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

Let $\Omega =\{1,2,\ldots ,d\}^{\mathbb{N}}$, $T$ be the shift acting on $\Omega$, $\mathcal{P}(T)$ the set of $T$-invariant probabilities. Given a H\"{o}lder potential $A$ and a continuous function $F$, we investigate the probabilities $\rho _{F,A}$ that are maximizers of the nonlinear pressure $\mathfrak{P}_{F,A}:=\sup_{\rho \in \mathcal{P}(T)}\{ F(\int A(x)\rho (\mathrm{d}x))+h(\rho )\} .$ $\rho _{F,A}$} 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 $\mu \in \mathcal{P}(T)$ be the maximal entropy measure, $\varphi _{n}(x)=n^{-1}(\varphi (x)+\varphi (T(x))+\cdots +\varphi (T^{n-1}(x)))$ and $\beta >0$.}\newline (I) We also consider the limit measure $\mathfrak{m}$ on $\Omega$, so that $\forall \psi \in C(\Omega )$, $\int \psi (x)\,\mathfrak{m}\,( \mathrm{d}x)\,\,=\lim_{n\rightarrow \infty }\frac{\,\int \,\psi (x)\,\,\,e^{ \frac{\beta n}{2}\,\,A_{n}((x)^{2}}\,\,\mu \,(\mathrm{d}x)\,}{\int e^{\frac{ \beta n}{2}\,\,A_{n}((x)^{2}}\mu \,(\mathrm{d}x)\,\,}.$ 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 $\Omega$, so that $\forall \psi \in C(\Omega )$, $\int \psi (x)\mathfrak{M}(\mathrm{d} x)=\lim_{k\rightarrow \infty }\frac{\,\int \psi _{n_{k}}(x)e^{\frac{\beta n_{k}}{2}A_{n_{k}}(x)^{2}}\mu (\mathrm{d}x)}{\int e^{\frac{\beta n_{k}}{2} A_{n_{k}}(x)^{2}}\mu (\mathrm{d}x)}.$ We call $\mathfrak{M}$ a quadratic mean-field equilibrium probability; it is shift-invariant. Explicit examples are given.