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

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.

Multiple-transit paths and density correlation functions in partially asymmetric simple exclusion process

“…for l = 1, · · · , N (17) in which only the lth element is non-zero and equal to one. If we write the partition function (14) as…”

“…Apart from these approaches one can try to connect the matrix representation of the algebra of a one-dimensional reaction-diffusive system directly to the transfer matrix of an equilibrium two-dimensional walk model. In this way the one can actually show that the normalization coefficient of the stationary distribution function of the non-equilibrium system coincides with the partition function of an equilibrium walk model [15][16][17]. In this paper we consider the PASEP with open boundaries where the steady-state of the system can be written in terms of a linear superposition of product measures with a finite number of shocks.…”

The PASEP at q=-1

“…It is also useful to interpret Z(z) as the generating function for a model of weighted lattice paths [31][32][33][34][35][36] , which gives some insight into the PASEP phase diagram. From this point of view the tridiagonal matrix C in the particular representation of equ.…”

Bose-Einstein condensation and a two-dimensional walk model

“…Over the last couple of years there has been a growing interest in studying of the connections between the one-dimensional driven-diffusive systems and the twodimensional walk models [1][2][3][4]. It has been shown that a properly defined steady-state normalization factor, called the partition function, of some of the one-dimensional driven-diffusive systems with open boundaries obtained using a matrix product method (reader can see [5] for a review) is equal to the partition function of a twodimensional walk model obtained using a transfer matrix method [1,6,7]. It is known that the one-dimensional driven-diffusive systems exhibit a variety of interesting critical behaviors, such as non-equilibrium phase transition and realspace condensation, by changing their microscopic reaction rates.…”