Questions tagged [thermodynamic-formalism]
Thermodynamic formalism is the study of equilibrium states, Gibbs measures and topological pressure for dynamical systems.
22 questions
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Information theoretical interpretation of Free Energy
When exploring the concept of free energy from an information-theoretic perspective, I often come across statements like:
"Free energy measures the degree of surprise an agent experiences when ...
- 253
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A (possible) generic spectral property in one dimensional dynamics
Context and Definitions
Consider the interval $I=[0,1]$. We say that $T:I\to I$ satisfies the axiom A (I am following [1]) if:
$T$ has a finite number of hyperbolic periodic attractors; and
defining $...
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Relationship between heat kernel and Maxwell-Boltzmann distribution
There appears to be a connection between the heat kernel and Maxwell-Boltzmann distribution, but I have not seen this in the literature before. I'd appreciate any kind comments or corrections/...
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Seeking for references - Bowen Formula and a link between dimension theory and thermodynamic formalism
I'm needing references - preferably published papers and books - about this subject. I'm relatively new to the state of the art of fractal geometry and am way too inexperienced to seek for myself at ...
- 165
2
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83
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Biased ensemble in the unitary group
I am interested in studying the ensemble of unitary random matrices in $U(L)$ made as follows
$$
\mu(U)=\frac{1}{\mathcal{Z}[\omega]}\mu_{\rm Haar}(U) e^{-\sum_{k=1}^L \sum_{l=1}^N \omega_k |U_{kl}|^2}...
- 121
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Generalized Ising Model
I am in very trouble with a particular expression. I leave the original pages in order to have everything available and what I am goin to leave are the first pages of nine chapter of Non Perturbative ...
1
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1
answer
107
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Lyapunov spectrum($h_{\mathrm{top}}(K(\alpha))$) achieves a positive value somewhere
$\DeclareMathOperator{\top}{\mathrm{top}}$Let $(\Sigma, T)$ be a topologically mixing subshift of finite type and $f:\Sigma \to \mathbb{R}$ be a Hölder continuous map.
Let
$$K(\alpha)=\Big\{x\in \...
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Explanation for why an ideal fluid doesn't have increasing entropy?
The equations of motion for a very simple ideal fluid (specifically a calorically perfect, monatomic, ideal gas) are \begin{align*}\dot{\rho}+\nabla \cdot (\rho u)=0 \;&\text{(mass conservation)} \...
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Factor map between subshifts preserving topological pressure (or measure-theoretic entropy)
Let $G$ be a countable amenable group and let $X,Y$ be subshifts with finite alphabet over $G$. Suppose that $h(X) = h(Y)$ (equal topological entropy). I am interested in continuous factor maps $\pi: ...
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Gurevich's entropy and topological entropy in a countable Markov shift
Good afternoon, I understand that Gurevich's entropy and topological entropy coincide when the countable Markov shift is topologically mixing (right?)
Does anyone know of an example or a reference ...
- 193
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Introduction to information geometry and/or geometric control theory
Some background: I'vebeen searching for a research project to work through my grad studies and I found information geometry like a strong candidate but the amount of work out there is overwhelming. I ...
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Examples of non-uniqueness of the equilibrium states
Let $f:X\rightarrow X$ be an Axiom $A$ diffeomorphism on a compact metric space $X$. Assume that $\phi:X\rightarrow \mathbb{R}$ is Hölder continuous. R. Bowen shows that there is a unique equilibrium ...
- 1,043
4
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1
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331
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Equation of state for hard rods
Some context:
For ideal gases, the thermodynamic equation of state is the well-known:
$$
pV = nRT \tag{1}
$$
where $n$ is the amount of substance, $R$ the universal gas constant and $P,V,T$ are ...
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Continuity of Lyapunov spaces
The following appears in the paper "Continuity properties of entropy" by Newhouse from 1989:
Let $M$ be some smooth Riemannian compact manifold (you may assume boundary-less), and let $f\in Diff^{1+\...
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4
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2
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Can a smooth diffeomorphisms of a Riemannian manifold have only positive Lyapunov exponents?
Let $\mu$ be some ergodic measure of our compact Riemannian manifold $M$, which is preserved by $f\in Diff^{1+\beta}(M)$. Is it possible that all the Lyapunov exponents of $\mu$ will be positive? ...
- 502
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Maximizing entropy under constraints
This question is about an extension of the variational principle in thermodynamical formalism when one adds linear constraints to the measures.
Consider the one-sided shift $\sigma:\mathcal{A}^\...
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Estimates of Hausdorff dimension (and its derivatives)
For example, the cookie cutter maps, say $T:I_1 \cup I_2 \subset [0,1] \to [0,1] $ is a $C^2$ map such that $|T'|>1$ and provided $I_1$ and $I_2$ are disjoint closed intervals and $T(I_i)=[0,1]$. ...
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Is $\text{Bow}(X,T)$ a Banach Space?
Let $X=\{0,1\}^{\mathbb{N}}$ be the sequence space and $T:X\to X$ the left shift mapping. Define the vector space $\text{Bow}(X,T)$ as
$$
\text{Bow}(X,T)=\{f\in C^{0}(X);~\sup_{n\in \mathbb{N}}\sup_{...
user11178
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Using topological pressure to determine a subshift of finite type
I am interested in recognising graphs (or matrices, or subshifts of finite type) using topological pressure. Suppose that we play the following game:
${\bf Step 1:}$ I write down an irreducible n x n ...
- 717
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Multifractal Analysis and Dimension Spectrum of Unions
Consider the classical Multifractal Analysis, and the decomposition of the state space $X$ into level sets
$$X=\bigcup_{\alpha}\left\{x\mid d_\mu(x)=\alpha\right\}\cup\left\{x\mid d_\mu(x) \,\mathrm{...
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Does equality of Hodge star and symplectic star imply Kähler structure?
Question
The question asked is:
On a manifold $M$ equipped with a Riemann metric $g$ and a symplectic structure $\omega$, with $\ast$ the Hodge star and $\ast_s$ the symplectic star, does $\ast=\...
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How to estimate the pressure?
I have a finite collection of diffeomorphisms $g_1,\cdots,g_n$ taking the unit interval $I$ to disjoint subintervals $I_1, I_2,\cdots,I_n$. If $G$ is the semigroup they generate, the limit set of $G$ (...
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