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

TL;DR

The paper develops a nonlinear thermodynamic formalism for symbolic dynamics, defining the nonlinear pressure $\mathfrak{P}_{F,A}=\sup_{\rho\in\mathcal{P}(T)}\{ F(\rho(A))+h(\rho)\}$ and studying when nonlinear equilibrium probabilities are unique. By bridging large-deviation theory (via $c$, $\hat c$, and $I_A$) with Bogoliubov's variational principle, it derives self-consistency conditions that identify nonlinear equilibria as linear equilibria for self-consistent effective potentials, with explicit treatment of convex and concave $F$. The quadratic case $F(x)=\beta x^{2}/2$ is analyzed in depth, revealing multiple nonlinear equilibria and phase transitions, and a mean-field free-energy framework that recasts the nonlinear problem in an affine variational form. The work provides explicit examples—including (anti)ferromagnetic-type and generalized Curie-Weiss models—and develops quadratic mean-field Gibbs probabilities, culminating in a tilting LDP property that connects phase transitions to the exponential tilting of large deviations.

Abstract

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

Figures (5)

• Figure 1: In blue is the graph of $t \to R_1(t)$ and in yellow is the graph of the identity. The two graphs intersect just at $t=0$; the only case which would correspond to $p^{\prime \prime}(0)=1$. No non-zero point satisfying $p^{\prime }(t)=t$ when $u=1$.
• Figure 2: In blue the graph of $t \to R_{1.2}(t)$ and in yellow the graph of the identity. Excluding $t=0$, we get two other symmetric solutions $t_{0}$ and $-t_{0}$ of the equation $t = R_{1.2}(t)$, when $u=1.2>1$.
• Figure 3: In blue the graph of $t\rightarrow R_{0.8}(t)$ and in yellow the graph of the identity. The two graphs intersect just at $t=0.$ This corresponds to the case where $u<1$, here when $u=0.8$.
• Figure 4: For $\beta =0.6$ and for $A$ as in \ref{['go']} we show: the blue line is the graph of $t\rightarrow \beta t$, the yellow curve is the graph of $t\rightarrow \frac{d}{dt}P(t\beta A)$, the green curve is the graph of $t\rightarrow \varphi (t)$ and the red curve is the graph of $t\rightarrow \varphi ^{\prime \prime }(t)$. The value $t=3$ gives an exact parameter where the self-consistency condition is true.
• Figure 5: Given the potential $\beta\, A(x)=\beta\,\frac{J}{2} \sum_{n=1}^\infty 2^{-n} x_n$, $F(x)=\frac{\beta J}{2} x^2$, $J=2$, and the maximal entropy measure $\mu$, we show above the graph of the function $y\to\mathcal{I}^{A, F}= I_{A}(y) - \, u_{A,J,\beta,0} \,\,\,y^2= I_{A}(y) - \,\frac{\beta \, 2}{2} \,\,\,y^2$, when $\beta = \frac{ 4^{1/3} + 0.2}{2}$.

Theorems & Definitions (36)

• Definition 2.1
• Theorem 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.5
• Remark 2.6
• Remark 2.7
• Proposition 2.8
• Remark 2.9
• Remark 2.10