In mathematics, especially in ergodic theory, ergodicity is a way of saying that a dynamical system behaves as one indivisible statistical system, rather than being composed of statistically distinguishable subsystems. More precisely, a measure-preserving dynamical system is ergodic if every invariant measurable set has either measure zero or full measure. Equivalently, the system cannot be decomposed, up to sets of measure zero, into two smaller invariant parts of positive measure.
Ergodic theorems relate this condition to the equality of time averages and space averages. Under suitable hypotheses, the time average of an observable along almost every orbit is equal to its space average. Ergodicity itself, however, is not the same as randomness, mixing, chaos, or the assertion that every individual orbit visits every part of the space.
Ergodicity may also be described in terms of ergodic measures: an invariant probability measure is ergodic if it cannot be decomposed into a nontrivial convex combination of other invariant probability measures.
Ergodic systems occur in many areas of physics, geometry, probability theory, and dynamical systems. The origins of the subject lie in statistical physics, where Ludwig Boltzmann formulated the ergodic hypothesis in connection with the foundations of statistical mechanics.
Informal explanation and motivation
Ergodicity is a way of saying that a system has no hidden division into smaller statistically separate parts. If the system is observed for a long time, its long-run statistical behavior is governed by one invariant measure, rather than by a choice among several different invariant components.
First example: coin tossing A simple example comes from coin tossing. Suppose the same biased coin is tossed forever, with probability of heads on each toss. The resulting stationary process is ergodic: apart from events of probability zero, there is no further permanent statistical label attached to the sequence. In particular, the long-run frequency of heads is for almost every sequence.
By contrast, suppose that before any tosses are made, one of two coins is chosen at random: one coin has probability of heads, and the other has probability . The chosen coin is then tossed forever. This process is stationary, but it is not ergodic. The hidden choice of coin is an invariant fact about the whole infinite sequence, and it separates the process into two statistically different components.
In a measure-preserving dynamical system, this idea is expressed by saying that every invariant measurable set has either measure zero or full measure. Equivalently, the system cannot be decomposed, up to sets of measure zero, into two invariant parts of positive measure.
Ergodicity is related to the equality of time averages and space averages through ergodic theorems. It should not be confused with randomness, mixing, chaos, or the claim that every individual orbit visits every part of the space.
Second example: irrational rotations of a circle
Another basic example comes from rotations of the circle. Let be the rotation of the circle through the angle radians. After iterations, a point has been rotated by .
If is rational, say in lowest terms, then every point is periodic: after iterations it returns to its original position. With respect to Lebesgue measure on the circle, this rotation is not ergodic. For example, the nonconstant function
is invariant under the rotation, since replacing by does not change . The circle can thus be divided into smaller invariant pieces, such as .
If is irrational, the orbit of every point is dense in the circle. In fact, the irrational rotation is ergodic with respect to Lebesgue measure: every measurable set that is invariant under the rotation has either measure zero or full measure. Thus the system has many different individual orbits, but it has no nontrivial measurable decomposition into invariant statistical components.
This example also shows that ergodicity is weaker than mixing. Irrational rotations are ergodic, but they are not mixing.
Measurable sets and invariant events
In the measure-theoretic definition of ergodicity, the basic data are a measure space , and a time-evolution of its points. In the coin-toss example, the state space may be taken to be the space of infinite sequences of outcomes, and the time evolution is the shift map on this space. In the rotation example, the state space is the circle, and the time evolution is repeated application of the rotation. Both are examples of discrete time dynamical systems. The elements of the space are possible states of the system. A measurable set is a collection of states to which the theory assigns a measure, such as length, area, volume, or probability. The space is equipped with a sigma-algebra of sets that are allowed to be measured.
When is a topological space, such as an interval, a circle, or a manifold, the usual choice is often the Borel sigma-algebra, generated by the open sets. In probability examples, measurable sets are events. For instance, in an infinite sequence of coin tosses, the event "the first toss is heads" is measurable, as is the event "the long-run frequency of heads exists and is equal to ".
A set is invariant if membership in it is unchanged by the time evolution, up to sets of measure zero. For a transformation , this is usually written as
or, more generally, denoting the symmetric difference of sets by :
A (discrete time) measure preserving system is thus the pair where is a measure space and is a measure-preserving transformation.
An invariant measurable set represents a statistical property of the state that persists under the dynamics. Ergodicity says that every such persistent measurable property is trivial: it holds either almost nowhere (i.e., only on a set of measure zero) or almost everywhere (i.e., everywhere except a set of measure zero).
Physical motivation
The original motivation for ergodicity came from statistical mechanics. In classical mechanics, the state of a system is represented by a point in phase space: for example, the positions and momenta of all particles in a gas. The time evolution of the system moves this point through phase space.
For an isolated Hamiltonian system, the energy is conserved, so the motion is confined to an energy surface
Hamiltonian flow preserves the natural phase-space volume, called Liouville measure. After restricting to an energy surface, this gives the measure underlying the microcanonical ensemble.
The physical question was whether long-time averages along the actual motion of one system can be replaced by averages over the corresponding energy surface. For an observable , this asks whether a time average such as
agrees, for typical initial states , representing the state of the large number of particles in the gas, with the average of over the energy surface. The ergodic hypothesis was an early attempt to justify this replacement.
In modern measure-theoretic language, ergodicity does not mean that a trajectory literally passes through every point of the energy surface. Rather, it means that, relative to the chosen invariant measure, there is no further invariant measurable decomposition of the system into smaller statistical components. Under suitable hypotheses, ergodic theorems then imply the equality of time averages and space averages for many observables.
History and etymology
The term ergodic is usually attributed to Ludwig Boltzmann and is commonly explained as a combination of the Greek words (ergon, "work") and (hodos, "path"). The exact historical development of the term is not entirely straightforward; it has also been connected with Boltzmann's term ergomonode, used in his work on statistical mechanics.
The concept arose in the attempt to justify the use of statistical methods in statistical mechanics. In that setting, one wants to relate time averages along the motion of a mechanical system to averages over a space of possible states. This motivation led to the ergodic hypothesis, according to which a mechanical system may, under suitable assumptions, be treated as if its long-time behavior samples the relevant state space.
The original physical form of the hypothesis was later replaced by more precise mathematical notions in ergodic theory, especially the measure-theoretic definition in terms of invariant measurable sets. In 1913, Michel Plancherel showed that one natural formulation of the classical mechanical ergodic hypothesis could not hold as originally stated.
Ergodicity in physics and geometry
Ergodicity appears in several related settings in physics and geometry. In each case, one must specify both the state space and the invariant measure. The common measure-theoretic question is whether the system has nontrivial invariant measurable subsets.
In statistical mechanics
In statistical mechanics, ergodicity is used to justify replacing long-time averages of observables by averages over an invariant ensemble. For isolated Hamiltonian systems, the relevant ensemble is usually the microcanonical ensemble on an energy surface. This is the setting of the classical ergodic hypothesis.
Rigorous proofs of ergodicity for realistic many-particle systems are difficult. Important mathematical examples come from dynamical billiards, where idealized particle collisions provide systems for which ergodicity can sometimes be proved.
Simple dynamical systems
Basic examples of ergodic and non-ergodic systems occur already in low-dimensional dynamics. Irrational rotations of the circle are ergodic with respect to Lebesgue measure, but they are not mixing. Other standard examples include the baker's map, Arnold's cat map, Bernoulli shifts, and certain billiard systems. These examples illustrate different relations between ergodicity, mixing, entropy, and chaotic behavior.
In geometry
Ergodic flows also arise naturally in geometry. A central example is the geodesic flow on a Riemannian manifold, usually considered on the unit tangent bundle with its natural invariant measure.
On a flat torus, motion in a fixed irrational direction is dense on the torus and is ergodic after restricting to that direction. The full geodesic flow on the unit tangent bundle is not ergodic, since the direction of motion is conserved. By contrast, the geodesic flow on a compact Riemannian manifold of negative curvature is ergodic with respect to the natural invariant measure. This provides one of the classical families of examples in ergodic theory and is closely related to Anosov flows.
Other geometric examples include horocycle flows, flows on translation surfaces, and billiard flows. In these settings, ergodicity concerns invariant measurable subsets of the appropriate phase space, not whether a single path literally passes through every point.
In quantum mechanics
In quantum mechanics, the word ergodicity is used more cautiously, since quantum evolution is not described by point trajectories in classical phase space. Related notions appear in quantum chaos and in the quantum ergodicity theorem, which concerns the distribution of eigenfunctions in a semiclassical limit. These ideas are related to, but not identical with, classical measure-theoretic ergodicity.
Definition for discrete-time systems Ergodic measures provide one of the cornerstones with which ergodicity is generally discussed. A formal definition follows.
Invariant measure Let be a measurable space. If is a measurable function from to itself and a probability measure on , then a measure-preserving dynamical system is defined as a dynamical system for which for all . Such a is said to preserve equivalently, that is -invariant.
Ergodic measure A measurable function is said to be -ergodic or that is an ergodic measure for if preserves and the following condition holds:
For any such that either or .
In other words, there are no -invariant subsets up to measure 0 (with respect to ).