Open two-species exclusion processes with integrable boundaries

Abstract

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 9 generators) is explicitly constructed and physical observables (such as currents, densities) calculated.

Matrix product formula for Macdonald polynomials

“…The results discussed in this paper have a direct generalisation to the inhomogeneous multi-species asymmetric exclusion process with boundaries [48,49,46,47,50] resulting in a matrix product formula for Koornwinder polynomials [51]. A connection between Koornwinder polynomials and the quantum XXZ chain, which is closely related to the exclusion process by a similarity transformation, was made in [52].…”

T-Q relations for the integrable two-species asymmetric simple exclusion process with open boundaries

“…For open systems, the integrability is guaranteed by the YBE and reflection equation, where the latter accounts for integrable boundaries [48,49]. The two boundary reflection matrices of the model described above are the third class of Markovian K-matrices in [30]…”

“…The multi-species ASEP (m-ASEP) is also exactly solvable with periodic boundaries [27,28] and a variety of open boundaries [20,[29][30][31]. While a number of results is known for the exact solution of m-ASEP with periodic boundary conditions, not much is known for open boundary conditions.…”

Matrix Product Solution of the Stationary State of Two-Species Open Zero Range Processes

“…While this proof assumes a finite number of local configurations on each site, recent works obtained the steady-state distributions of different types of generalized zero range processes (ZRP)-stochastic hopping models on a lattice with the hopping rate depending on the occupation number-with periodic boundary conditions [6,22,21] and a single-species ZRP with open boundary conditions [4] using the matrix product ansatz with an unbounded number of configurations. Given the previous works on models with multiple species of particles [17,27,3,7,11,15,10,28,13], a naturally arising question is the following: is it possible to obtain the steady-state distribution of the two-species open 1D ZRP using the matrix product method? The two-species model is important because it is closely related to many interesting statistical physics phenomena such as the behavior of shaken granular gases in which the grains come in one of two sizes, and the behavior of networks with directed edges (see the review article [12] and references therein).…”