Ergodicity

In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies that the average behavior of the system can be deduced from the trajectory of a "typical" point. Equivalently, a sufficiently large collection of random samples from a process can represent the average statistical properties of the entire process. Ergodicity is a property of the system; it is a statement that the system cannot be reduced or factored into smaller components. Ergodic theory is the study of systems possessing ergodicity.

"Ergodic" redirects here. For other uses, see Ergodic (disambiguation).

Ergodic systems occur in a broad range of systems in physics and in geometry. This can be roughly understood to be due to a common phenomenon: the motion of particles, that is, geodesics on a hyperbolic manifold are divergent; when that manifold is compact, that is, of finite size, those orbits return to the same general area, eventually filling the entire space.

Ergodic systems capture the common-sense, every-day notions of randomness, such that smoke might come to fill all of a smoke-filled room, or that a block of metal might eventually come to have the same temperature throughout, or that flips of a fair coin may come up heads and tails half the time. A stronger concept than ergodicity is that of mixing, which aims to mathematically describe the common-sense notions of mixing, such as mixing drinks or mixing cooking ingredients.

The proper mathematical formulation of ergodicity is founded on the formal definitions of measure theory and dynamical systems, and rather specifically on the notion of a measure-preserving dynamical system. The origins of ergodicity lie in statistical physics, where Ludwig Boltzmann formulated the ergodic hypothesis.

History and etymology[edit]

The term ergodic is commonly thought to derive from the Greek words ἔργον (ergon: "work") and ὁδός (hodos: "path", "way"), as chosen by Ludwig Boltzmann while he was working on a problem in statistical mechanics.^[2] At the same time it is also claimed to be a derivation of ergomonode, coined by Boltzmann in a relatively obscure paper from 1884. The etymology appears to be contested in other ways as well.^[3]

The idea of ergodicity was born in the field of thermodynamics, where it was necessary to relate the individual states of gas molecules to the temperature of a gas as a whole and its time evolution thereof. In order to do this, it was necessary to state what exactly it means for gases to mix well together, so that thermodynamic equilibrium could be defined with mathematical rigor. Once the theory was well developed in physics, it was rapidly formalized and extended, so that ergodic theory has long been an independent area of mathematics in itself. As part of that progression, more than one slightly different definition of ergodicity and multitudes of interpretations of the concept in different fields coexist.

For example, in classical physics the term implies that a system satisfies the ergodic hypothesis of thermodynamics,^[4] the relevant state space being position and momentum space.

In dynamical systems theory the state space is usually taken to be a more general phase space. On the other hand in coding theory the state space is often discrete in both time and state, with less concomitant structure. In all those fields the ideas of time average and ensemble average can also carry extra baggage as well—as is the case with the many possible thermodynamically relevant partition functions used to define ensemble averages in physics, back again. As such the measure theoretic formalization of the concept also serves as a unifying discipline. In 1913 Michel Plancherel proved the strict impossibility of ergodicity for a purely mechanical system.^[5]

for every $A\in {\mathcal {B}}$ with $\mu {\mathord {\left(T^{-1}(A)\bigtriangleup A\right)}}=0$ we have $\mu (A)=0$ or $\mu (A)=1\,$ (where $\bigtriangleup$ denotes the );

symmetric difference

for every $A\in {\mathcal {B}}$ with positive measure we have ${\textstyle \mu {\mathord {\left(\bigcup _{n=1}^{\infty }T^{-n}(A)\right)}}=1}$ ;

for every two sets $A,B\in {\mathcal {B}}$ of positive measure, there exists $n>0$ such that $\mu {\mathord {\left(\left(T^{-n}(A)\right)\cap B\right)}}>0$ ;

Every measurable function $f:X\to \mathbb {R}$ with $f\circ T=f$ is constant on a subset of full measure.

in convex Euclidean domains;

Billiards

the of a negatively curved Riemannian manifold of finite volume is ergodic (for the normalised volume measure);

geodesic flow

the on a hyperbolic manifold of finite volume is ergodic (for the normalised volume measure)

horocycle flow

Ergodicity of Markov chains[edit]

The dynamical system associated with a Markov chain[edit]

Let $S$ be a finite set. A Markov chain on $S$ is defined by a matrix $P\in [0,1]^{S\times S}$ , where $P(s_{1},s_{2})$ is the transition probability from $s_{1}$ to $s_{2}$ , so for every $s\in S$ we have ${\textstyle \sum _{s'\in S}P(s,s')=1}$ . A stationary measure for $P$ is a probability measure $\nu$ on $S$ such that $\nu P=\nu$ ; that is ${\textstyle \sum _{s'\in S}\nu (s')P(s',s)=\nu (s)}$ for all $s\in S$ .

Using this data we can define a probability measure $\mu _{\nu }$ on the set $X=S^{\mathbb {Z} }$ with its product σ-algebra by giving the measures of the cylinders as follows: $\mu _{\nu }(\cdots \times S\times \{(s_{n},\ldots ,s_{m})\}\times S\times \cdots )=\nu (s_{n})P(s_{n},s_{n+1})\cdots P(s_{m-1},s_{m}).$

Stationarity of $\nu$ then means that the measure $\mu _{\nu }$ is invariant under the shift map $T\left(\left(s_{k}\right)_{k\in \mathbb {Z} })\right)=\left(s_{k+1}\right)_{k\in \mathbb {Z} }$ .

Criterion for ergodicity[edit]

The measure $\mu _{\nu }$ is always ergodic for the shift map if the associated Markov chain is irreducible (any state can be reached with positive probability from any other state in a finite number of steps).^[24]

The hypotheses above imply that there is a unique stationary measure for the Markov chain. In terms of the matrix $P$ a sufficient condition for this is that 1 be a simple eigenvalue of the matrix $P$ and all other eigenvalues of $P$ (in $\mathbb {C}$ ) are of modulus <1.

Note that in probability theory the Markov chain is called ergodic if in addition each state is aperiodic (the times where the return probability is positive are not multiples of a single integer >1). This is not necessary for the invariant measure to be ergodic; hence the notions of "ergodicity" for a Markov chain and the associated shift-invariant measure are different (the one for the chain is strictly stronger).^[25]

Moreover the criterion is an "if and only if" if all communicating classes in the chain are recurrent and we consider all stationary measures.

Generalisations[edit]

The definition of ergodicity also makes sense for group actions. The classical theory (for invertible transformations) corresponds to actions of $\mathbb {Z}$ or $\mathbb {R}$ .

For non-abelian groups there might not be invariant measures even on compact metric spaces. However the definition of ergodicity carries over unchanged if one replaces invariant measures by quasi-invariant measures.

Important examples are the action of a semisimple Lie group (or a lattice therein) on its Furstenberg boundary.

A measurable equivalence relation it is said to be ergodic if all saturated subsets are either null or conull.

Walters, Peter (1982). . Springer. ISBN 0-387-95152-0.

An Introduction to Ergodic Theory

Brin, Michael; Garrett, Stuck (2002). Introduction to Dynamical Systems. Cambridge University Press. 0-521-80841-3.

ISBN

and Sjoerd Dirksin, "A Simple Introduction to Ergodic Theory"