In
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 ...
, 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
Classical thermodynamics considers three main kinds of thermodynamic process: (1) changes in a system, (2) cycles in a system, and (3) flow processes.
(1)A Thermodynamic process is a process in which the thermodynamic state of a system is change ...
of
mixing in the everyday world: mixing paint, mixing drinks,
industrial mixing, ''etc''.
The concept appears in
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 expres ...
—the study 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. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
es 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 ca ...
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 th ...
: 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 ca ...
, 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
In mathematics, 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 are ...
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). 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 named ...
—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 the ...
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
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 comp ...
and the
horseshoe map
In the mathematics of chaos theory, a horseshoe map is any member of a class of chaotic maps of the square into itself. It is a core example in the study of dynamical systems. The map was introduced by Stephen Smale while studying the behavi ...
, 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
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
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
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 ...
, 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
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 other ...
.
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
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 simil ...
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 o ...
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 ca ...
, 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 o ...
. 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, Unit ...
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 les ...
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 th ...
.
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 with ...
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
The dyadic transformation (also known as the dyadic map, bit shift map, 2''x'' mod 1 map, Bernoulli map, doubling map or sawtooth map) is the mapping (i.e., recurrence relation)
: T: , 1) \to , Arnold's cat map">, 1)^\infty
: x \mapsto (x_0, x_1, x_2, ...
s, Kolmogorov automorphisms, and the Anosov flow (the geodesic flow on the unit tangent bundle of compact manifolds of negative curvature.)
of the system. A
of ''f''. In the
. A related idea is expressed by the
, with ''g'' being the continuous parameter, with the requirement that a non-empty intersection hold for all
(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.
. The sequence space into which the process maps can be endowed with a topology, the
. The
s. These cylinder sets generate a
. The symbol
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
. That is to say, a strongly mixing process is such that, in a way that is uniform over all times
; more colloquially, the process, in a strong sense, forgets its history.
denote the space of Borel-measurable functions that are square-integrable with respect to the measure
. Also 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
'' may either decay at an exponential rate, or be always equal to one.
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
.