Phase diagram of the ABC model on an interval

TL;DR

This paper analyzes the phase diagram of the asymmetric ABC model on an interval, revealing a unique phase structure except at low temperature with equal densities, where multiple phases coexist, extending previous ring-based results.

Contribution

It extends the understanding of the ABC model to an interval geometry, proving the form of density profiles and phase uniqueness or coexistence in the thermodynamic limit.

Findings

Density profiles follow a periodic trajectory in a quartic potential.

Unique phase exists for most parameters, except at low temperature with equal densities.

Coexistence of multiple phases occurs at low temperature with equal densities.

Abstract

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.