In mathematics, more specifically in the theory of dynamical systems and probability theory, ergodicity is a property of a (discrete or continuous) dynamical system which expresses a form of irreducibility of the system, from a measure-theoretic viewpoint.
It includes the ergodicity of stochastic processes; though the language used for the study of ergodic processes is usually more probabilist. The origin of the notion and the nomenclature lie in statistical physics, where L. Boltzmann formulated the ergodic hypothesis. An informal way to phrase it is that the average behaviour over time on a trajectory does not depend on the particular trajectory chosen. The modern notion of ergodicity has found many applications in mathematics beyond this original motivation.
The study of ergodic systems is part of the larger field of ergodic theory.
Definition for discrete-time systemsEdit
Any precise definition of the phenomenon of ergodicity from a mathematical viewpoint requires measure theory. A more intuitive description, from a physical viewpoint, is the ergodic hypothesis. Ergodic processes give a more probabilistic formulation for certain cases.
For a discrete dynamical system , where the space is endowed with the additional structure of a probability measure space which is invariant under the transformation , ergodicity means that there is no way to measurably isolate a nontrivial part of which is invariant under . (Here "trivial" means that the subset or its complement has measure 0.) Figuratively one could say that the transformation mashes the space together in a (measurably) intractable way; a stronger, quantitative notion is that of mixing.
Let be a measurable space. If is a measurable function from to itself and a probability measure on then we say that is -ergodic or is an ergodic measure for if preserves and the following condition holds:
- For any such that either or .
The simplest example is when is a finite set and the counting measure. Then a self-map of preserves if and only if it is a bijection, and it is ergodic if and only if has only one orbit (that is, for every there exists such that ). For example, if then the cycle is ergodic, but the permutation is not (it has the two invariant subsets and ).
The definition given above admits the following immediate reformulations:
- for every with we have or (where denotes the symmetric difference);
- for every with positive measure we have ;
- for every two sets of positive measure, there exists such that ;
- Every measurable function with is constant on a subset of full measure.
Importantly for applications, the condition in the last characterisation can be restricted to square-integrable functions only:
- If and then is constant almost everywhere.
Bernoulli shifts and subshiftsEdit
There are many more ergodic measures for the shift map on . Periodic sequences give finitely supported measures. More interestingly, there are infinitely-supported ones which are subshifts of finite type.
Let be the unit circle , with its Lebesgue measure . For any the rotation of of angle is given by . If then is not ergodic for the Lebesgue measure as it has infinitely many finite orbits. On the other hand, if is irrational then is ergodic.
Let be the 2-torus. Then any element defines a self-map of since . When one obtains the so-called Arnold's cat map, which is ergodic for the Lebesgue measure on the torus.
If is a probability measure on a space which is ergodic for a transformation the pointwise ergodic theorem of G. Birkhoff states that for every measurable functions and for -almost every point the time average on the orbit of converges to the space average of . Formally this means that
The mean ergodic theorem of J. von Neumann is a similar, weaker statement about averaged translates of square-integrable functions.
An immediate consequence of the definition of ergodicity is that on a topological space , and if is the σ-algebra of Borel sets, if is -ergodic then -almost every orbit of is dense in the support of .
This is not an equivalence since for a transformation which is not uniquely ergodic, but for which there is an ergodic measure with full support , for any other ergodic measure the measure is not ergodic for but its orbits are dense in the support. Explicit examples can be constructed with shift-invariant measures.
A transformation of a probability measure space is said to be mixing for the measure if for any measurable sets the following holds:
This notion of mixing is sometimes called strong mixing, as opposed to weak mixing which means that
The transformation is said to be properly ergodic if it does not have an orbit of full measure. In the discrete case this means that the measure is not supported on a finite orbit of .
Definition for continous-time dynamical systemsEdit
The definition is essentially the same for continuous-time dynamical systems as for a single transformation. Let be a measurable space and for each , then such a system is given by a family of measurable functions from to itself, so that for any the relation holds (usually it is also asked that the orbit map from is also measurable). If is a probability measure on then we say that is -ergodic or is an ergodic measure for if each preserves and the following condition holds:
- For any and any such that either or .
As in the discrete case the simplest example is that of a transitive action, for instance the action on the circle given by is ergodic for Lebesgue measure.
An example with infinitely many orbits is given by the flow along an irrational slope on the torus: let and . Let ; then if this is ergodic for the Lebesgue measure.
Further examples of ergodic flows are:
Ergodicity in compact metric spacesEdit
If is a compact metric space it is naturally endowed with the σ-algebra of Borel sets. The additional structure coming from the topology then allows a much more detailed theory for ergodic transformations and measures on .
Functional analysis interpretationEdit
A very powerful alternate definition of ergodic measures can be given using the theory of Banach spaces. Radon measures on form a Banach space of which the set of probability measures on is a convex subset. Given a continuous transformation of the subset of -invariant measures is a closed convex subset, and a measure is ergodic for if and only if it is an extreme point of this convex.
Existence of ergodic measuresEdit
In the setting above it follows from the Banach-Alaoglu theorem that there always exists extremal points in . Hence a transformation of a compact metric space always admits ergodic measures.
In general an invariant measure need not be ergodic, but as a consequence of Choquet theory it can always be expressed as the barycenter of a probability measure on the set of ergodic measures. This is referred to as the ergodic decomposition of the measure.
In the case of and the counting measure is not ergodic. The ergodic measures for are the uniform measures supported on the subsets and and every -invariant probability measure can be written in the form for some . In particular is the ergodic decomposition of the counting measure.
Everything in this section transfers verbatim to continous actions of or on compact metric spaces.
The transformation is said to be uniquely ergodic if there is a unique Borel probability measure on which is ergodic for .
In the examples considered above, irrational rotations of the circle are uniquely ergodic; shift maps are not.
Probabilistic interpretation: ergodic processesEdit
If is a discrete-time stochastic process on a space it is said to be ergodic if the joint distribution of the variables on is invariant under the shift map . This is a particular case of the notions discussed above.
The simplest case is that of an independent and identically distributed process which corresponds to the shift map described above. Another important case is that of a Markov chain which is discussed in detail below.
A similar interpretation holds for continuous-time stochastic processes though the construction of the measurable structure of the action is more complicated.
Ergodicity of Markov chainsEdit
The dynamical system associated with a Markov chainEdit
Using this data we can define a probability measure on the set with its product σ-algebra by giving the measures of the cylinders as follows:
Criterion for ergodicityEdit
The hypotheses above imply that there is a unique stationary measure for the Markov chain. In terms of the matrix a sufficient condition for this is that 1 be a simple eigenvalue of the matrix and all other eigenvalues of (in ) 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 stricty stronger).
Moreover the criterion is an "if and only if" if all communicating classes in the chain are recurrent and we consider all stationary measures.
If for all then the stationary measure is the counting measure, the measure is the product of counting measures. The Markov chain is ergodic, so the shift example from above is a special case of the criterion.
Non-ergodic Markov chainsEdit
Markov chains with recurring communicating classes are not irreducible are not ergodic, and this can be seen immediately as follows. If are two distinct recurrent communicating classes there are nonzero stationary measures supported on respectively and the subsets and are both shift-invariant and of measure 1.2 for the invariant probability measure . A very simple example of that is the chain on given by the matrix (both states are stationary).
A periodic chainEdit
The Markov chain on given by the matrix is irreducible but periodic. Thus it is not ergodic in the sense of Markov chain though the associated measure on is ergodic for the shift map. However the shift is not mixing for this measure, as for the sets
Ergodic group actionsEdit
The definition of ergodicity also makes sense for group actions. The classical theory (for invertible transformations) corresponds to actions of or .
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.
A measurable equivalence relation it is said to be ergodic if all saturated subsets are either null or conull.
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, 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.
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. At the same time it is also claimed to be a be a derivation of ergomonode, coined by Boltzmann in a relatively obscure paper from 1884. The etymology appers to be contested in other ways as well.
- Walters 1982, p. 32.
- Walters 1982, p. 29.
- "Example of a measure-preserving system with dense orbits that is not ergodic". MathOverflow. September 1, 2011. Retrieved May 16, 2020.
- Walters 1982, p. 152.
- Walters 1982, p. 153.
- Walters 1982, p. 159.
- Walters 1982, p. 42.
- "Different uses of the word "ergodic"". MathOverflow. September 4, 2011. Retrieved May 16, 2020.
- Feller, William (1 August 2008). An Introduction to Probability Theory and Its Applications (2nd ed.). Wiley India Pvt. Limited. p. 271. ISBN 978-81-265-1806-7.
- Walters 1982, §0.1, p. 2
- Gallavotti, Giovanni (1995). "Ergodicity, ensembles, irreversibility in Boltzmann and beyond". Journal of Statistical Physics. 78 (5–6): 1571–1589. arXiv:chao-dyn/9403004. Bibcode:1995JSP....78.1571G. doi:10.1007/BF02180143.
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