The grand canonical ABC model: a reflection asymmetric mean field Potts model

TL;DR

This paper studies a reflection asymmetric mean field Potts model with three particle types, revealing phase transitions in the grand canonical ensemble and analyzing how chemical potentials influence the system's equilibrium states.

Contribution

It introduces a grand canonical ensemble analysis of a three-component reflection asymmetric mean field Potts model, identifying phase transition conditions and comparing with canonical ensemble results.

Findings

Phase transition from uniform to multiple phases at critical temperature

Unique equilibrium state for unequal chemical potentials

Critical temperature is three times the canonical ensemble's T_c

Abstract

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 $T\lambda_A$, $T\lambda_B$, and $T\lambda_C$. We find that for $\lambda_A=\lambda_B=\lambda_C$ the system undergoes a phase transition from a uniform density to a continuum of phases at a critical temperature $\hat T_c=(2\pi/\sqrt3)^{-1}$. 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 $\hat T_c=3T_c$, where $T_c$ is the critical temperature for a similar transition in the canonical ensemble at fixed equal densities $r_A=r_B=r_C=1/3$.