Bose-Einstein condensation and a two-dimensional walk model

Abstract

We introduce a two-dimensional walk model in which a random walker can only move on the first quarter of a two-dimensional plane. We calculate the partition function of this walk model using a transfer matrix method and show that the model undergoes a phase-transition. Surprisingly the partition function of this two-dimensional walk model is exactly equal to that of a driven-diffusive system defined on a discrete lattice with periodic boundary conditions in which a phase transition occurs from a high-density to a low-density phase. The driven-diffusive system can be mapped to a zero-range process where the particles can accumulate in a single lattice site in the low-density phase. This is very reminiscent of real-space Bose-Einstein condensation.

Non-equilibrium phase transition in a two-species driven-diffusive model of classical particles

“…Comparing this partition function with that of the drivendiffusive model we have shown that these two model are equivalent. It should be noted that the walk model introduced in [11] and the one introduced in present work can be mapped onto zero-range process.…”

“…The density profile of the model is calculated exactly and the spatial correlations of the model are obtained in terms of 1-point correlation function. We have introduced a two-dimensional walk model in which the random walker, in contrast with the lattice path introduced in [11], can start from any height upper than the origin and that the end point of the lattice path can be at any height upper than the start point. This type of lattice path is introduced in [19].…”

“…In [11], the author has introduced a one-dimensional driven-diffusive model of classical particles with hardcore interactions. The model consists of a single particle of type A (called the second-class particle) and M À 1 particles of type B (called the first-class particles).…”

“…It has been shown that the partition function of some of the onedimensional driven-diffusive models with open boundaries obtained using a matrix product method is equal to the partition function of a two-dimensional walk model obtained using a transfer matrix method [8,11,12]. In [11], the author has introduced a lattice path with specific dynamical rules where walker can start from origin and end at any height upper than origin and it has been shown that the partition function of this two-dimensional walk model is exactly equal to that of a driven-diffusive system defined on a discrete lattice with periodic boundary conditions that can be mapped to a zero-range process [13,14]. In this paper, a two-dimensional walk model is introduced in which the walker can only move on the first quarter of a two-dimensional plane.…”

A new family of exactly solvable disordered reaction–diffusion systems

“…We will show that our new family of exactly solvable systems can be mapped onto a ZRP. Similar systems have already been studied in related literature [4,18,19]. We start with a disordered exclusion process defined on a one-dimensional lattice of finite size with periodic boundary conditions.…”