Temporal evolution of product shock measures in the totally asymmetric simple exclusion process with sublattice-parallel update

Abstract

It is known that when the steady state of a one-dimensional multispecies system, which evolves via a random-sequential updating mechanism, is written in terms of a linear combination of Bernoulli shock measures with random-walk dynamics, it can be equivalently expressed as a matrix-product state. In this case the quadratic algebra of the system always has a two-dimensional matrix representation. Our investigations show that this equivalence exists at least for the systems with deterministic sublattice-parallel update. In this paper we consider the totally asymmetric simple exclusion process on a finite lattice with open boundaries and sublattice-parallel update as an example.

Discrete-time analysis of traveling wave solutions and steady state of a PASEP with open boundaries

“…It is shown that this steadystate can be also written in terms of a product of four non-commuting matrices. One of the main results obtained here is that these matrices have exactly the same generic structure of the matrices first introduced in [7] indicating that the steady-state of a onedimensional driven-diffusive system can be written as a linear superposition of product shock measures. It is easy now to explain the two-dimensional matrix representation of the PASEP with parallel dynamics introduced in [8,9].…”

“…The matrices L and R are also written in the basis (0, 1). Following [7] let us define two different product shock measures. We denote the shocks at even sites 2k (k = 1, · · · , L) as |µ 2k and at odd sites 2k + 1 (k = 0, · · · , L) as |µ 2k+1 respectively:…”

“…The matrix representation structure given in (13) was first introduced in [7] for the Totally Asymmetric Simple Exclusion Process (TASEP) with open boundaries in sublatticeparallel updating scheme. In the TASEP the particles enter the system only from the left boundary (the first lattice site).…”

“…For the same systems, but in discrete-time updating scheme, it seems that the two-dimensional matrix representation first proposed in [7] is sufficient in order to construct the steady-state. Considering the example studied here i.e.…”

Equivalence of a one-dimensional driven-diffusive system and an equilibrium two-dimensional walk model

“…Three families of two-state systems with nearest-neighbor interactions have been introduced and studied in [5]. This has been also generalized to the multi-species systems with long range interactions and even the systems with discrete time updating scheme [10]. The study of the steady-state properties of onedimensional driven-diffusive systems has a long history.…”