Nonlinear Thermodynamic Formalism: Mean-field Phase Transitions, Large Deviations and Bogoliubov's Variational Principle

Abstract

Let $\hat{I}\text{c◯}=\{1,2,\dots ,d{\}}^{\mathbb{N}}$, $T$ be the shift acting on $\hat{I}\text{c◯}$, $\mathcal{P}(T)$ the set of $T$-invariant probabilities. Given a Hölder potential $A$ and a continuous function $F$, we investigate the probabilities $\ddot{I}{}_{F,A}$ that are maximizers of the nonlinear pressure ${\mathfrak{P}}_{F,A}:={\sup }_{\ddot{I}\in \mathcal{P}(T)}\{F(\int A(x)\ddot{I}(\mathrm{d}x))+h(\ddot{I})\}.$ $\ddot{I}{}_{F,A}$} is called a nonlinear equilibrium; a nonlinear phase transition occurs when there is more than one. In the case $F$\ is convex or concave, we combine Varadhan's lemma and Bogoliubov's variational principle to characterize them via the linear pressure problem and self-consistency conditions. Let $\hat{I}¼\in \mathcal{P}(T)$ be the maximal entropy measure, $\ddot{I}{†}_n(x)=n^{-1}(\ddot{I}†(x)+\ddot{I}†(T(x))+\cdots +\ddot{I}†(T^{n-1}(x)))$ and ${\hat{I}}^2>0$.}\newline (I) We also consider the limit measure $\mathfrak{m}$ on $\hat{I}\text{c◯}$, so that $\forall \ddot{I}ˆ\in C(\hat{I}\text{c◯})$, $\int \ddot{I}ˆ(x)\mathfrak{m}(\mathrm{d}x)={\lim }_{n\to \infty }\frac{\int \ddot{I}ˆ(x)e^{\frac{{\hat{I}}^{2}n}{2}{A}_n((x{)}^2}\hat{I}¼(\mathrm{d}x)}{\int e^{\frac{{\hat{I}}^{2}n}{2}{A}_n((x{)}^2}\hat{I}¼(\mathrm{d}x)}.$ We call $\mathfrak{m}$ a \textit{quadratic mean-field Gibbs probability (II) Via subsequences $n_k$, $k\in \mathbb{N}$, we study the limit measure $\mathfrak{M}$ on $\hat{I}\text{c◯}$, so that $\forall \ddot{I}ˆ\in C(\hat{I}\text{c◯})$, $\int \ddot{I}ˆ(x)\mathfrak{M}(\mathrm{d}x)={\lim }_{k\to \infty }\frac{\int \ddot{I}{ˆ}_{n_k}(x)e^{\frac{{\hat{I}}^2{n}_k}{2}{A}_{n_k}(x{)}^2}\hat{I}¼(\mathrm{d}x)}{\int e^{\frac{{\hat{I}}^2{n}_k}{2}{A}_{n_k}(x{)}^2}\hat{I}¼(\mathrm{d}x)}.$ We call $\mathfrak{M}$ a quadratic mean-field equilibrium probability; it is shift-invariant. Explicit examples are given.