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

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 ${\hat{T}}_c=(2\pi /\sqrt{3}{)}^{-1}$. 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 ${\hat{T}}_c=3T_c$, where $T_c$ is the critical temperature for a similar transition in the canonical ensemble at fixed equal densities $r_A=r_B=r_C=1/3$.

24 pages, 3 figures

References in corpus (3)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• Long-range correlations and ensemble inequivalence in a generalized ABC model
• Notes on the Statistical Mechanics of Systems with Long-Range Interactions

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• Anomalous long-range correlations at a non-equilibrium phase transition
• Phase diagram of the repulsive Blume-Emery-Griffiths model in the presence of external magnetic field on a complete graph
• Relaxation processes in a system with logarithmic growth
• Phase diagram of a generalized ABC model on the interval