Phase diagram of the ABC model with nonequal densities

Abstract

The ABC model is a driven diffusive exclusion model, composed of three species of particles that hop on a ring with local asymmetric rates. In the weak asymmetry limit, where the asymmetry vanishes with the length of the system, the model exhibits a phase transition between a homogenous state and a phase separated state. We derive the exact solution for the density profiles of the three species in the hydrodynamic limit for arbitrary average densities. The solution yields the complete phase diagram of the model and allows the study of the nature of the first order phase transition found for average densities that deviate significantly from the equal densities point.PACS numbers: 05.50.+q, 05.70.Ln and 64.60.Cn

Aging processes in systems with anomalous slow dynamics

“…The ABC model has recently yielded a flurry of interesting studies [20][21][22][23][24][25][26][27][28][29][30][31][32][33] that helped in establishing it as a paradigm for systems far from equilibrium. Not only is the ABC model characterized by its anomalous slow dynamics, making it a representative for a larger class of systems with a similar coarsening process [34][35][36][37][38], it also exhibits a variety of interesting nonequilibrium phase transitions whose properties change dramatically when breaking certain conservation laws.…”

Drift of Phase Fluctuations in the ABC Model

“…Namely, for some β ∈ (0, β r ) there exist other stationary solutions to (1.1) which actually describe the typical profiles with respect to the invariant measure of the underlying microscopic dynamics. In [11] the stationary solutions to (1.1) are characterized and their explicit expression is written in terms of elliptic functions (see also [14], where similar results were obtained for some general exclusion processes, including ABC with not necessarily weakly asymmetric transition rates). This analysis reveals that for some r and β ∈ (0, β r ) there exist one-periodic stationary solutions to (1.1), in agreement with the conjectured occurrence of a first order transition.…”

“…This analysis reveals that for some r and β ∈ (0, β r ) there exist one-periodic stationary solutions to (1.1), in agreement with the conjectured occurrence of a first order transition. The next natural issue on (1.1) is whether it admits traveling wave solutions, this is indeed a side question both in [5] and [11]. A simple argument, discussed in [23] shows that such solutions do not occur.…”

“…In order to describe it, fix values of r and β in the low temperature part of the phase diagram. The analysis in [11] implies the existence of a one-periodic profile ρ = ρ A , ρ B , ρ C such that the one-periodic stationary solutions to (1.1) are given by the translations of ρ. Observe that, as discussed in [1,11], for larger values of β there exist also 1 2 -periodic solutions, 1 3periodic solutions,...…”

Simulations of driven and reconstituting lattice gases

“…However, in those systems all NN species are exchangeable [25] whereas in this mapping note that the ordering of characters A 0 , A 1 , · · · , A k−1 set by the initial conditions is conserved throughout all subsequent times, modulo eventual interposition of one or more adjacent k-mers between A's. Thus, the invariant IS of a given sector simply refers to the sequence emerged after deletion of all k-mers or 'reducible' characters appearing in any configuration of that sector.…”