Structural equivalence conjecture for dynamical and static Curie–Weiss Potts rate functionals

Let $\Delta^2$ be the unit simplex in the hyperplane $H$, and let $U$ be an open subset of $H$ containing $\Delta^2$. Define

$$D=\{(\alpha,\beta,t):\beta<3,\ t>0,\ \alpha\in U\}.$$

Let $G_{\alpha,\beta,t}$ denote the potential of the time-evolved model, and let $f_{\beta,\alpha}$ denote the potential of the static model. A smooth map $\psi_1$ maps $D$ to $(0,\infty)\times\Delta^2$, and a smooth map $\psi_2$ maps $D\times\Delta^2$ to the static state space $\Delta^2$.

Structural equivalence conjecture. There exist maps $\psi_1$ and $\psi_2$ with the stated diffeomorphism properties such that, for every $(\alpha,\beta,t)\in D$ and $m\in\Delta^2$,

$$G_{\alpha,\beta,t}(m)=f_{\psi_1(\alpha,\beta,t)}\circ\psi_2(\alpha,\beta,t,m),$$

and there exists a function $(\beta,t)\mapsto\beta_{\mathrm{st}}(\beta,t)$ on $(0,3)\times(0,\infty)$ satisfying

$$\operatorname{pr}_1\circ\psi_1(\alpha,\beta,t)=\beta_{\mathrm{st}}(\beta,t),$$

so that the effective static inverse temperature does not depend on the dynamical $\alpha$.

This conjecture asserts a smooth structural equivalence between the dynamical and static rate functionals, explaining the correspondence between their phase diagrams and stationary-point structures through a parameter-dependent change of variables. Its resolution is not supplied in the source.