Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

How is the theory of ergodic measure-preserving transformations related to ergodicity in the physical sense (which I understood as, very very roughly speaking, that a physical system is called ergodic if "averaging" over "states" of the physical system equals the "average" over time)?

I am sorry maybe the question is a bit unspecific for now, but I guess it's still of interest also for others who are about to dive into the subject.

share|cite|improve this question
up vote 3 down vote accepted

They couldn't be related more closely: they're exactly the transformations for which the physical average equals the time average almost everywhere.

Specifically, they observe Birkhoff's ergodic theorem, that $\int_X fd\mu=\lim_{n\rightarrow \infty} \frac{1}{n}\sum_{k=0}^{n-1}f(T^kx), \mu$-almost everywhere, for any integrable $f$. A nice special case is if $f$ is the characteristic function of some subspace of $A$, in which case this says that the sequence $T^kx$ gets into $A$ proportionally often to the measure of $A$ in $X$ for just about every $x$.

share|cite|improve this answer
You mean $\mathbb{P}((T^k \circ X) \in A) = \mathbb{P}(A)$? And with $k \to X_k = T^k X$, $\mathbb{N}$ is interpreted as (discrete) time? – Suedklee Aug 2 '12 at 19:31
If $f = \chi_{A}$ is the characteristic function of subspace $A$, the Birkhoff sum $\frac{1}{n}\sum_{k=0}^{n-1}f(T^kx)$ is precisely $|\{k \in [\![ 0, n-1 ]\!] \, | \, T^k(x) \in A\}|/n$, that is: the proportion of time the orbit of $x$ spent in $A$ during the first $n$ seconds. That this proportion tends to the relative measure of $A$ is the physical meaning of ergodicity, IIRC. – PseudoNeo Aug 2 '12 at 20:18
@Suedklee: if by $X$ you mean the whole space, then $\mathbb{P}(T^K\circ X)=\mathbb{P}(A)$ is just measure invariance, so yes, if that's what you were going for. My formulation does have $\mathbb{N}$-value time, which you could generalize to flows by taking an integral instead of a sum. – Kevin Carlson Aug 2 '12 at 21:06
The whole space should $(\Omega, \mathcal{F}, \mathbb{P})$ as usual in probability theory - I interpreted your small-x $x$ as a random variable which is usually denoted in capital letters, so I mean a random variable $X$. What do you mean by $x$? – Suedklee Aug 3 '12 at 7:09
With apologies for notational differences, I had $x$ as a point in $\Omega$, and $f$ as an $L^1$ function (complex or real valued) on that space. Your statement might hold, but you need an interpretation of "almost all" random variables. – Kevin Carlson Aug 3 '12 at 12:20

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.