Ergodic theory
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In mathematics, a measure-preserving transformation T on a probability space is said to be ergodic if the only measurable sets invariant under T have measure 0 or 1. An older term was metrically transitive. Ergodic theory, the study of ergodic transformations, grew out of an attempt to prove the ergodic hypothesis of statistical physics.
Consider the "time average" of a well-behaved function f. This is defined as the average (if it exists) over iterations of T starting from some initial point x.
- <math> \hat f(x) = \lim_{n\rightarrow\infty}\;
\frac{1}{n} \sum_{k=0}^n f\left(T^k x\right) <math>
Consider also the "space average" or "phase average" of f, defined as
- <math> \bar f = \int f\,d\mu <math>
where μ is the measure of the probability space.
In general the time mean and space mean may be different. For an ergodic transformation, the time mean is equal to the space mean almost everywhere. This is the celebrated ergodic theorem, in an abstract form due to George David Birkhoff.
The time spent in a measurable set A is called the sojourn time. An immediate consequence of the ergodic theorem is that the measure of A is equal to the mean sojourn time.
- <math> \mu(A) = \int \chi_A\, d\mu
= \lim_{n\rightarrow\infty}\; \frac{1}{n} \sum_{k=0}^n \chi_A\left(T^k x\right) <math>
where χA is the indicator function on A.
Let the occurrence times of a measurable set A be defined as the set k1, k2, k3, ..., of times k such that Tk(x) is in A, sorted in increasing order. The differences between consecutive occurrence times Ri = ki − ki−1 are called the recurrence times of A. Another consequence of the ergodic theorem is that the average recurrence time of A is inversely proportional to the measure of A, assuming that the initial point x is in A, so that k0 = 0.
- <math> \frac{R_1 + \cdots + R_n}{n} \rightarrow \frac{1}{\mu(A)}
\quad\mbox{(almost surely)}<math>
(See almost surely.) That is, the smaller A is, the longer it takes to return to it.
References
- Vladimir Igorevich Arnol'd and André Avez, Ergodic Problems of Classical Mechanics. New York: W.A. Benjamin. 1968.
- Leo Breiman, Probability. Original edition published by Addison-Wesley, 1968; reprinted by Society for Industrial and Applied Mathematics, 1992. ISBN 0-89871-296-3. (See Chapter 6.)
- Peter Walters, An introduction to ergodic theory, Springer, New York, 1982, ISBN 0-387-95152-0.ja:エルゴード理論