Mixing (mathematics)

It has been suggested that this article be merged with Mixing (physics). (Discuss) Proposed since March 2008.

In mathematics, mixing is an abstract concept originating from physics: the attempt to describe the irreversible thermodynamic process of mixing in the everyday world: mixing paint, mixing drinks, etc.

The concept appears in ergodic theory—the study of stochastic processes and measure-preserving dynamical systems. Several different definitions for mixing exist, including strong mixing, weak mixing and topological mixing, with the last not requiring a measure to be defined. Some of the different definitions of mixing can be arranged in a hierarchical order; thus, strong mixing implies weak mixing. In all cases, ergodicity is implied: that is, every system that is mixing is also ergodic (and so one says that mixing is a "stronger" notion than ergodicity).

Let ${\displaystyle \langle X_{t}\rangle =\{\ldots ,X_{t-1},X_{t},X_{t+1},\ldots \}}$ be a sequence of random variables. Such a sequence is naturally endowed with a topology, the product topology. The open sets of this topology are called cylinder sets. These cylinder sets generate a sigma algebra, the Borel sigma algebra; it is the smallest (coarsest) sigma algebra that contains the topology.

Define a function ${\displaystyle \alpha (s)}$, called the strong mixing coefficient, as

${\displaystyle \alpha (s)\equiv \sup \left\{\,|P(A\cap B)-P(A)P(B)|:-\infty <t<\infty ,A\in X_{-\infty }^{t},B\in X_{t+s}^{\infty }\,\right\}.}$

In this definition, P is the probability measure on the sigma algebra. The symbol ${\displaystyle X_{a}^{b}}$, with ${\displaystyle -\infty \leq a\leq b\leq \infty }$ denotes a subalgebra of the sigma algebra; it is the set of cylinder sets that are specified between times a and b. Given specific, fixed values ${\displaystyle X_{a}}$, ${\displaystyle X_{a+1}}$, etc., of the random variable, at times ${\displaystyle a}$, ${\displaystyle a+1}$, etc., then it may be thought of as the sigma-algebra generated by

${\displaystyle \{X_{a},X_{a+1},\ldots ,X_{b}\}.}$

The process ${\displaystyle \langle X_{t}\rangle }$ is strong mixing if ${\displaystyle \alpha (s)\rightarrow 0}$ as ${\displaystyle s\rightarrow \infty }$.

One way to describe this is that strong mixing implies that for any two possible states of the system (realizations of the random variable), when given a sufficient amount of time between the two states, the occurrence of the states is independent.

A similar definition can be given using the vocabulary of measure-preserving dynamical systems. Let ${\displaystyle (X,{\mathcal {A}},\mu ,T)}$ be a dynamical system, with T being the time-evolution or shift operator. Then, if for all ${\displaystyle A,B\in {\mathcal {A}}}$, if one has

${\displaystyle \lim _{n\to \infty }\mu (A\cap T^{-n}B)=\mu (A)\mu (B)}$

then the system is strong mixing. For shifts parametrized by a continuous variable instead of a discrete integer n, the same definition applies, with ${\displaystyle T^{-n}}$ replaced by ${\displaystyle T_{g}}$ with g being the continuous-time parameter.

A dynamical system is said to be weak mixing if

${\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\sum _{k=0}^{n}|\mu (A\cap T^{-k}B)-\mu (A)\mu (B)|=0.}$

Strong mixing implies weak mixing, and every weakly-mixing system is ergodic.

For a system that is weak mixing, the shift operator T will have no (non-constant) square-integrable eigenfunctions with associated eigenvalue of one.^{[citation needed]} In general, a shift operator will have a continuous spectrum, and thus will always have eigenfunctions that are generalized functions. However, for the system to be (at least) weak mixing, none of the eigenfunctions with associated eigenvalue of one can be square integrable.

A form of mixing may be defined without appeal to a measure, only using the topology of the system. A continuous map ${\displaystyle f:X\to X}$ is said to be topologically transitive if, for every pair of non-empty open sets ${\displaystyle A,B\subset X}$, there exists an integer n such that

${\displaystyle f^{n}(A)\cap B\neq \varnothing }$

where ${\displaystyle f^{n}}$ is the n 'th iterate of f. A related idea is expressed by the wandering set.

Lemma: If X is a compact metric space, then f is topologically transitive if and only if there exists a point ${\displaystyle x\in X}$ with a dense orbit, that is, an orbit such that the set ${\displaystyle \{f^{n}(x):n\in \mathbb {N} \}}$ is dense in X.

A system is said to be topologically mixing if there exists an integer N, such that, for all ${\displaystyle n>N}$, one has

${\displaystyle f^{n}(A)\cap B\neq \varnothing }$.

For a continuous-time system, ${\displaystyle f^{n}}$ is replaced by the flow ${\displaystyle \phi _{g}}$, with g being the continuous parameter, with the requirement that a non-empty intersection hold for all ${\displaystyle \Vert g\Vert >N}$.

A weak topological mixing is one that has no non-constant 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.

The definition given above is sometimes called strong 2-mixing, to distinguish it from a generalized definition.

Thus, for example, a strong-3-mixing system may be defined as a system for which

${\displaystyle \lim _{m,n\to \infty }\mu (A\cap T^{-m}B\cap T^{-m-n}C)=\mu (A)\mu (B)\mu (C)}$

holds for all measurable sets A, B, C. Strong n-mixing may be defined analogously.

It is unknown whether strong 2-mixing implies strong 3-mixing. It is known that strong m-mixing implies ergodicity.

• Achim Klenke, Probability Theory, (2006) Springer ISBN 978-1-84800-047-6

• V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics, (1968) W. A. Benjamin, Inc.