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

TL;DR

This paper demonstrates a direct mathematical connection between a one-dimensional driven-diffusive system's steady-state and an equilibrium two-dimensional walk model, linking their partition functions and physical quantities.

Contribution

It introduces an equilibrium 2D walk model and establishes a formal equivalence with a 1D driven-diffusive system's steady-state via transfer matrix and matrix-product approaches.

Findings

Partition functions of both models are directly connected.

Transfer matrix of the walk model relates to the algebra of the driven-diffusive system.

Physical quantities are related through a similarity transformation.

Abstract

It is known that a single product shock measure in some of one-dimensional driven-diffusive systems with nearest-neighbor interactions might evolve in time quite similar to a random walker moving on a one-dimensional lattice with reflecting boundaries. The non-equilibrium steady-state of the system in this case can be written in terms of a linear superposition of such uncorrelated shocks. Equivalently, one can write the steady-state of this system using a matrix-product approach with two-dimensional matrices. In this paper we introduce an equilibrium two-dimensional one-transit walk model and find its partition function using a transfer matrix method. We will show that there is a direct connection between the partition functions of these two systems. We will explicitly show that in the steady-state the transfer matrix of the one-transit walk model is related to the matrix representation of the algebra of the driven-diffusive model through a similarity transformation. The physical quantities are also related through the same transformation.