Abstract
We study the metastable minima of the Curie–Weiss Potts model with three states, as a function of the inverse temperature, and for arbitrary vector-valued external fields. Extending the classic work of Ellis and Wang (Stoch Process Appl 35(1):59–79, 1990) and Wang (Stoch Process Appl 50(2):245–252, 1994) we use singularity theory to provide the global structure of metastable (or local) minima. In particular, we show that the free energy has up to four local minimizers (some of which may at the same time be global) and describe the bifurcation geometry of their transitions under variation of the parameters.
Highlights
1.1 Research ContextThe Potts model [29], and in particular its Curie–Weiss version, is next to the Curie–Weiss Ising model, one of the most studied models in statistical mechanics
The phase-diagram for the stable states of the Curie–Weiss Potts model, that is the behaviour of global minimizers, is known and described by the Ellis–Wang theorem [8] in zero external field
Speaking, the metastable phase diagram is a partition of the parameter space whose cells contain parameter values (β, α) such that fβ,α has the same number of local minima
Summary
The Potts model [29], and in particular its Curie–Weiss version, is next to the Curie–Weiss Ising model, one of the most studied models in statistical mechanics. As a guiding principle for such analysis, singularity theory is very useful to understand the organization of stationary points for varying parameters It allows to understand and discover the types of local bifurcations which are present in the applied problem as (by Thom’s theorem) they must be related via local diffeomorphisms to (partial) unfoldings of elementary singularities (so-called catastrophes). The associated free energy in terms of the empirical spin distribution ν is given by fβ,α (ν ) This is a real-valued function on the unit simplex 2 with two parameters: the inverse temperature β and the external fields modeled by the a-priori measure α. By a phase (stable or metastable) we mean a (global or local) minimizer of the free energy fβ,α
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