In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
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 in a probability space, where the index of the family often has the interpretation of time. Sto ...
, 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 tra ...
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 is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, "statistical properties" refers to properties which are expressed through the behav ...
is the study of systems possessing ergodicity.
Ergodic systems occur in a broad range of systems in
physics
Physics is the scientific study of matter, its Elementary particle, 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 whi ...
and in
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
. 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 locally 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 conn ...
s on a
hyperbolic manifold 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, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
, 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 magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
and
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
s, and rather specifically on the notion of a
measure-preserving dynamical system. The origins of ergodicity lie in
statistical physics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
, where
Ludwig Boltzmann
Ludwig Eduard Boltzmann ( ; ; 20 February 1844 – 5 September 1906) was an Austrian mathematician and Theoretical physics, theoretical physicist. His greatest achievements were the development of statistical mechanics and the statistical ex ...
formulated the
ergodic hypothesis.
Informal explanation
Ergodicity occurs in broad settings in
physics
Physics is the scientific study of matter, its Elementary particle, 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 whi ...
and
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
. All of these settings are unified by a common mathematical description, that of the
measure-preserving dynamical system. Equivalently, ergodicity can be understood in terms of
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 in a probability space, where the index of the family often has the interpretation of time. Sto ...
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 definite pattern or predictability in information. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. ...
. 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 p ...
and
Brownian motion
Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
, 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
Saturn has the most extensive and complex ring system of any planet in the Solar System. The rings consist of particles in orbit around the planet made almost entirely of water ice, with a trace component of rocky material. Particles range fro ...
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
The set
is understood to be the total space to be filled: the mixing bowl, the smoke-filled room, ''etc.'' The
measure is understood to define the natural
volume
Volume is a measure of regions in 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
and of its subspaces. The collection of subspaces is denoted by
, and the size of any given
subset
In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), 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 a ...
is
; the size is its volume. Naively, one could imagine
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 po ...
of
; this doesn't quite work, as not all subsets of a space have a volume (famously, the
Banach–Tarski paradox
The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then ...
). Thus, conventionally,
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 subset of a topological space that can be formed from its open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets ...
—the collection of subsets that can be constructed by taking
intersections,
unions and
set complement
In set theory, the complement of a set , often denoted by A^c (or ), is the set of elements not in .
When all elements in the universe, i.e. all elements under consideration, are considered to be members of a given set , the absolute complemen ...
s of open sets; these can always be taken to be measurable.
The time evolution of the system is described by a
map
A map is a symbolic depiction of interrelationships, commonly spatial, between things within a space. A map may be annotated with text and graphics. Like any graphic, a map may be fixed to paper or other durable media, or may be displayed on ...
. Given some subset
, its image
will in general be a deformed version of
– it is squashed or stretched, folded or cut into pieces. Mathematical examples include the
baker's map and the
horseshoe map, both inspired by
bread
Bread is a baked food product made from water, flour, and often yeast. It is a staple food across the world, particularly in Europe and the Middle East. Throughout recorded history and around the world, it has been an important part of many cu ...
-making. The set
must have the same volume as
; 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
with
. Worse, a single point
has no size. These difficulties can be avoided by working with the inverse map
; it will map any given subset
to the parts that were assembled to make it: these parts are
. 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
is the inverse of some map
. The proper definition of a volume-preserving map is one for which
because
describes all the pieces-parts that
came from.
One is now interested in studying the time evolution of the system. If every set
eventually comes to fill all of
over a long period of time (that is, if
approaches all of
for large
), the system is said to be
ergodic. If every set
satisfies , the system is a
conservative system In mathematics, a conservative system is a dynamical system which stands in contrast to a dissipative system. Roughly speaking, such systems have no friction or other mechanism to dissipate the dynamics, and thus, their phase space does not shrink o ...
, placed in contrast to a
dissipative system
A dissipative system is a thermodynamically open system which is operating out of, and often far from, thermodynamic equilibrium in an environment with which it exchanges energy and matter. A tornado may be thought of as a dissipative system. Di ...
, where some subsets
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 states that every conservative system can be decomposed into a family of ergodic components.
Mixing is a stronger statement than ergodicity. Mixing asks for this ergodic property to hold between any two sets
, and not just between some set
and
. That is, given any two sets
, a system is said to be (topologically) mixing if there is an integer
such that, for all
and
, one has that
. Here,
denotes
set intersection
In set theory, the intersection of two sets A and B, denoted by A \cap B, is the set containing all elements of A that also belong to B or equivalently, all elements of B that also belong to A.
Notation and terminology
Intersection is writt ...
and
is the
empty set
In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...
. 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.
Ergodic 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; 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 or probability calculus 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 expre ...
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 magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
; these are the
Kolmogorov axioms.
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
coin-flips; but I want the
'th of them to be heads, and then I don't care about what comes after that". This can be written as the set
where
is "don't care" and
is "heads". The volume of this space is again one-half.
The above is enough to build up a measure-preserving dynamical system, in its entirety. The sets of
or
occurring in the
'th place are called
cylinder sets. 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 subset of a topological space that can be formed from its open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets ...
defined above. In formal terms, the cylinder sets form the
base for a
topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
on the
space
Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...
of all possible infinite-length coin-flips. The measure
has all of the common-sense properties one might hope for: the measure of a cylinder set with
in the
'th position, and
in the
'th position is obviously 1/4, and so on. These common-sense properties persist for set-complement and set-union: everything except for
and
in locations
and
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.
For the coin-flip process, the time-evolution operator
is the
shift operator
In mathematics, and in particular functional analysis, the shift operator, also known as the translation operator, is an operator that takes a function
to its translation . In time series analysis, the shift operator is called the '' lag opera ...
that says "throw away the first coin-flip, and keep the rest". Formally, if
is a sequence of coin-flips, then
. The measure is obviously shift-invariant: as long as we are talking about some set
where the first coin-flip
is the "don't care" value, then the volume
does not change:
. In order to avoid talking about the first coin-flip, it is easier to define
as inserting a "don't care" value into the first position:
. With this definition, one obviously has that
with no constraints on
. This is again an example of why
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
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 in a probability space, where the index of the family often has the interpretation of time. Sto ...
. Thus, an informal definition of ergodicity is that a sequence is ergodic if it visits all of
; 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
), 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
are representative of
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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s. Explicitly, given a sequence
, the corresponding real number is
:
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:
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 mentioned by German mathematician Georg Cantor in 1883.
Throu ...
, 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
:
In the end, these are all "the same thing".
The Cantor set plays key roles in many branches of mathematics. In
recreational mathematics
Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research-and-application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited ...
, 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 (38 ...
, it appears in a vast variety of theorems. A key one for stochastic processes is the
Wold decomposition, which states that any
stationary process
In mathematics and statistics, a stationary process (also called a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose statistical properties, such as mean and variance, do not change over time. M ...
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 (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
In mathematics, symbolic dynamics is the study of dynamical systems defined on a discrete space consisting of infinite sequences of abstract symbols. The evolution of the dynamical system is defined as a simple shift of the sequence.
Because of t ...
. Important special cases include
subshifts of finite type and
sofic systems.
History and etymology
The term ''ergodic'' is commonly thought to derive from the
Greek
Greek may refer to:
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 of all kno ...
words (''ergon'': "work") and (''hodos'': "path", "way"), as chosen by
Ludwig Boltzmann
Ludwig Eduard Boltzmann ( ; ; 20 February 1844 – 5 September 1906) was an Austrian mathematician and Theoretical physics, theoretical physicist. His greatest achievements were the development of statistical mechanics and the statistical ex ...
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. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
. 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 (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
, 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 a notion of thermodynamics with axiomatic status referring to an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable ...
could be defined with
mathematical rigor
Rigour (British English) or rigor (American English; see spelling differences) describes a condition of stiffness or strictness. These constraints may be environmentally imposed, such as "the rigours of famine"; logically imposed, such as mat ...
. Once the theory was well developed in
physics
Physics is the scientific study of matter, its Elementary particle, 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 whi ...
, it was rapidly formalized and extended, so that
ergodic theory
Ergodic theory is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, "statistical properties" refers to properties which are expressed through the behav ...
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 refers to physics theories that are non-quantum or both non-quantum and non-relativistic, depending on the context. In historical discussions, ''classical physics'' refers to pre-1900 physics, while '' modern physics'' refers to ...
the term implies that a system satisfies the
ergodic hypothesis of
thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
,
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 systems, complex dynamical systems, usually by employing differential equations by nature of the ergodic theory, ergodicity of dynamic systems. When differ ...
the state space is usually taken to be a more general
phase space
The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...
. On the other hand in
coding theory
Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and computer data storage, data sto ...
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
Michel Plancherel (; 16 January 1885 – 4 March 1967) was a Swiss people, Swiss mathematician.
Biography
He was born in Bussy, Fribourg, Bussy (Canton of Fribourg, Switzerland) and obtained his Diplom in mathematics from the University of Fribou ...
proved the strict impossibility of ergodicity for a purely mechanical system.
Ergodicity in physics and geometry
A review of ergodicity in physics, and in
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
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.
Physical systems can be split into three categories:
classical mechanics
Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...
, which describes machines with a finite number of moving parts,
quantum mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
, 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. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
, 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 State of matter, phases, that arise from electromagnetic forces between atoms and elec ...
. These presented below.
In statistical mechanics
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 scientific study of matter, its Elementary particle, 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 whi ...
. Consider a container of
liquid
Liquid is a state of matter with a definite volume but no fixed shape. Liquids adapt to the shape of their container and are nearly incompressible, maintaining their volume even under pressure. The density of a liquid is usually close to th ...
, or
gas, or
plasma, or other collection of
atom
Atoms are the basic particles of the chemical elements. An atom consists of a atomic nucleus, nucleus of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished fr ...
s or
particle
In the physical sciences, a particle (or corpuscle 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 ...
s. Each and every particle
has a 3D position, and a 3D velocity, and is thus described by six numbers: a point in six-dimensional space
If there are
of these particles in the system, a complete description requires
numbers. Any one system is just a single point in
The
physical system
A physical system is a collection of physical objects under study. The collection differs from a set: all the objects must coexist and have some physical relationship.
In other words, it is a portion of the physical universe chosen for analys ...
is not all of
, of course; if it's a box of width, height and length
then a point is in
Nor can velocities be infinite: they are scaled by some probability measure, for example the
Boltzmann–Gibbs measure for a gas. Nonetheless, for
close to the
Avogadro number
The Avogadro constant, commonly denoted or , is an SI defining constant with an exact value of when expressed in reciprocal moles.
It defines the ratio of the number of constituent particles to the amount of substance in a sample, where th ...
, 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
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 states that physical systems actually are ergodic. Multiple time scales are at work: gases 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
A phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. In the context of optically trapped objects, the quantized vibration mode can be defined a ...
s, as obviously the atoms in a solid do not exchange locations.
Glass
Glass is an amorphous (non-crystalline solid, non-crystalline) solid. Because it is often transparency and translucency, transparent and chemically inert, glass has found widespread practical, technological, and decorative use in window pane ...
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 pro ...
-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 ...
or plasma. The first hard-sphere ergodicity theorem was for
Sinai's billiards, 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 and chemistry, the van der Waals force (sometimes 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 elec ...
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.
Simple dynamical systems
The formal study of ergodicity can be approached by examining fairly simple dynamical systems. Some of the primary ones are listed here.
The
irrational rotation of a circle is ergodic: the
orbit
In celestial mechanics, an orbit (also known as orbital revolution) 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 ...
of a point is such that eventually, every other point in the circle is visited. Such rotations are a special case of the
interval exchange map. The
beta expansions of a number are ergodic: beta expansions of a real number are done not in base-''N'', but in base-
for some
The reflected version of the beta expansion is
tent map; there are a variety of other ergodic maps of the unit interval. Moving to two dimensions, the
arithmetic billiards with irrational angles are ergodic. One can also take a flat rectangle, squash it, cut it and reassemble it; this is the previously-mentioned
baker's map. Its points can be described by the set of bi-infinite strings in two letters, that is, extending to both the left and right; as such, it looks like two copies of the Bernoulli process. If one deforms sideways during the squashing, one obtains
Arnold's cat map. In most ways, the cat map is prototypical of any other similar transformation.
In classical mechanics and geometry
Ergodicity is a widespread phenomenon in the study of
symplectic manifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sy ...
s and
Riemannian manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
s. Symplectic manifolds provide the generalized setting for
classical mechanics
Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...
, where the motion of a mechanical system is described by a
geodesic
In geometry, a geodesic () is a curve representing in some sense the locally 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 conn ...
. Riemannian manifolds are a special case: the
cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This m ...
of a Riemannian manifold is always a symplectic manifold. In particular, the geodesics on a Riemannian manifold are given by the solution of the
Hamilton–Jacobi equations.
The
geodesic flow of a
flat torus 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.
For non-flat surfaces, one has that the
geodesic flow of any negatively curved
compact Riemann surface is ergodic. A surface is "compact" in the sense that it has finite surface area. The geodesic flow is a generalization of the idea of moving in a "straight line" on a curved surface: such straight lines are
geodesic
In geometry, a geodesic () is a curve representing in some sense the locally 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 conn ...
s. One of the earliest cases studied is
Hadamard's billiards, which describes geodesics on the
Bolza surface, topologically equivalent to a donut with two holes. Ergodicity can be demonstrated informally, if one has a sharpie and some reasonable example of a two-holed donut: starting anywhere, in any direction, one attempts to draw a straight line; rulers are useful for this. It doesn't take all that long to discover that one is not coming back to the starting point. (Of course, crooked drawing can also account for this; that's why we have proofs.)
These results extend to higher dimensions. The geodesic flow for negatively curved compact
Riemannian manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
s is ergodic. A classic example for this is the
Anosov flow
In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold ''M'' is a certain type of mapping, from ''M'' to itself, with rather clearly marked local directions of "expansion" and "contr ...
, which is the
horocycle flow on a
hyperbolic manifold. This can be seen to be a kind of
Hopf fibration. Such flows commonly occur in
classical mechanics
Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...
, which is the study in
physics
Physics is the scientific study of matter, its Elementary particle, 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 whi ...
of finite-dimensional moving machinery, e.g. the
double pendulum
In physics and mathematics, in the area of dynamical systems, a double pendulum, also known as a chaotic pendulum, is a pendulum with another pendulum attached to its end, forming a simple physical system that exhibits rich dynamical systems, dy ...
and so-forth. Classical mechanics is constructed on
symplectic manifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sy ...
s. The flows on such systems can be deconstructed into
stable and unstable manifolds; as a general rule, when this is possible, chaotic motion results. That this is generic can be seen by noting that the
cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This m ...
of a
Riemannian manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
is (always) a symplectic manifold; the geodesic flow is given by a solution to the
Hamilton–Jacobi equation
In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mecha ...
s for this manifold. In terms of the
canonical coordinates
on the cotangent manifold, the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
or
energy
Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
is given by
:
with
the (inverse of the)
metric tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
and
the
momentum
In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. ...
. The resemblance to the
kinetic energy
In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion.
In classical mechanics, the kinetic energy of a non-rotating object of mass ''m'' traveling at a speed ''v'' is \fracmv^2.Resnick, Rober ...
of a
point particle
A point particle, ideal particle or point-like particle (often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take ...
is hardly accidental; this is the whole point of calling such things "energy". In this sense, chaotic behavior with ergodic orbits is a more-or-less generic phenomenon in large tracts of geometry.
Ergodicity results have been provided in
translation surfaces,
hyperbolic groups and
systolic geometry
In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and ...
. Techniques include the study of
ergodic flows, the
Hopf decomposition, and the
Ambrose–Kakutani–Krengel–Kubo theorem. An important class of systems are the
Axiom A systems.
A number of both classification and "anti-classification" results have been obtained. The
Ornstein isomorphism theorem applies here as well; again, it states that most of these systems are isomorphic to some
Bernoulli scheme. This rather neatly ties these systems back into the definition of ergodicity given for a stochastic process, in the previous section. The anti-classification results state that there are more than a
countably infinite
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
number of inequivalent ergodic measure-preserving dynamical systems. This is perhaps not entirely a surprise, as one can use points in the Cantor set to construct similar-but-different systems. See
measure-preserving dynamical system for a brief survey of some of the anti-classification results.
In wave mechanics
All of the previous sections considered ergodicty either from the point of view of a measurable dynamical system, or from the dual notion of tracking the motion of individual particle trajectories. A closely related concept occurs in (non-linear)
wave mechanics. There, the
resonant interaction allows for the mixing of
normal mode
A normal mode of a dynamical system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at fixed frequencies ...
s, often (but not always) leading to the eventual
thermalization of the system. One of the earliest systems to be rigorously studied in this context is the
Fermi–Pasta–Ulam–Tsingou problem, a string of weakly coupled oscillators.
A resonant interaction is possible whenever the
dispersion relation
In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the ...
s for the wave media allow three or more normal modes to sum in such a way as to conserve both the total momentum and the total energy. This allows energy concentrated in one mode to bleed into other modes, eventually distributing that energy uniformly across all interacting modes.
Resonant interactions between waves helps provide insight into the distinction between
high-dimensional chaos (that is,
turbulence
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to laminar flow, which occurs when a fluid flows in parallel layers with no disruption between ...
) and thermalization. When normal modes can be combined so that energy and momentum are exactly conserved, then the theory of resonant interactions applies, and energy spreads into all of the interacting modes. When the dispersion relations only allow an approximate balance, turbulence or chaotic motion results. The turbulent modes can then transfer energy into modes that do mix, eventually leading to thermalization, but not before a preceding interval of chaotic motion.
In quantum mechanics
As to quantum mechanics, there is no universal quantum definition of ergodicity 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
. 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 does 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.
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
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.
It captures and generalises intuitive notions such as length, area, an ...
. If
is a measurable function from
to itself and
a
probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...
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
).
Some authors relax the requirement that
preserves
to the requirement that
is a non-singular transformation with respect to
, meaning that if
is a subset so that
has zero measure, then so does
.
Examples
The simplest example is when
is a finite set and
the
counting measure
In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinit ...
. 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
In celestial mechanics, an orbit (also known as orbital revolution) 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 ...
(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
).
Equivalent formulations
The definition given above admits the following immediate reformulations:
* for every
with
we have
or
(where
denotes the
symmetric difference
In mathematics, the symmetric difference of two sets, also known as the disjunctive union and set sum, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets \ and ...
);
* 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 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
and
then
is constant almost everywhere.
Further examples
Bernoulli shifts and subshifts
Let
be a finite set and
with
the
product measure
In mathematics, given two measurable spaces and measures on them, one can obtain a product measurable space and a product measure on that space. Conceptually, this is similar to defining the Cartesian product of sets and the product topology o ...
(each factor
being endowed with its counting measure). Then the
shift operator
In mathematics, and in particular functional analysis, the shift operator, also known as the translation operator, is an operator that takes a function
to its translation . In time series analysis, the shift operator is called the '' lag opera ...
defined by
is .
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.
Irrational rotations
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.
Arnold's cat map
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.
Ergodic theorems
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.
Related properties
Dense orbits
An immediate consequence of the definition of ergodicity is that on a topological space
, and if
is the σ-algebra of
Borel set
In mathematics, a Borel set is any subset of a topological space that can be formed from its open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets ...
s, 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.
Mixing
A transformation
of a probability measure space
is said to be mixing for the measure
if for any measurable sets
the following holds:
Proper ergodicity
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 continuous-time dynamical systems
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
, if for all
we have
then either
or
.
Examples
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.
Ergodic flows
Further examples of ergodic flows are:
*
Billiards
Cue sports are a wide variety of games of skill played with a cue stick, which is used to strike billiard balls and thereby cause them to move around a cloth-covered table bounded by elastic bumpers known as . Cue sports, a category of stic ...
in convex Euclidean domains;
* the
geodesic flow of a negatively curved Riemannian manifold of finite volume is ergodic (for the normalised volume measure);
* the
horocycle flow on a
hyperbolic manifold of finite volume is ergodic (for the normalised volume measure)
Ergodicity in compact metric spaces
If
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, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
it is naturally endowed with the σ-algebra of
Borel set
In mathematics, a Borel set is any subset of a topological space that can be formed from its open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets ...
s. The additional structure coming from the topology then allows a much more detailed theory for ergodic transformations and measures on
.
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 (, ) 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 vectors and ...
s.
Radon measure
In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the -algebra of Borel sets of a Hausdorff topological space that is finite on all compact sets, outer regular on all Borel sets, and ...
s on
form a Banach space of which the set
of probability measures on
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
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
In mathematics, an extreme point of a convex set S in a Real number, real or Complex number, complex vector space is a point in S that does not lie in any open line segment joining two points of S. The extreme points of a line segment are calle ...
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
. 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 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
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