Ergodicity

Prerequisites: Statistical Mechanics, Phase Space. Full path (10 pages)

Intuitive
Concrete
Abstract

Intuitive

A billiard ball in a rectangle with two circular ends: slight changes in initial conditions make such a big difference that eventually it could be anywhere. To be precise, we say its motion is "ergodic".

ergodicity.mp4

For the precise definition of "ergodic" read my diary:

(link: https://tinyurl.com/baez-bunimovich) tinyurl.com/baez-bunimovich

great example of the evolution of apparent randomness from a completely deterministic process

Yes, it sort of demystifies how that works.

https://mobile.twitter.com/johncarlosbaez/status/1023252524248137728

Concrete

Statistical mechanics replaces the long-time behaviour of a system by an average over an ensemble. Ergodicity is the property that justifies this replacement.

Consider a Hamiltonian system with phase space coordinates $x = (q, p)$ and flow $\phi_t$ . Energy is conserved, so a trajectory stays on the energy surface $\Sigma_E = \{x : H(x) = E\}$ . Liouville's theorem implies that the microcanonical measure $d\mu_E = \frac{\delta(H(x) - E)\, d^{n}q\, d^{n}p}{\Omega(E)}, \qquad \Omega(E) = \int \delta(H(x) - E)\, d^{n}q\, d^{n}p,$ is invariant under the flow. For an observable $f$ define the time average along the trajectory starting at $x$ , $\bar f(x) = \lim_{T\to\infty} \frac{1}{T}\int_0^T f(\phi_t x)\, dt ,$ and the ensemble average $\langle f \rangle = \int f\, d\mu_E$ . The system is ergodic on $\Sigma_E$ if $\bar f(x) = \langle f\rangle$ for every integrable $f$ and for almost every starting point $x$ (all except a set of measure zero). An equivalent condition: every measurable subset of $\Sigma_E$ that is invariant under the flow has measure $0$ or $1$ . A trajectory then cannot be confined to a region of nonzero but incomplete measure.

Example: rotation on a circle. Take $x \in [0,1)$ and the discrete map $x_{n+1} = x_n + \alpha \bmod 1$ , which preserves the length measure $dx$ . Expand an observable in Fourier modes, $f(x) = \sum_k c_k e^{2\pi i k x}$ . For $k \neq 0$ the time average of one mode is a geometric sum, $\frac{1}{N}\sum_{n=0}^{N-1} e^{2\pi i k (x + n\alpha)} = \frac{e^{2\pi i k x}}{N}\,\frac{1 - e^{2\pi i k N\alpha}}{1 - e^{2\pi i k\alpha}} \xrightarrow{N\to\infty} 0 ,$ provided $k\alpha$ is not an integer. So if $\alpha$ is irrational, every mode except $k = 0$ averages to zero and $\bar f = c_0 = \int_0^1 f\, dx$ : the map is ergodic. If $\alpha = p/q$ is rational, every orbit is periodic with $q$ points and the map is not ergodic.

Physical systems. The same argument shows that two uncoupled harmonic oscillators with frequencies $\omega_1, \omega_2$ are not ergodic on their energy surface, since each energy is conserved separately. Each trajectory fills a torus, densely if $\omega_1/\omega_2$ is irrational, but the torus has measure zero on $\Sigma_E$ . In general a system with additional conserved quantities, such as an integrable system, is not ergodic on the energy surface. Chaotic billiards such as the Bunimovich stadium mentioned above are proven to be ergodic. For systems close to integrable, KAM theory shows that a set of invariant tori of positive measure survives, so these systems are not ergodic either.

Abstract

Setting. A measure-preserving dynamical system is a probability space $(X, \mathcal F, \mu)$ with a measurable map $T: X \to X$ (or a flow $\phi_t$ ) such that $\mu(T^{-1}A) = \mu(A)$ for all $A \in \mathcal F$ . It is ergodic if every invariant set, $T^{-1}A = A$ , has $\mu(A) \in \{0, 1\}$ .

Ergodic theorems. Birkhoff's pointwise ergodic theorem states that for every $f \in L^1(\mu)$ the averages $\frac{1}{N}\sum_{n=0}^{N-1} f(T^n x)$ converge for $\mu$ -almost every $x$ to the conditional expectation $\mathbb E[f \mid \mathcal I](x)$ , where $\mathcal I$ is the $\sigma$ -algebra of invariant sets. For an ergodic system $\mathcal I$ is trivial and the limit is $\int_X f\, d\mu$ . Von Neumann's mean ergodic theorem is the Hilbert space version: the Koopman operator $U f = f \circ T$ is an isometry of $L^2(\mu)$ (unitary if $T$ is invertible), and the averages $\frac1N\sum_{n<N} U^n f$ converge in $L^2$ to the orthogonal projection of $f$ onto the subspace of $U$ -invariant functions. Ergodicity is equivalent to this subspace being the constants, that is, the eigenvalue $1$ of $U$ being simple.

Invariant measures. The set of $T$ -invariant probability measures is convex. A measure is ergodic if and only if it is an extreme point of this set, and every invariant measure decomposes into ergodic ones (ergodic decomposition, under mild regularity assumptions on $X$ ). In statistical mechanics this mirrors the decomposition of a Gibbs state into pure phases.

Stronger properties. Ergodicity is the weakest in a hierarchy of randomness properties: $\text{Bernoulli} \Rightarrow \text{K-system} \Rightarrow \text{mixing} \Rightarrow \text{weak mixing} \Rightarrow \text{ergodic}.$ Mixing means $\mu(T^{-n}A \cap B) \to \mu(A)\mu(B)$ for all measurable $A, B$ , so correlations decay and an initial non-equilibrium distribution relaxes towards $\mu$ in the weak sense. Ergodicity alone only controls time averages. The circle rotation by an irrational angle is ergodic but not weakly mixing, because $U$ has the eigenfunctions $e^{2\pi i k x}$ with eigenvalues $e^{2\pi i k\alpha}$ . Weak mixing is equivalent to $U$ having no eigenfunctions other than the constants.

Ergodicity

Intuitive

Concrete

Abstract

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