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

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. Furthermore, weak mixing (and thus also strong mixing) implies ergodicity: that is, every system that is weakly mixing is also ergodic (and so one says that mixing is a "stronger" condition 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, written as .

is understood to be the total space to be filled: the mixing bowl, the smoke-filled room, etc. The measure and of its subspaces. The collection of subspaces is denoted by , and the size of any given subset is to be the power set of ; this doesn't quite work, as not all subsets of a space have a volume (famously, the Banach–Tarski paradox consists of the measurable subsets—the subsets that do have a volume. It is always taken to be a Borel set—the collection of subsets that can be constructed by taking intersections, unions and set complements; these can always be taken to be measurable.

The time evolution of the system is described by a map – 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; the squashing/stretching does not alter the volume of the space, only its distribution. Such a system is "measure-preserving" (area-preserving, volume-preserving).

with has no size. These difficulties can be avoided by working with the inverse map 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 is the inverse of some map came from.

), the system is said to be ergodic behaves in this way, the system is a conservative system, placed in contrast to a dissipative system 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 hopf decomposition theorem states that every measure space 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 . That is, given any two sets and denotes set intersection is the empty set.

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 with the same proportion as it does elsewhere?

One phrases the definition of strong mixing as the requirement that

This product-of-volumes has more than passing resemblance to Bayes' theorem in probabilities; this is not an accident, but rather a consequence that measure theory and probability theory are the same theory: they share the same axioms (the Kolmogorov axioms), even as they use different notation.

.

Mixing in dynamical systems

be a measure-preserving dynamical system, with T being the time-evolution or shift operator. The system is said to be strong mixing, one has

For shifts parametrized by a continuous variable instead of a discrete integer n with g being the continuous-time parameter.

A dynamical system is said to be weak mixing if one has

in the usual sense, weak mixing if

in the Cesàro sense in the Cesàro sense. Hence, strong mixing implies weak mixing, which implies ergodicity. However, the converses are 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 was historically the first example given of a system that is weak mixing but not strong mixing.

Theorem. Weak mixing implies ergodicity.

Proof. If the action of the map decomposes into two components , then we have , so weak mixing implies has zero measure, and the other one has full measure.

Covering families

Given a topological space, such as the unit interval (whether it has its end points or not), we can construct a measure on it by taking the open sets, then take their unions, complements, unions, complements, and so on to infinity, to obtain all the Borel sets on the Borel sets, then add in all the subsets of measure-zero ("negligible sets"). This is how we obtain the Lebesgue measure and the Lebesgue measurable sets.

, and so it is a Lebesgue measure space. Verifying strong-mixing can be simplified if we only need to check a smaller set of measurable sets.

is a set of measurable sets, such that any open set is a disjoint union of sets in it. Compare this with base in topology, which is less restrictive as it allows non-disjoint unions.

Theorem. is strong mixing.

Proof. for all measurable .

L2 formulation

converges strongly and in the sense of Cesàro to , i.e.,

is strongly mixing if, for any function converges weakly to

Since the system is assumed to be measure preserving, this last line is equivalent to saying that the covariance grows. Actually, since this works for any function grows.

Products of dynamical systems

We then have the following characterizations of weak mixing:

1. Proposition. is weakly mixing if and only if, for any ergodic dynamical system is also ergodic.

   Proposition. 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.

Examples

Irrational rotations 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 automorphisms, and the Anosov flow (the geodesic flow on the unit tangent bundle of compact manifolds of negative curvature.)

, the "shift to left in base is strongly mixing on the covering family , therefore it is strongly mixing on , and therefore it is strongly mixing on .

Similarly, for any finite or countable alphabet , we can impose a discrete probability distribution on it, then consider the probability distribution on the "coin flip" space, where each "coin flip" can take results from or the doubly-infinite space . In both cases, the shift map (one letter to the left) is strongly mixing, since it is strongly mixing on the covering family of cylinder sets. The Baker's map is isomorphic to a shift map, so it is strongly mixing.

Topological mixing

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

is the nth iterate of f. In the operator theory, a topologically transitive bounded linear operator (a continuous linear map on a topological vector space) is usually called hypercyclic operator.

A related idea is expressed by the wandering set, in fact a point is non wandering if

Lemma: If X is a complete metric space with no isolated point, then f is topologically transitive if and only if there exists a hypercyclic point, that is, a point x is dense in X.

A system is said to be topologically mixing and , there exists an integer N, such that, for all , one has

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 (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

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. The open sets of this topology are called cylinder sets. These cylinder sets generate a σ-algebra, the Borel σ-algebra; this is the smallest σ-algebra that contains the topology.

, called the strong mixing coefficient, as

for all 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 .

is said to be strongly mixing as tend towards being independent; more colloquially, the process, in a strong sense, forgets its history.

Mixing in Markov processes

were a stationary Markov process. Also let

denote the conditional expectation Finally, let

denote the space of square-integrable functions with mean zero.

The ρ-mixing coefficients of the process {xt} 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 ρt 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 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 satisfying

.

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.

• Manfred Einsiedler and Thomas Ward, Ergodic theory with a view towards number theory, (2011) Springer

• Achim Klenke, Probability Theory, (2006) Springer