Remarks on the multi-species exclusion process with reflective boundaries

TL;DR

This paper studies a multi-species exclusion process with reflective boundaries, providing a simplified matrix product approach to understand the system's block structure and dynamics.

Contribution

It introduces a simplified generalized matrix ansatz for multi-species exclusion processes with reflective boundaries, extending previous methods from periodic cases.

Findings

Block matrices connect different particle sectors

Simpler solution for reflective boundary conditions

Matrix product form for the Markov matrix

Abstract

We investigate one of the simplest multi-species generalizations of the one dimensional 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, which starts with a local relation [Arita et al, J. Phys. A 44, 335004 (2011)]. The solution to this relation for the reflective boundary condition is much simpler than that for the periodic boundary condition.