Phase diagram of the ABC model on an interval

Details Phase diagram of the ABC model on an interval Phase diagram of the ABC model on an interval

2009-05-29

arxiv.org/abs/0905.4849v2

JSP, Volume 137, Numbers 5-6, 1166-1204 (2009)

Physics

Condensed Matter

Statistical Mechanics

52 pages, AMS-LaTeX, 8 figures from 10 eps figure files. Revision: minor changes in response to referee reports; paper to appe

Scientific paper

10.1007/s10955-009-9834-x

The three species asymmetric ABC model was initially defined on a ring by Evans, Kafri, Koduvely, and Mukamel, and the weakly asymmetric version was later studied by Clincy, Derrida, and Evans. Here the latter model is studied on a one-dimensional lattice of N sites with closed (zero flux) boundaries. In this geometry the local particle conserving dynamics satisfies detailed balance with respect to a canonical Gibbs measure with long range asymmetric pair interactions. This generalizes results for the ring case, where detailed balance holds, and in fact the steady state measure is known only for the case of equal densities of the different species: in the latter case the stationary states of the system on a ring and on an interval are the same. We prove that in the N to infinity limit the scaled density profiles are given by (pieces of) the periodic trajectory of a particle moving in a quartic confining potential. We further prove uniqueness of the profiles, i.e., the existence of a single phase, in all regions of the parameter space (of average densities and temperature) except at low temperature with all densities equal; in this case a continuum of phases, differing by translation, coexist. The results for the equal density case apply also to the system on the ring, and there extend results of Clincy et al.

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