Energy Landscape of the Two-Component Curie–Weiss–Potts Model with Three Spins

Abstract

In this paper, we investigate the energy landscape of the two-component spin systems, known as the Curie–Weiss–Potts model, which is a generalization of the Curie–Weiss model consisting of $q\ge 3$ spins. In the energy landscape of a multi-component model, the most important element is the relative strength between the inter-component interaction strength and the component-wise interaction strength. If the inter-component interaction is stronger than the component-wise interaction, we can expect all the components to be synchronized in the course of metastable transition. However, if the inter-component interaction is relatively weaker, then the components will be desynchronized in the course of metastable transition. For the two-component Curie–Weiss model, the phase transition from synchronization to desynchronization has been precisely characterized in studies owing to its mean-field nature. The purpose of this paper is to extend this result to the Curie–Weiss–Potts model with three spins. We observe that the nature of the phase transition for the three-spin case is entirely different from the two-spin case of the Curie–Weiss model, and the proof as well as the resulting phase diagram is fundamentally different and exceedingly complicated.

Appendix A: Proof of Proposition 2.2

Here, we present a proof of Proposition 2.2.

Proof of Proposition 2.2

For $\varvec{x}\in \Xi _{N}^{2}$, from the definition of ${\mathbb {H}}_{N}$,

$$\begin{aligned}&\sum _{\varvec{\sigma }\in \Omega :\varvec{r}(\varvec{\sigma })=\varvec{x}}\frac{1}{Z_{N,\beta }}e^{-\beta {\mathbb {H}}_{N}(\varvec{\sigma })}, =\frac{N!}{(Nx_{1}^{(1)})!\cdots (Nx_{q}^{(1)})!}\cdot {{\frac{N!}{(Nx_{1}^{(2)})!\cdots (Nx_{q}^{(2)})!}}}\\&\quad \times \frac{1}{Z_{N,\beta }}\exp \left[ {{\frac{\beta }{N(1+J)}}}\left\{ \sum _{k=1,2}\sum _{i=1}^{q}\frac{1}{2}\left\{ (Nx_{i}^{(k)})(Nx_{i}^{(k)}-1)\right\} +J\sum _{i=1}^{q}Nx_{i}^{(1)}Nx_{i}^{(2)}\right\} \right] , \end{aligned}$$

and by Stirling’s formula, we have

$$\begin{aligned}&\approx \frac{\exp \left\{ -\beta /(1+J)\right\} }{(\sqrt{2\pi N})^{2(q-1)}\sqrt{\prod _{k=1,2}\prod _{i=1}^{q}x_{i}^{(k)}}Z_{N,\beta }}\\&\qquad \times \exp \left[ N\frac{\beta }{1+J}\left\{ \sum _{k=1,2}\sum _{i=1}^{q}\frac{1}{2}(x_{i}^{(k)})^{2}+J\sum _{i=1}^{q}x_{i}^{(1)}x_{i}^{(2)}-\frac{1+J}{\beta }\left( \sum _{k=1,2}\sum _{i=1}^{q}x_{i}^{(k)}\log x_{i}^{(k)}\right) \right\} \right] ,\\&\quad =\frac{1}{{\widehat{Z}}_{N,\beta ,J}}\exp \left\{ -N\frac{\beta }{1+J}\left( F_{\beta ,J}(\varvec{x})+\frac{1}{N}G_{N,\,\beta }(\varvec{x})\right) \right\} , \end{aligned}$$

where

$$\begin{aligned} F_{\beta ,J}(\varvec{x})&=-\sum _{k=1,2}\sum _{i=1}^{q}\frac{1}{2}(x_{i}^{(k)})^{2}-J\sum _{i=1}^{q}x_{i}^{(1)}x_{i}^{(2)}+\frac{1+J}{\beta }\left( \sum _{k=1,2}\sum _{i=1}^{q}x_{i}^{(k)}\log x_{i}^{(k)}\right) ,\\ G_{N,\beta ,J}(\varvec{x})&=\frac{1+J}{2\beta }\log \left( \prod _{k=1,2}\prod _{j=1}^{q}x_{i}^{(k)}\right) +O\left( N^{-(q-1)}\right) . \end{aligned}$$

We decompose the function $F_{\beta ,J}$ into an energy part H and entropy part S. That is,

$$\begin{aligned} F_{\beta ,J}(\varvec{x})=H(\varvec{x})+\frac{1+J}{\beta }S(\varvec{x}), \end{aligned}$$

where

$$\begin{aligned} H(\varvec{x})=-\sum _{k=1,2}\sum _{i=1}^{q}\frac{1}{2}(x_{i}^{(k)})^{2}-J\sum _{i=1}^{q}x_{i}^{(1)}x_{i}^{(2)}\quad \text {and}\quad S(\varvec{x})=\sum _{k=1,2}\sum _{i=1}^{q}x_{i}^{(k)}\log x_{i}^{(k)}. \end{aligned}$$

$\square$