非线性热力学形式：平均场相变、大偏差与 Bogoliubov 变分原理
Nonlinear Thermodynamic Formalism: Mean-field Phase Transitions, Large Deviations and Bogoliubov's Variational Principle
AI总结：
本文研究移位系统上由非线性压力定义的平衡态概率，利用 Varadhan 引理与 Bogoliubov 变分原理刻画凸或凹情形下的非线性相变，并构造二次平均场 Gibbs 概率与移位不变的二次平均场平衡态概率。
摘要：
设 $Ω=\{1,2,\ldots ,d\}^{\mathbb{N}}$，$T$ 为作用于 $Ω$ 的移位映射，$\mathcal{P}(T)$ 为所有 $T$-不变概率的集合。给定一个 Hölder 势 $A$ 和一个连续函数 $F$，我们研究作为非线性压力 $\mathfrak{P}_{F,A}:=\sup_{ρ\in \mathcal{P}(T)}\{ F(\int A(x)ρ(\mathrm{d}x))+h(ρ)\}$ 的极大化子的概率 $ρ_{F,A}$。$ρ_{F,A}$ 被称为非线性平衡态；当存在多于一个非线性平衡态时，就发生非线性相变。在 $F$ 为凸或凹的情形下，我们结合 Varadhan 引理和 Bogoliubov 变分原理，通过线性压力问题和自洽条件来刻画这些非线性平衡态。设 $μ\in \mathcal{P}(T)$ 为最大熵测度，$φ_{n}(x)=n^{-1}(φ(x)+φ(T(x))+\cdots +φ(T^{n-1}(x)))$，且 $β>0$。（I）我们还考虑 $Ω$ 上的极限测度 $\mathfrak{m}$，使得对任意 $ψ\in C(Ω)$，$\int ψ(x)\\,\mathfrak{m}\\,( \mathrm{d}x)\\,\\,=\lim_{n\rightarrow \infty }\frac{\\,\int \\,ψ(x)\\,\\,\\,e^{ \frac{βn}{2}\\,\\,A_{n}((x)^{2}}\\,\\,μ\\,(\mathrm{d}x)\\,}{\int e^{\frac{ βn}{2}\\,\\,A_{n}((x)^{2}}μ\\,(\mathrm{d}x)\\,\\,}.$ 我们称 $\mathfrak{m}$ 为二次平均场 Gibbs 概率。（II）通过子序列 $n_{k}$，$k\in \mathbb{N}$，我们研究 $Ω$ 上的极限测度 $\mathfrak{M}$，使得对任意 $ψ\in C(Ω)$，$\int ψ(x)\mathfrak{M}(\mathrm{d} x)=\lim_{k\rightarrow \infty }\frac{\\,\int ψ_{n_{k}}(x)e^{\frac{βn_{k}}{2}A_{n_{k}}(x)^{2}}μ(\mathrm{d}x)}{\int e^{\frac{βn_{k}}{2} A_{n_{k}}(x)^{2}}μ(\mathrm{d}x)}.$ 我们称 $\mathfrak{M}$ 为二次平均场平衡态概率；它是移位不变的。文中给出了显式例子。
英文摘要：
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, $φ_{n}(x)=n^{-1}(φ(x)+φ(T(x))+\cdots +φ(T^{n-1}(x)))$ 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.
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