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Pressure and Equilibrium States in Ergodic Theory

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Encyclopedia of Complexity and Systems Science

Definition of the Subject

Gibbs and equilibrium states of one-dimensional lattice models in statistical physics play a prominent role in the statisticaltheory of chaotic dynamics. They first appear in the ergodictheory of certain differentiable dynamical systems, called “uniformly hyperbolic systems”, mainly Anosov and Axiom A diffeomorphisms (andflows). The central idea is to “code” the orbits of these systems into (infinite) symbolic sequences of symbols by following their history ona finite partition of their phase space. This defines a nice shift dynamical system called a subshift of finite type or a topologicalMarkov chain. Then the construction of their “natural” invariant measures and the study of their properties are carried out at the symboliclevel by constructing certain equilibrium states in the sense of statistical mechanics which turn out to be also Gibbs states. The study of uniformlyhyperbolic systems brought out several...

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Abbreviations

Dynamical system:

In this article: a continuous transformation T of a compact metric space X. For each \( { x\in X } \), the transformation T generates a trajectory \( { (x,Tx,T^2x,\dots) } \).

Invariant measure:

In this article: a probability measure µ on X which is invariant under the transformation T, i.?e., for which \( { \langle f\circ T,\mu\rangle=\langle f,\mu\rangle } \) for each continuous \( { f\colon X\to\mathbb{R} } \). Here \( { \langle f,\mu\rangle } \) is a short-hand notation for \( { \int_X f\,\mathrm{d}\mu } \). The triple \( { (X,T,\mu) } \) is called a measure-preserving dynamical system.

Ergodic theory:

Ergodic theory is the mathematical theory of measure-preserving dynamical systems.

Entropy:

In this article: the maximal rate of information gain per time that can be achieved by coarse-grained observations on a measure-preserving dynamical system. This quantity is often denoted \( { h(\mu) } \).

Equilibrium state:

In general, a given dynamical system \( { T\colon X\to X } \) admits a huge number of invariant measures. Given some continuous \( { \phi \colon X\to\mathbb{R} } \) (“potential”), those invariant measures which maximize a functional of the form \( { F(\mu)=h(\mu)+\langle \phi,\mu\rangle } \) are called “equilibrium states” for ?.

Pressure:

The maximum of the functional \( { F(\mu) } \) is denoted by \( { P(\phi) } \) and called the “topological pressure” of ?, or simply the “pressure” of ?.

Gibbs state:

In many cases, equilibrium states have a local structure that is determined by the local properties of the potential ?. They are called “Gibbs states”.

Sinai–Ruelle–Bowen measure:

Special equilibrium or Gibbs states that describe the statistics of the attractor of certain smooth dynamical systems.

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Author information

Authors and Affiliations

  1. Centre de Physique Théorique, CNRS/École Polytechnique, Palaiseau, France

    Jean-René Chazottes

  2. Department Mathematik, Universität Erlangen-Nürnberg, Erlangen, Germany

    Gerhard Keller

Authors
  1. Jean-René Chazottes
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  2. Gerhard Keller
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Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Chazottes, JR., Keller, G. (2009). Pressure and Equilibrium States in Ergodic Theory . In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_414

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