Mixing (mathematics)

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In mathematics, mixing is a concept applied in ergodic theory, that is, the study of stochastic processes and measure-preserving dynamical systems. Several different definitions for mixing can be made, including strong mixing, weak mixing and topological mixing, with the last not even requiring a concept of measure to be defined. Applications of mixing are often seen in the physics of mixing.

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[edit] Mixing in stochastic processes

Let

\langle X_t \rangle = \{ \ldots, X_{t-1}, X_t, X_{t+1}, \ldots \}

be a sequence of random variables, and

X_a^b

the sigma-algebra generated by

\{X_a, X_{a+1},\ldots, X_b \}

for

-\infty \leq a \leq b \leq \infty.

The process \langle X_t \rangle is strong mixing if

\alpha(s)\downarrow 0

as

s\rightarrow \infty,

where

\alpha(s) := \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\}

is the so-called strong mixing coefficient. Here, P is the probability measure.

[edit] Mixing in dynamical systems

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

\lim_{n\to\infty} \mu (A \cap T^{-n}B) = \mu(A)\mu(B)

then the system is called strong mixing. For shifts parameterized by a continuous variable instead of a discrete integer n, the same definition applies, with T − n replaced by Tg with g being the continuous-time parameter.

A dynamical system is said to be weak mixing if

\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. 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, then none of the eigenfunctions can be square integrable.

[edit] Topological mixing

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

f^n(A) \cap B \ne \varnothing

where fn 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 x\in X with a dense orbit, that is, an orbit such that the set \{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 n > N, one has

f^n(A) \cap B \neq \varnothing.

For a continuous-time system, fn is replaced by the flow φg, with g the continuous parameter, with the requirement that a non-empty intersection hold for all \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.

There are examples of systems that are weak mixing but not topologically mixing, and examples that are topologically mixing but not strong mixing.

[edit] Generalizations

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

\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 not known if strong 2-mixing implies strong 3-mixing. It is known that strong m-mixing implies ergodicity.

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