User:Eugene M. Izhikevich/Proposed/Pressure and equilibrium states

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Let \(T:X\to X\) be a continuous transformation of a compact metric space \((X,d)\ .\) Let \(C^0(X,\mathbb{R})\) denote the Banach algebra of real-valued continuous functions of \(X\) equipped with the supremum norm. The topological pressure of \(T\) will be a map \(P(T,\cdot):C^0(X,\mathbb{R})\to\mathbb{R}\cup\{\infty\}\ .\) It contains topological entropy in the sense that \(P(T,0)=h(T)\) where \(0\) denotes the member of \(C^0(X,\mathbb{R})\) identically equal to zero.



Equilibrium States in Finite Systems

The following elementary setting contains germs of notions and results described hereafter.

Let \(\Omega\) be a finite set, a configuration space without any specific structure.

A state is a probability vector \(\mu=(\mu(\omega)| \omega\in\Omega)\ .\) The set of states is denoted by \(\mathcal{M}\ .\)

The entropy of the state \(\mu\) is defined as \(H(\mu):=-\sum_{\omega\in \Omega} \mu(\omega) \log\mu(\omega)\ .\)

Each configuration \(\omega\in\Omega\) is assigned an energy value \(u(\omega)\in\mathbb{R}\) such that in state \(\mu\) the system has mean energy \(\mu(u):=\sum_{\omega\in\Omega} \mu(\omega)u(\omega)\ .\)

\(Z(\beta):=\sum_{\omega\in\Omega} \exp(\beta u(\omega))\) is the partition function of \(u\) where \(\beta\in\mathbb{R}\ .\)

For each \(\beta\in\mathbb{R}\) the Gibbs measure \(\mu_\beta\) on \(\Omega\) is defined by \[ \mu_\beta:=\frac{1}{Z(\beta)} \exp(\beta u(\omega)). \] The following theorem is an elementary prototype of a much more general statement given later on.

Variational principle(elementary version):

Each Gibbs measure \(\mu_\beta\) with \(\beta\in\mathbb{R}\) satisfies \[ H(\mu_\beta)+\mu_\beta(\beta u) = \log Z(\beta)=\sup\{H(\nu)+\nu(\beta u)\big|\ \nu \in\mathcal{M}\}. \] A measure \(\nu\) for which this supremum is attained is called an equilibrium state for \(\beta u\ .\) Thus Gibbs measures are equilibrium states. In fact, \(\mu_\beta\) is the only equilibrium state for \(\beta u\ .\)

Proof of the elementary version of the variational principle

Topological Pressure

A set \(E \subset X\) is said to be \((n,\varepsilon)\)-separated, if for every \(x, y\in E\) with \(x\neq y\) there is \(i\in\{0,1,\dots,n-1\}\) such that \(d(T^ix,T^iy)\ge\varepsilon\ .\) Let \(s(n,\varepsilon)\) be the maximal cardinality of an \((n,\varepsilon)\)-separated set in \(X\ .\) Again, by compactness, this number is always finite. For \(f\in C^0(X,\mathbb{R})\ ,\) \(x\in X\) and \(n\in \mathbb{N}_0\) define \(S_n f(x):= \sum_{i=0}^{n-1} f(T^i(x))\ .\)

For \(\varepsilon>0\ ,\) \(n\in\mathbb{N}\ ,\) let \[ Z(T,f,\varepsilon,n):=\sup\left\{\sum_{x\in E} e^{S_n f(x)}\ \big| \ E\subset X\;\mathrm{is}\;(n,\varepsilon)-\mathrm{separated}.\right\} \] Then \[ P(T,f):=\lim_{\varepsilon\to 0}\limsup_{n\to\infty}\frac{1}{n}\log Z(T,f,\varepsilon,n) \] is called the topological pressure of \(T\) with respect to \(f\ .\)

Properties of Pressure

We give some properties of \(P(T,\cdot):C^0(X,\mathbb{R})\to\mathbb{R}\cup\{\infty\}\ .\)

If \(f,g \in C^0(X,\mathbb{R})\ ,\) \(\varepsilon>0\) and \(c\in\mathbb{R}\) then the following are true.

  • \(f\leq g\) implies \(P(T,f)\leq P(T,g)\ .\) In particular \(h(T)+\inf f \leq P(T,f)\leq h(T)+ \sup f\ .\)
  • \(P(T,\cdot)\) is either finite valued or constantly \(\infty\ .\)
  • \(|P(T,f)-P(T,g)|\leq \|f-g\|\ .\)
  • \(P(T,\cdot)\) is convex.
  • \(P(T,f+c)=P(T,f)+c\ .\)
  • \(P(T,f+g\circ T-g)=P(T,f)\)

Now we look at how \(P(T,\cdot)\) depends on \(T\ .\)

  • If \(k>0\) \(P(T^k,S_k f)=k P(T,f)\ .\)
  • If \(T\) is a homeomorphism \(P(T^{-1},f)\ .\)
  • If \(Y\) is a closed subset of \(X\) with \(TY\subset Y\) then \(P(T|_{Y},f|_{Y})\leq P(T,f)\ .\)
  • If \(T_i:X_i\to X_i\) (\(i=1,2\)) is a continuous map of a compact of a compact metric space \((X_i,d_i)\) and if \(\phi:X_1\to X_2\) is a surjective continuous map with \(\phi\circ T_1=T_2\circ\phi\) then \(P(T_2,f)\leq P(T_1,f\circ\phi)\) \(\forall f \in C^{0}(X_2,\mathbb{R})\ .\) If \(\phi\) is a homeomorphism then \(P(T_2,f)=P(T_1,f\circ\phi)\ .\)

The Variational Principle

Denote by \(\mathcal{M}(X,T)\) the set of \(T\)-invariant probability measures on \(X\) (equipped with weak\(^*\) or vague topology). We have the following theorem:

Let \(T:X\to X\) be a continuous transformation of a compact metric space \((X,d)\) and let \(f\in C^0(X,\mathbb{R})\ .\) Then \[ P(T,f)=\sup\left\{h_\mu(T)+\int f d\mu\ \Big| \ \mu\in \mathcal{M}(X,T)\right\} \] where \(h_\mu(T)\) is the Kolmogorov-Sinai entropy of \(\mu\ .\)

Equilibrium States

The variational principle gives a natural way of selecting members of \(\mathcal{M}(X,T)\ .\) The concept extends the idea of measure with maximal entropy.

Let \(T:X\to X\) be a continuous map of a compact metric space \(X\) and let \(f \in C^0(X,\mathbb{R})\ .\)

A member of \(\mathcal{M}(X,T)\) is called an equilibrium state for \(f\) if \(P(T,f)=h_\mu(T)+\int f d\mu\ .\)

Let \(\mathcal{E\!S}(f)\) denote the collection of all equilibrium states for \(f\ .\) Notice that this set can be empty but if the entropy map is upper semi-continuous then \(\mathcal{E\!S}(f)\) is a non-empty compact subset of \(\mathcal{M}(X,T)\ .\)


  • \(\mathcal{E\!S}(f)\) is a convex set.
  • If \(f,g\in C^0(X,\mathbb{R})\) and if there exists \(c\in\mathbb{R}\) such that \(f-g-c\) belongs to the closure of the set \(\{h\circ T-h| h\in C^0(X,\mathbb{R})\) in \(C^0(X,\mathbb{R})\ ,\) then \(\mathcal{E\!S}(f)=\mathcal{E\!S}(g)\ .\)

The notion of equilibrium state is tied in with the notion of tangent functional to the convex function \(P(T,\cdot):C^0(X,\mathbb{R})\to\mathbb{R}\ .\) See Tangent Functional to the Pressure.

Equilibrium States on shift spaces

We consider a configuration space \(\Omega:=A^{\mathbb{N}}\ .\)

Application to Differentiable Dynamics

Connection with Statistical Mechanics

Recommended reading

[B] R. Bowen: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Second revised edition. Lecture Notes in Mathematics, 470. Springer-Verlag, Berlin, 2008

[K] G. Keller: Equilibrium States in Ergodic Theory. London Mathematical Society Student Texts 42 Cambridge University Press, 1998

[R1] D. Ruelle: Thermodynamic Formalism: The Mathematical Structures of Equilibrium Statistical Mechanics. Second revised edition. Cambridge Mathematical Library. Cambridge University Press, 2004

[W] P. Walters: An introduction to ergodic theory. Graduate Texts in Mathematics. Springer, 2000.

[Z] M. Zinsmeister: Thermodynamic Formalism and Holomorphic Dynamical Systems. SMF/AMS Texts and Monographs 2 (2000) [Publié en français dans le numéro 4 (1996) de la série Panoramas et Synthèses]

Further reading

[G] H.-O. Georgii: Gibbs measures and phase transitions. Studies in Mathematics 9. De Gruyter, Berlin, 1988

[I] R.B. Israel: Convexity in the theory of lattice gases. Princeton Series in Physics. Princeton University Press, 1979

[R2] D. Ruelle: Statistical Mechanics: Rigorous Results. World Scientific, 1999 [First edition: Benjamin, N.Y., 1969]

[S] Ya. Sinai: Gibbs measures in ergodic theory. Russian Mathematical Surveys (1972) Vol. 27 (4), 21-69.

See also

Topological entropy, Kolmogorov-Sinai Entropy, Hyperbolic dynamics, Anosov diffeomorphism, Axiom A systems, Symbolic dynamics, Sinai-Ruelle-Bowen measure

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