Editing Mixing (mathematics) (section)

== Informal explanation ==
The mathematical definition of mixing aims to capture the ordinary every-day process of mixing, such as mixing paints, drinks, cooking ingredients, [[Mixing (process engineering)|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 {{tmath|1= (X, \mathcal{A}, \mu, T) }}.

The set <math>X</math> is understood to be the total space to be filled: the mixing bowl, the smoke-filled room, ''etc.'' The [[measure (mathematics)|measure]] <math>\mu</math> is understood to define the natural volume of the space <math>X</math> and of its subspaces.  The collection of subspaces is denoted by {{tmath|1= \mathcal{A} }}, and the size of any given [[subset]] <math>A\subset X</math> is {{tmath|1= \mu(A) }}; the size is its volume.  Naively, one could imagine <math>\mathcal{A}</math> to be the [[power set]] of {{tmath|1= X }}; this doesn't quite work, as not all subsets of a space have a volume (famously, the [[Banach–Tarski paradox]]). Thus, conventionally, <math>\mathcal{A}</math> 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 [[set intersection|intersections]], [[set union|unions]] and [[set complement]]s; these can always be taken to be measurable. 

The time evolution of the system is described by a [[map (mathematics)|map]] <math>T:X\to X</math>. Given some subset <math>A\subset X</math>, its map <math>T(A)</math> will in general be a deformed version of <math>A</math> – 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 <math>T(A)</math> must have the same volume as <math>A</math>; 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 <math>x\ne y</math> with {{tmath|1= T(x)=T(y) }}. Worse, a single point <math>x\in X</math> has no size. These difficulties can be avoided by working with the inverse map {{tmath|1= T^{-1}:\mathcal{A}\to\mathcal{A} }}; it will map any given subset <math>A\subset X</math> to the parts that were assembled to make it: these parts are {{tmath|1= T^{-1}(A)\in\mathcal{A} }}. 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 <math>\mathcal{A}\to\mathcal{A}</math> is the inverse of some map {{tmath|1= X\to X }}. The proper definition of a volume-preserving map is one for which <math>\mu(A)=\mu(T^{-1}(A))</math> because <math>T^{-1}(A)</math> describes all the pieces-parts that <math>A</math> came from.

One is now interested in studying the time evolution of the system. If a set <math>A\in\mathcal{A}</math> eventually visits all of <math>X</math> over a long period of time (that is, if <math>\cup_{k=1}^n T^k(A)</math> approaches all of <math>X</math> for large <math>n</math>), the system is said to be [[ergodic system|ergodic]]. If every set <math>A</math> behaves in this way, the system is a [[conservative system]], placed in contrast to a [[dissipative system]], where some subsets <math>A</math> [[wandering set|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 {{tmath|1= A,B }}, and not just between some set <math>A</math> and {{tmath|1= X }}. That is, given any two sets {{tmath|1= A,B\in\mathcal{A} }}, a system is said to be (topologically) mixing if there is an integer <math>N</math> such that, for all <math>A, B</math> and {{tmath|1= n>N }}, one has that <math>T^n(A)\cap B\ne\varnothing</math>.  Here, <math>\cap</math> denotes [[set intersection]] and <math>\varnothing</math> 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 <math>A</math> and {{tmath|1= B }}, 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 {{tmath|1= A,B,C\in\mathcal{A} }}, 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 <math>A</math> 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 <math>B</math> that hard-to-reach corner. The question of mixing is then, can <math>A</math>, after a long enough period of time, not only penetrate into <math>B</math> but also fill <math>B</math> with the same proportion as it does elsewhere?

One phrases the definition of strong mixing as the requirement that 
:<math>\lim_{n\to\infty} \mu\left( T^{-n} A \cap B\right) = \mu(A)\mu(B).</math>
The time parameter <math>n</math> serves to separate <math>A</math> and <math>B</math> in time, so that one is mixing <math>A</math> while holding the test volume <math>B</math> fixed.  The product <math>\mu(A)\mu(B)</math> is a bit more subtle. Imagine that the volume <math>B</math> is 10% of the total volume, and that the volume of dye <math>A</math> will also be 10% of the grand total. If <math>A</math> is uniformly distributed, then it is occupying 10% of <math>B</math>, which itself is 10% of the total, and so, in the end, after mixing, the part of <math>A</math> that is in <math>B</math> is 1% of the total volume. That is, <math>\mu\left(\mbox{after-mixing}(A) \cap B\right) = \mu(A)\mu(B).</math> 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 [[Probability axioms|Kolmogorov axioms]]), even as they use different notation.

The reason for using <math>T^{-n} A</math> instead of <math>T^n A</math> in the definition is a bit subtle, but it follows from the same reasons why 
<math>T^{-1} A</math> was used to define the concept of a measure-preserving map. When looking at how much dye got mixed into the corner <math>B</math>, 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 <math>B</math>.