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arXiv:1112.5585 · Journal of Physics A Mathematical and Theoretical

Remarks on the multi-species exclusion process with reflective boundaries

Chikashi Arita

Published Mar 30, 2012• 6 citations
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Abstract

We investigate one of the simplest multi-species generalizations of the onedimensional exclusion process with reflective boundaries.The Markov matrix governing the dynamics of the system splits into blocks (sectors) specified by the number of particles of each kind.We find matrices connecting the blocks in a matrix product form.The procedure (generalized matrix ansatz) to verify that a matrix intertwines blocks of the Markov matrix was introduced in the periodic boundary condition (Arita et al 2011 J. Phys.A 44 335004), which starts with a local relation.The solution to this relation for the reflective boundary condition is much simpler than that for the periodic boundary condition.

DOI
10.1088/1751-8113/45/15/155001

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We investigate one of the simplest multi-species generalizations of the onedimensional exclusion process with reflective boundaries.The Markov matrix governing the dynamics of the system splits into blocks (sectors) specified by the number of particles of each kind.We find matrices connecting the blocks in a matrix product form.The procedure (generalized matrix ansatz) to verify that a matrix intertwines blocks of the Markov matrix was introduced in the periodic boundary condition (Arita et al 2011 J. Phys.A 44 335004), which starts with a local relation.The solution to this relation for the reflective boundary condition is much simpler than that for the periodic boundary condition.
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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.
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The spectrum of Hamiltonian (Markov matrix) of a multi-species asymmetric simple exclusion process on a ring is studied. The dynamical exponent concerning the relaxation time is found to coincide with the one-species case. It implies that the system belongs to the Kardar-Parisi-Zhang or Edwards-Wilkinson universality classes depending on whether the hopping rate is asymmetric or symmetric, respectively. Our derivation exploits a poset structure of the particle sectors, leading to a new spectral duality and inclusion relations. The Bethe ansatz integrability is also demonstrated.
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.