Nonlinear Thermodynamic Formalism: Mean-field Phase Transitions, Large Deviations and Bogoliubov's Variational Principle
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912698437992448 |
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| author | Bru, Jean-Bernard Pedra, Walter de Siqueira Lopes, Artur O. |
| author_facet | Bru, Jean-Bernard Pedra, Walter de Siqueira Lopes, Artur O. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_06975 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonlinear Thermodynamic Formalism: Mean-field Phase Transitions, Large Deviations and Bogoliubov's Variational Principle Bru, Jean-Bernard Pedra, Walter de Siqueira Lopes, Artur O. Dynamical Systems Statistical Mechanics Mathematical Physics Probability 37D35, 82B30, 82C26 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. |
| title | Nonlinear Thermodynamic Formalism: Mean-field Phase Transitions, Large Deviations and Bogoliubov's Variational Principle |
| topic | Dynamical Systems Statistical Mechanics Mathematical Physics Probability 37D35, 82B30, 82C26 |
| url | https://arxiv.org/abs/2511.06975 |