4 research outputs found
Phase diagram of a generalized ABC model on the interval
We study the equilibrium phase diagram of a generalized ABC model on an
interval of the one-dimensional lattice: each site is occupied by a
particle of type \a=A,B,C, with the average density of each particle species
N_\a/N=r_\a fixed. These particles interact via a mean field
non-reflection-symmetric pair interaction. The interaction need not be
invariant under cyclic permutation of the particle species as in the standard
ABC model studied earlier. We prove in some cases and conjecture in others that
the scaled infinite system N\rw\infty, i/N\rw x\in[0,1] has a unique
density profile \p_\a(x) except for some special values of the r_\a for
which the system undergoes a second order phase transition from a uniform to a
nonuniform periodic profile at a critical temperature .Comment: 25 pages, 6 figure
The grand canonical ABC model: a reflection asymmetric mean field Potts model
We investigate the phase diagram of a three-component system of particles on
a one-dimensional filled lattice, or equivalently of a one-dimensional
three-state Potts model, with reflection asymmetric mean field interactions.
The three types of particles are designated as , , and . The system is
described by a grand canonical ensemble with temperature and chemical
potentials , , and . We find that for
the system undergoes a phase transition from a
uniform density to a continuum of phases at a critical temperature . For other values of the chemical potentials the system
has a unique equilibrium state. As is the case for the canonical ensemble for
this model, the grand canonical ensemble is the stationary measure
satisfying detailed balance for a natural dynamics. We note that , where is the critical temperature for a similar transition in
the canonical ensemble at fixed equal densities .Comment: 24 pages, 3 figure
Phase diagram of the ABC model with nonequal densities
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.Comment: 19 pages, 6 figures, submitted to J. Phys.
Phase diagram of the ABC model with nonconserving processes
The three species ABC model of driven particles on a ring is generalized to
include vacancies and particle-nonconserving processes. The model exhibits
phase separation at high densities. For equal average densities of the three
species, it is shown that although the dynamics is {\it local}, it obeys
detailed balance with respect to a Hamiltonian with {\it long-range
interactions}, yielding a nonadditive free energy. The phase diagrams of the
conserving and nonconserving models, corresponding to the canonical and
grand-canonical ensembles, respectively, are calculated in the thermodynamic
limit. Both models exhibit a transition from a homogeneous to a phase-separated
state, although the phase diagrams are shown to differ from each other. This
conforms with the expected inequivalence of ensembles in equilibrium systems
with long-range interactions. These results are based on a stability analysis
of the homogeneous phase and exact solution of the hydrodynamic equations of
the models. They are supported by Monte-Carlo simulations. This study may serve
as a useful starting point for analyzing the phase diagram for unequal
densities, where detailed balance is not satisfied and thus a Hamiltonian
cannot be defined.Comment: 32 page, 7 figures. The paper was presented at Statphys24, held in
Cairns, Australia, July 201