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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, ergodicity expresses the idea that a point of a moving system, either a
dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
or a
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that ap ...
, 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 A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete tr ...
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 Ergodic theory ( Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expr ...
is the study of systems possessing ergodicity. Ergodic systems occur in a broad range of systems in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
and in
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
. This can be roughly understood to be due to a common phenomenon: the motion of particles, that is,
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connecti ...
s on a
hyperbolic manifold In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, r ...
are divergent; when that manifold is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
, 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 In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
and
dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
s, and rather specifically on the notion of a measure-preserving dynamical system. The origins of ergodicity lie in
statistical physics Statistical physics is a branch of physics that evolved from a foundation of statistical mechanics, which uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approxim ...
, where
Ludwig Boltzmann Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of ther ...
formulated the
ergodic hypothesis In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent by a system in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e., t ...
.


Informal explanation

Ergodicity occurs in broad settings in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
and
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
. All of these settings are unified by a common mathematical description, that of the measure-preserving dynamical system. An informal description of this, and a definition of ergodicity with respect to it, is given immediately below. This is followed by a description of ergodicity in
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that ap ...
es. They are one and the same, despite using dramatically different notation and language.


Measure-preserving dynamical systems

The mathematical definition of ergodicity aims to capture ordinary every-day ideas about
randomness In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual rand ...
. This includes ideas about systems that move in such a way as to (eventually) fill up all of space, such as
diffusion Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical ...
and
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
, as well as common-sense notions of mixing, such as mixing paints, drinks, cooking ingredients, industrial process mixing, smoke in a smoke-filled room, the dust in Saturn's rings and so on. To provide a solid mathematical footing, descriptions of ergodic systems begin with the definition of a measure-preserving dynamical system. This is written as (X, \mathcal, \mu, T). The set X is understood to be the total space to be filled: the mixing bowl, the smoke-filled room, ''etc.'' The measure \mu is understood to define the natural
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
of the space X and of its subspaces. The collection of subspaces is denoted by \mathcal, and the size of any given
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
A\subset X is \mu(A); the size is its volume. Naively, one could imagine \mathcal to be the
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...
of X; this doesn't quite work, as not all subsets of a space have a volume (famously, the Banach-Tarski paradox). Thus, conventionally, \mathcal consists of the measurable subsets—the subsets that do have a volume. It is always taken to be a
Borel set In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are na ...
—the collection of subsets that can be constructed by taking
intersections In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
, unions and set complements of open sets; these can always be taken to be measurable. The time evolution of the system is described by a map T:X\to X. Given some subset A\subset X, its map T(A) will in general be a deformed version of A – it is squashed or stretched, folded or cut into pieces. Mathematical examples include the
baker's map In dynamical systems theory, the baker's map is a chaotic map from the unit square into itself. It is named after a kneading operation that bakers apply to dough: the dough is cut in half, and the two halves are stacked on one another, and compr ...
and the horseshoe map, both inspired by
bread Bread is a staple food prepared from a dough of flour (usually wheat) and water, usually by baking. Throughout recorded history and around the world, it has been an important part of many cultures' diet. It is one of the oldest human-made f ...
-making. The set T(A) must have the same volume as A; the squashing/stretching does not alter the volume of the space, only its distribution. Such a system is "measure-preserving" (area-preserving, volume-preserving). A formal difficulty arises when one tries to reconcile the volume of sets with the need to preserve their size under a map. The problem arises because, in general, several different points in the domain of a function can map to the same point in its range; that is, there may be x \ne y with T(x) = T(y). Worse, a single point x \in X has no size. These difficulties can be avoided by working with the inverse map T^: \mathcal\to\mathcal; it will map any given subset A \subset X to the parts that were assembled to make it: these parts are T^(A)\in\mathcal. It has the important property of not losing track of where things came from. More strongly, it has the important property that ''any'' (measure-preserving) map \mathcal\to\mathcal is the inverse of some map X\to X. The proper definition of a volume-preserving map is one for which \mu(A) = \mu\mathord\left(T^(A)\right) because T^(A) describes all the pieces-parts that A came from. One is now interested in studying the time evolution of the system. If a set A\in\mathcal eventually comes to fill all of X over a long period of time (that is, if T^n(A) approaches all of X for large n), the system is said to be
ergodic 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 tha ...
. If every set A behaves in this way, the system is a conservative system, placed in contrast to a dissipative system, where some subsets A wander away, never to be returned to. An example would be water running downhill: once it's run down, it will never come back up again. The lake that forms at the bottom of this river can, however, become well-mixed. The
ergodic decomposition theorem 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 th ...
states that every ergodic system can be split into two parts: the conservative part, and the dissipative part. Mixing is a stronger statement than ergodicity. Mixing asks for this ergodic property to hold between any two sets A, B, and not just between some set A and X. That is, given any two sets A, B\in\mathcal, a system is said to be (topologically) mixing if there is an integer N such that, for all A, B and n>N, one has that T^n(A) \cap B \ne \varnothing. Here, \cap denotes set intersection and \varnothing is the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in othe ...
. Other notions of mixing include strong and weak mixing, which describe the notion that the mixed substances intermingle everywhere, in equal proportion. This can be non-trivial, as practical experience of trying to mix sticky, gooey substances shows.


Processes

The above discussion appeals to a physical sense of a volume. The volume does not have to literally be some portion of
3D space Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informa ...
; it can be some abstract volume. This is generally the case in statistical systems, where the volume (the measure) is given by the probability. The total volume corresponds to probability one. This correspondence works because the
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
s of
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
are identical to those of
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
; these are the
Kolmogorov axioms The Kolmogorov axioms are the foundations of probability theory introduced by Russian mathematician Andrey Kolmogorov in 1933. These axioms remain central and have direct contributions to mathematics, the physical sciences, and real-world probabili ...
. The idea of a volume can be very abstract. Consider, for example, the set of all possible coin-flips: the set of infinite sequences of heads and tails. Assigning the volume of 1 to this space, it is clear that half of all such sequences start with heads, and half start with tails. One can slice up this volume in other ways: one can say "I don't care about the first n - 1 coin-flips; but I want the n'th of them to be heads, and then I don't care about what comes after that". This can be written as the set (*, \cdots, *, h, *, \cdots) where * is "don't care" and h is "heads". The volume of this space is again (obviously!) one-half. The above is enough to build up a measure-preserving dynamical system, in its entirety. The sets of h or t occurring in the n'th place are called
cylinder set In mathematics, the cylinder sets form a basis of the product topology on a product of sets; they are also a generating family of the cylinder σ-algebra. General definition Given a collection S of sets, consider the Cartesian product X = \prod_ ...
s. The set of all possible intersections, unions and complements of the cylinder sets then form the
Borel set In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are na ...
\mathcal defined above. In formal terms, the cylinder sets form the base for a
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
on the
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consi ...
X of all possible infinite-length coin-flips. The measure \mu has all of the common-sense properties one might hope for: the measure of a cylinder set with h in the m'th position, and t in the k'th position is obviously 1/4, and so on. These common-sense properties persist for set-complement and set-union: everything except for h and t in locations m and k obviously has the volume of 3/4. All together, these form the axioms of a sigma-additive measure; measure-preserving dynamical systems always use sigma-additive measures. For coin flips, this measure is called the
Bernoulli measure In probability and statistics, a Bernoulli process (named after Jacob Bernoulli) is a finite or infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes only two values, canonically 0 and 1. ...
. For the coin-flip process, the time-evolution operator T is the shift operator that says "throw away the first coin-flip, and keep the rest". Formally, if (x_1, x_2, \cdots) is a sequence of coin-flips, then T(x_1, x_2, \cdots) = (x_2, x_3, \cdots). The measure is obviously shift-invariant: as long as we are talking about some set A\in\mathcal where the first coin-flip x_1 = * is the "don't care" value, then the volume \mu(A) does not change: \mu(A) = \mu(T(A)). In order to avoid talking about the first coin-flip, it is easier to define T^ as inserting a "don't care" value into the first position: T^(x_1, x_2, \cdots) = (*, x_1, x_2, \cdots). With this definition, one obviously has that \mu\mathord\left(T^(A)\right) = \mu(A) with no constraints on A. This is again an example of why T^ is used in the formal definitions. The above development takes a random process, the Bernoulli process, and converts it to a measure-preserving dynamical system (X, \mathcal, \mu, T). The same conversion (equivalence, isomorphism) can be applied to any
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that ap ...
. Thus, an informal definition of ergodicity is that a sequence is ergodic if it visits all of X; such sequences are "typical" for the process. Another is that its statistical properties can be deduced from a single, sufficiently long, random sample of the process (thus uniformly sampling all of X), or that any collection of random samples from a process must represent the average statistical properties of the entire process (that is, samples drawn uniformly from X are representative of X as a whole.) In the present example, a sequence of coin flips, where half are heads, and half are tails, is a "typical" sequence. There are several important points to be made about the Bernoulli process. If one writes 0 for tails and 1 for heads, one gets the set of all infinite strings of binary digits. These correspond to the base-two expansion of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s. Explicitly, given a sequence (x_1, x_2, \cdots), the corresponding real number is :y=\sum_^\infty \frac The statement that the Bernoulli process is ergodic is equivalent to the statement that the real numbers are uniformly distributed. The set of all such strings can be written in a variety of ways: \^\infty = \^\omega = \^\omega = 2^\omega = 2^\mathbb. This set is the
Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. T ...
, sometimes called the
Cantor space In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ω is called "the ...
to avoid confusion with the Cantor function :C(x) = \sum_^\infty \frac In the end, these are all "the same thing". The Cantor set plays key roles in many branches of mathematics. In recreational mathematics, it underpins the period-doubling fractals; in
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...
, it appears in a vast variety of theorems. A key one for stochastic processes is the Wold decomposition, which states that any stationary process can be decomposed into a pair of uncorrelated processes, one deterministic, and the other being a moving average process. The Ornstein isomorphism theorem states that every stationary stochastic process is equivalent to a
Bernoulli scheme In mathematics, the Bernoulli scheme or Bernoulli shift is a generalization of the Bernoulli process to more than two possible outcomes. Bernoulli schemes appear naturally in symbolic dynamics, and are thus important in the study of dynamical sy ...
(a Bernoulli process with an ''N''-sided (and possibly unfair) gaming die). Other results include that every non-dissipative ergodic system is equivalent to the Markov odometer, sometimes called an "adding machine" because it looks like elementary-school addition, that is, taking a base-''N'' digit sequence, adding one, and propagating the carry bits. The proof of equivalence is very abstract; understanding the result is not: by adding one at each time step, every possible state of the odometer is visited, until it rolls over, and starts again. Likewise, ergodic systems visit each state, uniformly, moving on to the next, until they have all been visited. Systems that generate (infinite) sequences of ''N'' letters are studied by means of symbolic dynamics. Important special cases include
subshifts of finite type In mathematics, subshifts of finite type are used to model dynamical systems, and in particular are the objects of study in symbolic dynamics and ergodic theory. They also describe the set of all possible sequences executed by a finite state machin ...
and sofic systems.


History and etymology

The term ''ergodic'' is commonly thought to derive from the
Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...
words (''ergon'': "work") and (''hodos'': "path", "way"), as chosen by
Ludwig Boltzmann Ludwig Eduard Boltzmann (; 20 February 1844 – 5 September 1906) was an Austrian physicist and philosopher. His greatest achievements were the development of statistical mechanics, and the statistical explanation of the second law of ther ...
while he was working on a problem in
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic b ...
. 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. The idea of ergodicity was born in the field of
thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws ...
, 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 Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In the ...
could be defined with mathematical rigor. Once the theory was well developed in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, it was rapidly formalized and extended, so that
ergodic theory Ergodic theory ( Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expr ...
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 Classical physics is a group of physics theories that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the ...
the term implies that a system satisfies the
ergodic hypothesis In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent by a system in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e., t ...
of
thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws ...
, the relevant state space being position and momentum space. In
dynamical systems theory Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called '' ...
the state space is usually taken to be a more general
phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...
. 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 In physics, specifically statistical mechanics, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents ...
can also carry extra baggage as well—as is the case with the many possible thermodynamically relevant partition functions used to define
ensemble average In physics, specifically statistical mechanics, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents ...
s 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 for ergodicity for a purely mechanical system.


Occurrence

A review of ergodicity in physics, and in
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
follows. In all cases, the notion of ergodicity is ''exactly'' the same as that for dynamical systems; ''there is no difference'', except for outlook, notation, style of thinking and the journals where results are published.


In physics

Physical systems can be split into three categories:
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
, which describes machines with a finite number of moving parts,
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, which describes the structure of atoms, and
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic b ...
, which describes gases, liquids, solids; this includes
condensed matter physics Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the su ...
. The case of classical mechanics is discussed in the next section, on ergodicity in geometry. As to quantum mechanics, there is no universal quantum definition of ergodocity or even chaos (see quantum chaos). However, there is a quantum ergodicity theorem stating that the expectation value of an operator converges to the corresponding microcanonical classical average in the semiclassical limit \hbar \rightarrow 0. Nevertheless, the theorem does not imply that ''all'' eigenstates of the Hamiltionian whose classical counterpart is chaotic are features and random. For example, the quantum ergodicity theorem do not exclude the existence of non-ergodic states such as quantum scars. In addition to the conventional scarring, there are two other types of quantum scarring, which further illustrate the weak-ergodicity breaking in quantum chaotic systems: perturbation-induced and many-body quantum scars. This section reviews ergodicity in statistical mechanics. The above abstract definition of a volume is required as the appropriate setting for definitions of ergodicity in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
. Consider a container of
liquid A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. As such, it is one of the four fundamental states of matter (the others being solid, gas, ...
, or gas, or plasma, or other collection of
atom Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, a ...
s or
particle In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from ...
s. Each and every particle x_i has a 3D position, and a 3D velocity, and is thus described by six numbers: a point in six-dimensional space \mathbb^6. If there are N of these particles in the system, a complete description requires 6N numbers. Any one system is just a single point in \mathbb^. The physical system is not all of \mathbb^, of course; if it's a box of width, height and length W\times H\times L then a point is in \left(W \times H \times L \times \mathbb^3\right)^N. Nor can velocities be infinite: they are scaled by some probability measure, for example the Boltzmann–Gibbs measure for a gas. None-the-less, for N close to
Avogadro's number The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining con ...
, this is obviously a very large space. This space is called the
canonical ensemble In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature. The system can exchange energy with the hea ...
. A physical system is said to be ergodic if any representative point of the system eventually comes to visit the entire volume of the system. For the above example, this implies that any given atom not only visits every part of the box W \times H \times L with uniform probability, but it does so with every possible velocity, with probability given by the Boltzmann distribution for that velocity (so, uniform with respect to that measure). The
ergodic hypothesis In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent by a system in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e., t ...
states that physical systems actually are ergodic. Multiple time scales are at work: gasses and liquids appear to be ergodic over short time scales. Ergodicity in a solid can be viewed in terms of the vibrational modes or
phonon In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phonon is an excited state in the quantum mechani ...
s, as obviously the atoms in a solid do not exchange locations.
Glass Glass is a non- crystalline, often transparent, amorphous solid that has widespread practical, technological, and decorative use in, for example, window panes, tableware, and optics. Glass is most often formed by rapid cooling (quenchin ...
es present a challenge to the ergodic hypothesis; time scales are assumed to be in the millions of years, but results are contentious. Spin glasses present particular difficulties. Formal mathematical proofs of ergodicity in statistical physics are hard to come by; most high-dimensional many-body systems are assumed to be ergodic, without mathematical proof. Exceptions include the dynamical billiards, which model
billiard ball A billiard ball is a small, hard ball used in cue sports, such as carom billiards, pool, and snooker. The number, type, diameter, color, and pattern of the balls differ depending upon the specific game being played. Various particular ball ...
-type collisions of atoms in an
ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is a ...
or plasma. The first hard-sphere ergodicity theorem was for
Sinai's billiards A dynamical billiard is a dynamical system in which a particle alternates between free motion (typically as a straight line) and specular reflections from a boundary. When the particle hits the boundary it reflects from it without loss of speed ( ...
, which considers two balls, one of them taken as being stationary, at the origin. As the second ball collides, it moves away; applying periodic boundary conditions, it then returns to collide again. By appeal to homogeneity, this return of the "second" ball can instead be taken to be "just some other atom" that has come into range, and is moving to collide with the atom at the origin (which can be taken to be just "any other atom".) This is one of the few formal proofs that exist; there are no equivalent statements ''e.g.'' for atoms in a liquid, interacting via
van der Waals force In molecular physics, the van der Waals force is a distance-dependent interaction between atoms or molecules. Unlike ionic or covalent bonds, these attractions do not result from a chemical electronic bond; they are comparatively weak and ...
s, even if it would be common sense to believe that such systems are ergodic (and mixing). More precise physical arguments can be made, though.


In geometry

Ergodicity is a wide-spread phenomenon in the study of
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
s. A quick sequence of examples, from simple to complicated, illustrates this point. The
geodesic flow In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
of a
flat torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...
following any irrational direction is ergodic; informally this means that when drawing a straight line in a square starting at any point, and with an irrational angle with respect to the sides, if every time one meets a side one starts over on the opposite side with the same angle, the line will eventually meet every subset of positive measure. More generally on any flat surface there are many ergodic directions for the geodesic flow. There are similar results for negatively curved compact
Riemann surfaces In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
; note that in this case the definition of
geodesic flow In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
is much more involved since there is no notion of constant direction on a non-flat surface. More generally the geodesic flow on a negatively curved compact
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...
s is ergodic, in fact it satisfies the stronger property of being an Anosov flow.


In finance

Models used in
finance Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of f ...
and
investment Investment is the dedication of money to purchase of an asset to attain an increase in value over a period of time. Investment requires a sacrifice of some present asset, such as time, money, or effort. In finance, the purpose of investing is ...
assume ergodicity, explicitly or implicitly. The ergodic assumption is prevalent in
modern portfolio theory Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk. It is a formalization and extension of diversificati ...
,
discounted cash flow The discounted cash flow (DCF) analysis is a method in finance of valuing a security, project, company, or asset using the concepts of the time value of money. Discounted cash flow analysis is widely used in investment finance, real estate de ...
(DCF) models, and aggregate indicator models that infuse
macroeconomics Macroeconomics (from the Greek prefix ''makro-'' meaning "large" + ''economics'') is a branch of economics dealing with performance, structure, behavior, and decision-making of an economy as a whole. For example, using interest rates, taxes, and ...
, among others. The situations modeled by these theories can be useful. But often they are only useful during much, but not all, of any particular time period under study. They can therefore miss some of the largest deviations from the standard model, such as financial crises, debt crises and
systemic risk In finance, systemic risk is the risk of collapse of an entire financial system or entire market, as opposed to the risk associated with any one individual entity, group or component of a system, that can be contained therein without harming the ...
in the banking system that occur only infrequently.
Nassim Nicholas Taleb Nassim Nicholas Taleb (; alternatively ''Nessim ''or'' Nissim''; born 12 September 1960) is a Lebanese-American essayist, mathematical statistician, former option trader, risk analyst, and aphorist whose work concerns problems of randomness ...
has argued that a very important part of empirical reality in finance and investment is non-ergodic. An even statistical distribution of probabilities, where the system returns to every possible state an infinite number of times, is simply not the case we observe in situations where “absorbing states" are reached, a state where ''ruin'' is seen. The death of an individual, or total loss of everything, or the devolution or dismemberment of a
nation state A nation state is a political unit where the state and nation are congruent. It is a more precise concept than "country", since a country does not need to have a predominant ethnic group. A nation, in the sense of a common ethnicity, may ...
and the legal regime that accompanied it, are all absorbing states. Thus, in finance,
path dependence Path dependence is a concept in economics and the social sciences, referring to processes where past events or decisions constrain later events or decisions. It can be used to refer to outcomes at a single point in time or to long-run equilibri ...
matters. A path where an individual, firm or country hits a "stop"—an absorbing barrier, "anything that prevents people with skin in the game from emerging from it, and to which the system will invariably tend. Let us call these situations ''ruin'', as the entity cannot emerge from the condition. The central problem is that if there is a possibility of ruin, cost benefit analyses are no longer possible."—will be non-ergodic. All traditional models based on standard probabilistic statistics break down in these extreme situations.


Definition for discrete-time systems


Formal definition

Let (X, \mathcal B) be a
measurable space In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. Definition Consider a set X and a σ-algebra \mathcal A on X. Then the ...
. If T is a measurable function from X to itself and \mu a
probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more ge ...
on (X, \mathcal B) then we say that T is \mu-ergodic or \mu is an ergodic measure for T if T preserves \mu and the following condition holds: : For any A \in \mathcal B such that T^(A) = A either \mu(A) = 0 or \mu(A) = 1. In other words there are no T-invariant subsets up to measure 0 (with respect to \mu). Recall that T preserving \mu (or \mu being T- invariant) means that \mu\mathord\left(T^(A)\right) = \mu(A) for all A \in \mathcal B (see also measure-preserving dynamical system). Note that some authors (eg, "An introduction to infinite ergodic theory" by Anderson, p. 21) relax the requirement that \mu is T-invariant to the requirement that pullbacks of measure-zero sets are measure-zero, i.e., the pushforward measure T_*\mu is singular with respect to \mu.


Examples

The simplest example is when X is a finite set and \mu the counting measure. Then a self-map of X preserves \mu if and only if it is a bijection, and it is ergodic if and only if T has only one
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
(that is, for every x, y \in X there exists k \in \mathbb N such that y = T^k(x)). For example, if X = \ then the cycle (1\, 2\, \cdots \, n) is ergodic, but the permutation (1\, 2)(3\, 4\, \cdots\, n) is not (it has the two invariant subsets \ and \).


Equivalent formulations

The definition given above admits the following immediate reformulations: * for every A \in \mathcal B with \mu\mathord\left(T^(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 \mu\mathord\left(\bigcup_^\infty T^(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^(A)\right) \cap B\right) > 0; * Every measurable function f: X\to\mathbb with f \circ T = f is constant on a subset of full measure. Importantly for applications, the condition in the last characterisation can be restricted to
square-integrable function In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value ...
s only: * If f \in L^2(X, \mu) and f \circ T = f then f is constant almost everywhere.


Further examples


Bernoulli shifts and subshifts

Let S be a finite set and X = S^\mathbb with \mu the product measure (each factor S being endowed with its counting measure). Then the shift operator T defined by T\left((s_k)_)\right) = (s_)_ is . There are many more ergodic measures for the shift map T on X. Periodic sequences give finitely supported measures. More interestingly, there are infinitely-supported ones which are
subshifts of finite type In mathematics, subshifts of finite type are used to model dynamical systems, and in particular are the objects of study in symbolic dynamics and ergodic theory. They also describe the set of all possible sequences executed by a finite state machin ...
.


Irrational rotations

Let X be the unit circle \, with its Lebesgue measure \mu. For any \theta \in \mathbb R the rotation of X of angle \theta is given by T_\theta(z) = e^z. If \theta \in \mathbb Q then T_\theta is not ergodic for the Lebesgue measure as it has infinitely many finite orbits. On the other hand, if \theta is irrational then T_\theta is ergodic.


Arnold's cat map

Let X = \mathbb^2/\mathbb^2 be the 2-torus. Then any element g \in \mathrm_2(\mathbb Z) defines a self-map of X since g\left(\mathbb^2\right) = \mathbb^2. When g = \left(\begin 2 & 1 \\ 1 & 1 \end\right) one obtains the so-called Arnold's cat map, which is ergodic for the Lebesgue measure on the torus.


Ergodic theorems

If \mu is a probability measure on a space X which is ergodic for a transformation T the pointwise ergodic theorem of G. Birkhoff states that for every measurable functions f: X \to \mathbb R and for \mu-almost every point x \in X the time average on the orbit of x converges to the space average of f. Formally this means that \lim_ \left( \frac 1 \sum_^k f\left(T^i(x)\right) \right) = \int_X fd\mu. The mean ergodic theorem of J. von Neumann is a similar, weaker statement about averaged translates of square-integrable functions.


Related properties


Dense orbits

An immediate consequence of the definition of ergodicity is that on a topological space X, and if \mathcal B is the σ-algebra of
Borel set In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are na ...
s, if T is \mu-ergodic then \mu-almost every orbit of T is dense in the support of \mu. 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 \mu_0, for any other ergodic measure \mu_1 the measure \frac(\mu_0 + \mu_1) is not ergodic for T but its orbits are dense in the support. Explicit examples can be constructed with shift-invariant measures.


Mixing

A transformation T of a probability measure space (X, \mu) is said to be mixing for the measure \mu if for any measurable sets A, B \subset X the following holds: \lim_ \frac 1 n \sum_^n \left, \mu(T^A \cap B) - \mu(A)\mu(B) \ = 0


Proper ergodicity

The transformation T 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 \mu is not supported on a finite orbit of T.


Definition for continuous-time dynamical systems

The definition is essentially the same for
continuous-time dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a ...
s as for a single transformation. Let (X, \mathcal B) be a measurable space and for each t \in \mathbb R_+, then such a system is given by a family T_t of measurable functions from X to itself, so that for any t, s \in \mathbb R_+ the relation T_ = T_s \circ T_t holds (usually it is also asked that the orbit map from \mathbb R_+ \times X \to X is also measurable). If \mu is a probability measure on (X, \mathcal B) then we say that T_t is \mu-ergodic or \mu is an ergodic measure for T if each T_t preserves \mu and the following condition holds: : For any A \in \mathcal B, if for all t \in \mathbb R_+ we have T_t^(A) \subset A then either \mu(A) = 0 or \mu(A) = 1.


Examples

As in the discrete case the simplest example is that of a transitive action, for instance the action on the circle given by T_t(z) = e^z is ergodic for Lebesgue measure. An example with infinitely many orbits is given by the flow along an irrational slope on the torus: let X = \mathbb S^1 \times \mathbb S^1 and \alpha \in \mathbb R. Let T_t(z_1, z_2) = \left(e^z_1, e^z_2\right); then if \alpha \not\in \mathbb Q this is ergodic for the Lebesgue measure.


Ergodic flows

Further examples of ergodic flows are: *
Billiards Cue sports are a wide variety of games of skill played with a cue, which is used to strike billiard balls and thereby cause them to move around a cloth-covered table bounded by elastic bumpers known as . There are three major subdivisions ...
in convex Euclidean domains; * the
geodesic flow In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
of a negatively curved Riemannian manifold of finite volume is ergodic (for the normalised volume measure); * the
horocycle In hyperbolic geometry, a horocycle (), sometimes called an oricycle, oricircle, or limit circle, is a curve whose normal or perpendicular geodesics all converge asymptotically in the same direction. It is the two-dimensional case of a horosph ...
flow on a
hyperbolic manifold In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, r ...
of finite volume is ergodic (for the normalised volume measure)


Ergodicity in compact metric spaces

If X is a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Britis ...
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
it is naturally endowed with the σ-algebra of
Borel set In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are na ...
s. The additional structure coming from the topology then allows a much more detailed theory for ergodic transformations and measures on X.


Functional analysis interpretation

A very powerful alternate definition of ergodic measures can be given using the theory of
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s. Radon measures on X form a Banach space of which the set \mathcal P(X) of probability measures on X is a
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
subset. Given a continuous transformation T of X the subset \mathcal P(X)^T of T-invariant measures is a closed convex subset, and a measure is ergodic for T if and only if it is an
extreme point In mathematics, an extreme point of a convex set S in a real or complex vector space is a point in S which does not lie in any open line segment joining two points of S. In linear programming problems, an extreme point is also called vertex ...
of this convex.


Existence of ergodic measures

In the setting above it follows from the Banach-Alaoglu theorem that there always exists extremal points in \mathcal P(X)^T. Hence a transformation of a compact metric space always admits ergodic measures.


Ergodic decomposition

In general an invariant measure need not be ergodic, but as a consequence of
Choquet theory In mathematics, Choquet theory, named after Gustave Choquet, is an area of functional analysis and convex analysis concerned with measures which have support on the extreme points of a convex set ''C''. Roughly speaking, every vector of ''C'' sho ...
it can always be expressed as the
barycenter In astronomy, the barycenter (or barycentre; ) is the center of mass of two or more bodies that orbit one another and is the point about which the bodies orbit. A barycenter is a dynamical point, not a physical object. It is an important con ...
of a probability measure on the set of ergodic measures. This is referred to as the ''ergodic decomposition'' of the measure.


Example

In the case of X = \ and T = (1\, 2)(3\, 4\, \cdots\, n) the counting measure is not ergodic. The ergodic measures for T are the uniform measures \mu_1, \mu_2 supported on the subsets \ and \ and every T-invariant probability measure can be written in the form t\mu_1 + (1 - t)\mu_2 for some t \in , 1/math>. In particular \frac\mu_1 + \frac\mu_2 is the ergodic decomposition of the counting measure.


Continuous systems

Everything in this section transfers verbatim to continuous actions of \mathbb R or \mathbb R_+ on compact metric spaces.


Unique ergodicity

The transformation T is said to be uniquely ergodic if there is a unique Borel probability measure \mu on X which is ergodic for T. In the examples considered above, irrational rotations of the circle are uniquely ergodic; shift maps are not.


Probabilistic interpretation: ergodic processes

If \left(X_n\right)_ is a discrete-time stochastic process on a space \Omega, it is said to be ergodic if the
joint distribution Given two random variables that are defined on the same probability space, the joint probability distribution is the corresponding probability distribution on all possible pairs of outputs. The joint distribution can just as well be considered ...
of the variables on \Omega^\mathbb is invariant under the shift map \left(x_n\right)_ \mapsto \left(x_\right)_. This is a particular case of the notions discussed above. The simplest case is that of an
independent and identically distributed In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usual ...
process which corresponds to the shift map described above. Another important case is that of a
Markov chain A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happen ...
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 chains


The dynamical system associated with a Markov chain

Let S be a finite set. A
Markov chain A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happen ...
on S is defined by a matrix P \in , 1, where P(s_1, s_2) is the transition probability from s_1 to s_2, so for every s \in S we have \sum_ P(s, s') = 1. A stationary measure for P is a probability measure \nu on S such that \nu P = \nu ; that is \sum_ \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 with its product σ-algebra by giving the measures of the
cylinders A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an in ...
as follows: \mu_\nu(\cdots \times S \times \ \times S \times \cdots) = \nu(s_n) P(s_n, s_) \cdots P(s_, s_m). Stationarity of \nu then means that the measure \mu_\nu is invariant under the shift map T\left(\left(s_k\right)_)\right) = \left(s_\right)_.


Criterion for ergodicity

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). 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 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 tha ...
if in addition each state is
aperiodic A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to desc ...
(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). Moreover the criterion is an "if and only if" if all communicating classes in the chain are recurrent and we consider all stationary measures.


Examples


Counting measure

If P(s, s') = 1/, S, for all s, s' \in S then the stationary measure is the counting measure, the measure \mu_P 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 chains

Markov chains with recurring communicating classes are not irreducible are not ergodic, and this can be seen immediately as follows. If S_1 \subsetneq S are two distinct recurrent communicating classes there are nonzero stationary measures \nu_1, \nu_2 supported on S_1, S_2 respectively and the subsets S_1^\mathbb and S_2^\mathbb are both shift-invariant and of measure 1.2 for the invariant probability measure \frac(\nu_1 + \nu_2). A very simple example of that is the chain on S = \ given by the matrix \left(\begin 1 & 0 \\ 0 & 1 \end\right) (both states are stationary).


A periodic chain

The Markov chain on S = \ given by the matrix \left(\begin 0 & 1 \\ 1 & 0 \end\right) is irreducible but periodic. Thus it is not ergodic in the sense of Markov chain though the associated measure \mu on \^ is ergodic for the shift map. However the shift is not mixing for this measure, as for the sets A = \cdots \times \ \times 1 \times \ \times 1 \times \ \cdots and B = \cdots \times \ \times 2 \times \ \times 2 \times \ \cdots we have \mu(A) = \frac = \mu(B) but \mu\left(T^A \cap B\right) = \begin \frac \text n \text \\ 0 \text n \text \end


Generalisations

The definition of ergodicity also makes sense for
group action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
s. 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 measure In mathematics, a quasi-invariant measure ''μ'' with respect to a transformation ''T'', from a measure space ''X'' to itself, is a measure which, roughly speaking, is multiplied by a numerical function of ''T''. An important class of examples ...
s. Important examples are the action of a semisimple Lie group (or a lattice therein) on its
Furstenberg boundary In potential theory, a discipline within applied mathematics, the Furstenberg boundary is a notion of boundary associated with a group. It is named for Harry Furstenberg, who introduced it in a series of papers beginning in 1963 (in the case of ...
. A measurable equivalence relation it is said to be ergodic if all saturated subsets are either null or conull.


Notes


References

* *


External links

{{wiktionary, ergodic * Karma Dajani and Sjoerd Dirksin
"A Simple Introduction to Ergodic Theory"
Ergodic theory