probabilities to their rightmost empty site and fast particles overtake slow ones with a definite probability. Using the matrix product ansatz technique, we obtain the relevant algebra and study the uncorrelated stationary state of the model both for an open system and on a ring. A complete comparison between the physical results in these updates and those of random sequential introduced in Karimipour V 1999 A multi-species ASEP and its relation to traffic flow Phys. Rev. E 59 205 and Karimipour V 1998 A multi-species ASEP, steady state and correlation functions on a periodic lattice Preprint cond-mat/9809193, is made.