![Baker's map mixing](https://upload.wikimedia.org/wikipedia/commons/8/8c/Baker%27s_map_mixing.gif)
In
mathematics, mixing is an abstract concept originating from
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 r ...
: the attempt to describe the irreversible
thermodynamic process of
mixing in the everyday world: mixing paint, mixing drinks,
industrial mixing, ''etc''.
The concept appears in
ergodic theory—the study of
stochastic processes and
measure-preserving dynamical system
In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special cas ...
s. Several different definitions for mixing exist, including ''strong mixing'', ''weak mixing'' and ''topological mixing'', with the last not requiring a
measure
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Mea ...
to be defined. Some of the different definitions of mixing can be arranged in a hierarchical order; thus, strong mixing implies weak mixing. Furthermore, weak mixing (and thus also strong mixing) implies
ergodicity
In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies tha ...
: that is, every system that is weakly mixing is also ergodic (and so one says that mixing is a "stronger" notion than ergodicity).
Informal explanation
The mathematical definition of mixing aims to capture the ordinary every-day process of mixing, such as mixing paints, drinks, cooking ingredients,
industrial process mixing, smoke in a smoke-filled room, and so on. To provide the mathematical rigor, such descriptions begin with the definition of a
measure-preserving dynamical system
In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special cas ...
, written as
.
The set
is understood to be the total space to be filled: the mixing bowl, the smoke-filled room, ''etc.'' The
measure
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Mea ...
is understood to define the natural volume of the space
and of its subspaces. The collection of subspaces is denoted by
, and the size of any given
subset 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 post ...
of
; this doesn't quite work, as not all subsets of a space have a volume (famously, the
Banach-Tarski paradox). 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 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,
unions and
set complement
In set theory, the complement of a set , often denoted by (or ), is the set of elements not in .
When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set , the absolute complement of is th ...
s; 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 emphasizing relationships between elements of some space, such as objects, regions, or themes.
Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
. Given some subset
, its map
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-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 a set
eventually visits 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
behaves in this way, 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. Dis ...
, 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 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
, 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.
The above definition of topological mixing should be enough to provide an informal idea of mixing (it is equivalent to the formal definition, given below). However, it made no mention of the volume of
and
, and, indeed, there is another definition that explicitly works with the volume. Several, actually; one has both strong mixing and weak mixing; they are inequivalent, although a strong mixing system is always weakly mixing. The measure-based definitions are not compatible with the definition of topological mixing: there are systems which are one, but not the other. The general situation remains cloudy: for example, given three sets
, one can define 3-mixing. As of 2020, it is not known if 2-mixing implies 3-mixing. (If one thinks of ergodicity as "1-mixing", then it is clear that 1-mixing does not imply 2-mixing; there are systems that are ergodic but not mixing.)
The concept of ''strong mixing'' is made in reference to the volume of a pair of sets. Consider, for example, a set
of colored dye that is being mixed into a cup of some sort of sticky liquid, say, corn syrup, or shampoo, or the like. Practical experience shows that mixing sticky fluids can be quite hard: there is usually some corner of the container where it is hard to get the dye mixed into. Pick as set
that hard-to-reach corner. The question of mixing is then, can
, after a long enough period of time, not only penetrate into
but also fill
with the same proportion as it does elsewhere?
One phrases the definition of strong mixing as the requirement that
:
The time parameter
serves to separate
and
in time, so that one is mixing
while holding the test volume
fixed. The product
is a bit more subtle. Imagine that the volume
is 10% of the total volume, and that the volume of dye
will also be 10% of the grand total. If
is uniformly distributed, then it is occupying 10% of
, which itself is 10% of the total, and so, in the end, after mixing, the part of
that is in
is 1% of the total volume. That is,
This product-of-volumes has more than passing resemblance to
Bayes theorem
In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For examp ...
in probabilities; this is not an accident, but rather a consequence that
measure theory and
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 the same theory: they share the same axioms (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 ...
), even as they use different notation.
The reason for using
instead of
in the definition is a bit subtle, but it follows from the same reasons why
was used to define the concept of a measure-preserving map. When looking at how much dye got mixed into the corner
, one wants to look at where that dye "came from" (presumably, it was poured in at the top, at some time in the past). One must be sure that every place it might have "come from" eventually gets mixed into
.
Mixing in dynamical systems
Let
be a
measure-preserving dynamical system
In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special cas ...
, with ''T'' being the time-evolution or
shift operator
In mathematics, and in particular functional analysis, the shift operator also known as translation operator is an operator that takes a function
to its translation . In time series analysis, the shift operator is called the lag operator.
Shift ...
. The system is said to be strong mixing if, for any
, one has
:
For shifts parametrized by a continuous variable instead of a discrete integer ''n'', the same definition applies, with
replaced by
with ''g'' being the continuous-time parameter.
A dynamical system is said to be weak mixing if one has
:
In other words,
is strong mixing if
in the usual sense, weak mixing if
:
in the
Cesàro sense, and ergodic if
in the Cesàro sense. Hence, strong mixing implies weak mixing, which implies ergodicity. However, the converse is not true: there exist ergodic dynamical systems which are not weakly mixing, and weakly mixing dynamical systems which are not strongly mixing. The
Chacon system Chacon may refer to:
* Chacón, a list of people with the surname Chacón or Chacon
* Captain Trudy Chacon, a fictional character in the 2009 film ''Avatar''
* Chacon, New Mexico, United States, a town
* Chacon Creek, a small stream in Texas, U ...
was historically the first example given of a system that is weak-mixing but not strong-mixing.
[
Matthew Nicol and Karl Petersen, (2009)]
Ergodic Theory: Basic Examples and Constructions
,
''Encyclopedia of Complexity and Systems Science'', Springer https://doi.org/10.1007/978-0-387-30440-3_177
''L''2 formulation
The properties of ergodicity, weak mixing and strong mixing of a measure-preserving dynamical system can also be characterized by the average of observables. By von Neumann's ergodic theorem, ergodicity of a dynamical system
is equivalent to the property that, for any function
, the sequence
converges strongly and in the sense of Cesàro to
, i.e.,
:
A dynamical system
is weakly mixing if, for any functions
and
:
A dynamical system
is strongly mixing if, for any function
the sequence
converges weakly to
i.e., for any function
:
Since the system is assumed to be measure preserving, this last line is equivalent to saying that the
covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the ...
so that the random variables
and
become orthogonal as
grows. Actually, since this works for any function
one can informally see mixing as the property that the random variables
and
become independent as
grows.
Products of dynamical systems
Given two measured dynamical systems
and
one can construct a dynamical system
on the Cartesian product by defining
We then have the following characterizations of weak mixing:
:Proposition. A dynamical system
is weakly mixing if and only if, for any ergodic dynamical system
, the system
is also ergodic.
:Proposition. A dynamical system
is weakly mixing if and only if
is also ergodic. If this is the case, then
is also weakly mixing.
Generalizations
The definition given above is sometimes called strong 2-mixing, to distinguish it from higher orders of mixing. A strong 3-mixing system may be defined as a system for which
:
holds for all measurable sets ''A'', ''B'', ''C''. We can define strong k-mixing similarly. A system which is strong ''k''-mixing for all ''k'' = 2,3,4,... is called mixing of all orders.
It is unknown whether strong 2-mixing implies strong 3-mixing. It is known that strong ''m''-mixing implies
ergodicity
In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies tha ...
.
Examples
Irrational rotation
In the mathematical theory of dynamical systems, an irrational rotation is a map
: T_\theta : ,1\rightarrow ,1\quad T_\theta(x) \triangleq x + \theta \mod 1
where ''θ'' is an irrational number. Under the identification of a circle wit ...
s of the circle, and more generally irreducible translations on a torus, are ergodic but neither strongly nor weakly mixing with respect to the Lebesgue measure.
Many maps considered as chaotic are strongly mixing for some well-chosen invariant measure, including: the
dyadic map,
Arnold's cat map,
horseshoe maps,
Kolmogorov automorphism In mathematics, a Kolmogorov automorphism, ''K''-automorphism, ''K''-shift or ''K''-system is an invertible, measure-preserving automorphism defined on a standard probability space that obeys Kolmogorov's zero–one law.Peter Walters, ''An Introduct ...
s, and 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 ...
(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. ...
on the unit
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
of
compact manifold
In mathematics, a closed manifold is a manifold without boundary that is compact.
In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components.
Examples
The only connected one-dimensional example is ...
s of
negative curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canoni ...
.)
Topological mixing
A form of mixing may be defined without appeal to a
measure
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Mea ...
, using only the
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 ...
of the system. A
continuous map
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
is said to be topologically transitive if, for every pair of non-empty
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
s
, there exists an integer ''n'' such that
:
where
is the
''n''th iterate of ''f''. In the
operator theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators ...
, a topologically transitive
bounded linear operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vect ...
(a continuous linear map on a
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
) is usually called
hypercyclic operator. A related idea is expressed by the
wandering set
In dynamical systems and ergodic theory, the concept of a wandering set formalizes a certain idea of movement and mixing. When a dynamical system has a wandering set of non-zero measure, then the system is a dissipative system. This is the opposit ...
.
Lemma: If ''X'' is a
complete metric space
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in .
Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
with no
isolated point
]
In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equiva ...
, then ''f'' is topologically transitive if and only if there exists a
hypercyclic vector, hypercyclic point , that is, a point ''x'' such that its orbit
is
dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
in ''X''.
A system is said to be topologically mixing if, given open sets
and
, there exists an integer ''N'', such that, for all
, one has
:
For a continuous-time system,
is replaced by the
flow , with ''g'' being the continuous parameter, with the requirement that a non-empty intersection hold for all
.
A weak topological mixing is one that has no non-constant
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
(with respect to the topology) eigenfunctions of the shift operator.
Topological mixing neither implies, nor is implied by either weak or strong mixing: there are examples of systems that are weak mixing but not topologically mixing, and examples that are topologically mixing but not strong mixing.
Mixing in stochastic processes
Let
be a
stochastic process on a probability space
. The sequence space into which the process maps can be endowed with a topology, the
product topology
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more natural-s ...
. The
open set
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
s of this topology are called
cylinder sets. These cylinder sets generate a
σ-algebra
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set ''X'' is a collection Σ of subsets of ''X'' that includes the empty subset, is closed under complement, and is closed under countable unions and countabl ...
, the
Borel σ-algebra
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 nam ...
; this is the smallest σ-algebra that contains the topology.
Define a function
, called the strong mixing coefficient, as
:
for all
. The symbol
, with
denotes a sub-σ-algebra of the σ-algebra; it is the set of cylinder sets that are specified between times ''a'' and ''b'', i.e. the σ-algebra generated by
.
The process
is said to be strongly mixing if
as
. That is to say, a strongly mixing process is such that, in a way that is uniform over all times
and all events, the events before time
and the events after time
tend towards being
independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independ ...
as
; more colloquially, the process, in a strong sense, forgets its history.
Mixing in Markov processes
Suppose
were a stationary
Markov process
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 happe ...
with stationary distribution
and let
denote the space of Borel-measurable functions that are square-integrable with respect to the measure
. Also let
:
denote the conditional expectation operator on
Finally, let
:
denote the space of square-integrable functions with mean zero.
The ''ρ''-mixing coefficients of the process are
:
The process is called ''ρ''-mixing if these coefficients converge to zero as , and “''ρ''-mixing with exponential decay rate” if for some . For a stationary Markov process, the coefficients ''ρ
t'' may either decay at an exponential rate, or be always equal to one.
The ''α''-mixing coefficients of the process are
:
The process is called ''α''-mixing if these coefficients converge to zero as , it is “α-mixing with exponential decay rate” if for some , and it is α-mixing with a sub-exponential decay rate if for some non-increasing function
satisfying
:
as
.
[
The ''α''-mixing coefficients are always smaller than the ''ρ''-mixing ones: , therefore if the process is ''ρ''-mixing, it will necessarily be ''α''-mixing too. However, when , the process may still be ''α''-mixing, with sub-exponential decay rate.
The ''β''-mixing coefficients are given by
:
The process is called ''β''-mixing if these coefficients converge to zero as , it is β-mixing with an exponential decay rate if for some , and it is β-mixing with a sub-exponential decay rate if as for some non-increasing function satisfying
:
as .][
A strictly stationary Markov process is ''β''-mixing if and only if it is an aperiodic recurrent Harris chain. The ''β''-mixing coefficients are always bigger than the ''α''-mixing ones, so if a process is ''β''-mixing it will also be ''α''-mixing. There is no direct relationship between ''β''-mixing and ''ρ''-mixing: neither of them implies the other.
]
References
* V. I. Arnold and A. Avez, ''Ergodic Problems of Classical Mechanics'', (1968) W. A. Benjamin, Inc.
* Achim Klenke, ''Probability Theory'', (2006) Springer
*
{{reflist
Ergodic theory