Coexistence of Ordered and Disordered Phases in Potts Models in the Continuum

Abstract

This is the second of two papers on a continuum version of the Potts model, where particles are points in R d , d ≥ 2, with a spin which may take S ≥ 3 possible values. Particles with different spins repel each other via a Kac pair potential of range γ −1 , γ > 0. In this paper we prove phase transition, namely we prove that if the scaling parameter of the Kac potential is suitably small, given any temperature there is a value of the chemical potential such that at the given temperature and chemical potential there exist S + 1 mutually distinct DLR measures.

Potts Models in the Continuum. Uniqueness and Exponential Decay in the Restricted Ensembles

“…If γ is small one may think that (1.1) "simulates mean field". Indeed we will prove in [8] that Theorem 1.1. For any d ≥ 2, S > 2 and β > 0 there is γ * > 0 such that for any γ ≤ γ * there exist λ β,γ and S + 1 mutually distinct, extremal DLR measures at (β, λ β,γ ).…”

“…To keep the statement simple we have not reported all the information we have on the structure of the DLR measures referring to [8] for the full result. In particular we know that the particles densities are close to their mean field values (for γ small).…”

Dynamical Widom–Rowlinson Model and Its Mesoscopic Limit

“…For a large class of potentials (with or without soft-core), Georgii and Häggström [13] established the phase transition. Further activity in this area concerns a mean-field theory for the Potts model in the continuum and, in particular, for the Widom-Rowlinson model, without hard-core, see [14] for the most general case (that is, two or more different types) and [4,5] (for three or more different types).…”

A Note on Reduced Strain Gradient Elasticity

“…In this context, it is also worth mentioning continuum bodies, characterized at micro scale, by interacting spins in a crystalline lattice (see e.g. Grimmett (2016);de Masi et al (2008de Masi et al ( , 2009) for some relevant results using the Potts model). As some other generalized models it can predict size-effect since it includes the characteristic length observed at the nanoscale and can describe dispersion for waves propagation in microstructured solids.…”