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 A, B, and C. The system is
described by a grand canonical ensemble with temperature T and chemical
potentials TλA​, TλB​, and TλC​. We find that for
λA​=λB​=λC​ the system undergoes a phase transition from a
uniform density to a continuum of phases at a critical temperature T^c​=(2π/3​)−1. For other values of the chemical potentials the system
has a unique equilibrium state. As is the case for the canonical ensemble for
this ABC model, the grand canonical ensemble is the stationary measure
satisfying detailed balance for a natural dynamics. We note that T^c​=3Tc​, where Tc​ is the critical temperature for a similar transition in
the canonical ensemble at fixed equal densities rA​=rB​=rC​=1/3.Comment: 24 pages, 3 figure