Ergodicity

Ergodicity

In mathematics, the term ergodic is used to describe a dynamical system which, broadly speaking, has the same behavior averaged over time as averaged over space. In physics the term is used to imply that a system satisfies the ergodic hypothesis of thermodynamics.

Contents

Etymology

The word ergodic is derived from the Greek words έργον and οδός, work and path. This was chosen by Boltzmann while working on a problem in statistical mechanics.[1]

Formal definition

Let (X,\; \Sigma ,\; \mu\,) be a probability space, and T:X \to X be a measure-preserving transformation. We say that T is ergodic with respect to μ (or alternatively that μ is ergodic with respect to T) if one of the following equivalent definitions is true: [2]

  • for every  E \in \Sigma with T^{-1}(E)=E\, either \mu(E)=0\, or \mu(E)=1\,.
  • for every  E \in \Sigma with \mu(T^{-1}(E)\bigtriangleup E)=0 either \mu(E)=0\, or \mu(E)=1\, (where \bigtriangleup denotes the symmetric difference).
  • for every  E \in \Sigma with positive measure we have \mu(\cup_{n=1}^\infty T^{-n}E) = 1.
  • for every two sets E and H of positive measure, there exists an n > 0 such that \mu(T^{-n}E\cap H)>0.

Measurable flows

These definitions have natural analogues for the case of measurable flows and, more generally, measure-preserving semigroup actions. Let {Tt} be a measurable flow on (X, Σ, μ). An element A of Σ is invariant mod 0 under {Tt} if

\mu(T^{t}(A)\bigtriangleup A)=0

for each tR. Measurable sets invariant mod 0 under a flow or a semigroup action form the invariant subalgebra of Σ, and the corresponding measure-preserving dynamical system is ergodic if the invariant subalgebra is the trivial σ-algebra consisting of the sets of measure 0 and their complements in X.

Markov chains

In a Markov chain, a state i is said to be ergodic if it is aperiodic and positive recurrent. If all states in a Markov chain are ergodic, then the chain is said to be ergodic.

Ergodic decomposition

Conceptually, ergodicity of a dynamical system is a certain irreducibility property, akin to the notions of irreducible representation in algebra and prime number in arithmetic. A general measure-preserving transformation or flow on a Lebesgue space admits a canonical decomposition into its ergodic components, each of which is ergodic.

See also

Notes

  1. ^ Walters 1982, §0.1, p. 2
  2. ^ Walters 1982, §1.5, p. 27

References

  • Walters, Peter (1982), An Introduction to Ergodic Theory, Springer, ISBN 0387951520 
  • Brin, Michael; Garrett, Stuck (2002), Introduction to Dynamical Systems, Cambridge University Press, ISBN 0521808413 

External links


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • ergodicity — ergodiškumas statusas T sritis fizika atitikmenys: angl. ergodicity vok. Ergodizität, f rus. эргодичность, f pranc. ergodicité, f …   Fizikos terminų žodynas

  • ergodicity condition — ergodiškumo sąlyga statusas T sritis fizika atitikmenys: angl. ergodic condition; ergodicity condition vok. Ergodenbedingung, f rus. условие эргодичности, n pranc. condition d’ergodicité, f; condition ergodique, f …   Fizikos terminų žodynas

  • ergodicity — noun see ergodic …   New Collegiate Dictionary

  • ergodicity — noun a) The condition of being ergodic b) The extent to which something is ergodic Syn: stochasticity …   Wiktionary

  • ergodicity — er·go·dic·i·ty …   English syllables

  • ergodicity — noun an attribute of stochastic systems; generally, a system that tends in probability to a limiting form that is independent of the initial conditions • Hypernyms: ↑randomness, ↑haphazardness, ↑stochasticity, ↑noise * * * noun see ergodic * * *… …   Useful english dictionary

  • ergodic — ergodicity /err geuh dis i tee/, n. /err god ik/, adj. Math., Statistics. of or pertaining to the condition that, in an interval of sufficient duration, a system will return to states that are closely similar to previous ones: the assumption of… …   Universalium

  • Ergodic theory — is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics. A central concern of ergodic theory is the behavior of a dynamical …   Wikipedia

  • Critical phenomena — In physics, critical phenomena is the collective name associated with the physics of critical points. Most of them stem from the divergence of the correlation length, but also the dynamics slows down. Critical phenomena include scaling relations… …   Wikipedia

  • Equipartition theorem — [ Thermal motion of an α helical peptide. The jittery motion is random and complex, and the energy of any particular atom can fluctuate wildly. Nevertheless, the equipartition theorem allows the average kinetic energy of each atom to be computed …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”