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Pressure and Equilibrium States in Ergodic Theory

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Mathematics of Complexity and Dynamical Systems

Article Outline

Glossary

Definition of the Subject

Introduction

Warming Up: Thermodynamic Formalism for Finite Systems

Shift Spaces, Invariant Measures and Entropy

The Variational Principle: A Global Characterization of Equilibrium

The Gibbs Property: A Local Characterization of Equilibrium

Examples on Shift Spaces

Examples from Differentiable Dynamics

Nonequilibrium Steady States and Entropy Production

Some Ongoing Developments and Future Directions

Bibliography

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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 ϕ.

Pressure:

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

Gibbs state:

In many cases, equilibrium states have a local structure that is determined by the local properties of the potential ϕ. 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. (2012). Pressure and Equilibrium States in Ergodic Theory . In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_90

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