Mixing (mathematics)

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, industrial mixing, "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. 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" 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, written as formula_1.
The set formula_2 is understood to be the total space to be filled: the mixing bowl, the smoke-filled room, "etc." The measure formula_3 is understood to define the natural volume of the space formula_2 and of its subspaces. The collection of subspaces is denoted by formula_5, and the size of any given subset formula_6 is formula_7; the size is its volume. Naively, one could imagine formula_5 to be the power set of formula_2; this doesn't quite work, as not all subsets of a space have a volume (famously, the Banach-Tarski paradox). Thus, conventionally, formula_5 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 formula_11. Given some subset formula_6, its map formula_13 will in general be a deformed version of formula_14 – 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 formula_13 must have the same volume as formula_14; 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 formula_17 with formula_18. Worse, a single point formula_19 has no size. These difficulties can be avoided by working with the inverse map formula_20; it will map any given subset formula_6 to the parts that were assembled to make it: these parts are formula_22. 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 formula_23 is the inverse of some map formula_24. The proper definition of a volume-preserving map is one for which formula_25 because formula_26 describes all the pieces-parts that formula_14 came from.
One is now interested in studying the time evolution of the system. If a set formula_28 eventually visits all of formula_2 over a long period of time (that is, if formula_30 approaches all of formula_2 for large formula_32), the system is said to be ergodic. If every set formula_14 behaves in this way, the system is a conservative system, placed in contrast to a dissipative system, where some subsets formula_14 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 formula_35, and not just between some set formula_14 and formula_2. That is, given any two sets formula_38, a system is said to be (topologically) mixing if there is an integer formula_39 such that, for all formula_40 and formula_41, one has that formula_42. Here, formula_43 denotes set intersection and formula_44 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 formula_14 and formula_46, 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 formula_47, 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 formula_14 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 formula_46 that hard-to-reach corner. The question of mixing is then, can formula_14, after a long enough period of time, not only penetrate into formula_46 but also fill formula_46 with the same proportion as it does elsewhere?
One phrases the definition of strong mixing as the requirement that 
The time parameter formula_32 serves to separate formula_14 and formula_46 in time, so that one is mixing formula_14 while holding the test volume formula_46 fixed. The product formula_59 is a bit more subtle. Imagine that the volume formula_46 is 10% of the total volume, and that the volume of dye formula_14 will also be 10% of the grand total. If formula_14 is uniformly distributed, then it is occupying 10% of formula_46, which itself is 10% of the total, and so, in the end, after mixing, the part of formula_14 that is in formula_46 is 1% of the total volume. That is, formula_66 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.
The reason for using formula_67 instead of formula_68 in the definition is a bit subtle, but it follows from the same reasons why 
formula_69 was used to define the concept of a measure-preserving map. When looking at how much dye got mixed into the corner formula_46, 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 formula_46.
Mixing in dynamical systems.
Let formula_1 be a measure-preserving dynamical system, with "T" being the time-evolution or shift operator. The system is said to be strong mixing if, for any formula_73, one has
For shifts parametrized by a continuous variable instead of a discrete integer "n", the same definition applies, with formula_75 replaced by formula_76 with "g" being the continuous-time parameter.
A dynamical system is said to be weak mixing if one has
In other words, formula_78 is strong mixing if formula_79 in the usual sense, weak mixing if
in the Cesàro sense, and ergodic if formula_81 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 was historically the first example given of a system that is weak-mixing but not strong-mixing.
formula_82 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 formula_1 is equivalent to the property that, for any function formula_84, the sequence formula_85 converges strongly and in the sense of Cesàro to formula_86, i.e.,
A dynamical system formula_1 is weakly mixing if, for any functions formula_89 and formula_90
A dynamical system formula_1 is strongly mixing if, for any function formula_93 the sequence formula_85 converges weakly to formula_95 i.e., for any function formula_90
Since the system is assumed to be measure preserving, this last line is equivalent to saying that the covariance formula_98 so that the random variables formula_99 and formula_100 become orthogonal as formula_32 grows. Actually, since this works for any function formula_102 one can informally see mixing as the property that the random variables formula_99 and formula_100 become independent as formula_32 grows.
Products of dynamical systems.
Given two measured dynamical systems formula_106 and formula_107 one can construct a dynamical system formula_108 on the Cartesian product by defining formula_109 We then have the following characterizations of weak 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.)
Topological mixing.
A form of mixing may be defined without appeal to a measure, using only the topology of the system. A continuous map formula_117 is said to be topologically transitive if, for every pair of non-empty open sets formula_118, there exists an integer "n" such that
where formula_120 is the "n"th 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.
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 formula_19, that is, a point "x" such that its orbit formula_122 is dense in "X".
A system is said to be topologically mixing if, given open sets formula_14 and formula_46, there exists an integer "N", such that, for all formula_41, one has
For a continuous-time system, formula_120 is replaced by the flow formula_128, with "g" being the continuous parameter, with the requirement that a non-empty intersection hold for all formula_129.
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.
Let formula_130 be a stochastic process on a probability space formula_131. 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.
Define a function formula_132, called the strong mixing coefficient, as
for all formula_134. The symbol formula_135, with formula_136 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 formula_137.
The process formula_130 is said to be strongly mixing if formula_139 as formula_140. That is to say, a strongly mixing process is such that, in a way that is uniform over all times formula_141 and all events, the events before time formula_141 and the events after time formula_143 tend towards being independent as formula_144; more colloquially, the process, in a strong sense, forgets its history.
Mixing in Markov processes.
Suppose formula_145 were a stationary Markov process with stationary distribution formula_146 and let formula_147 denote the space of Borel-measurable functions that are square-integrable with respect to the measure formula_146. Also let
denote the conditional expectation operator on formula_150 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 {"xt"} 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 for some non-increasing function formula_154 satisfying
as formula_156.
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 for some non-increasing function formula_154 satisfying
as formula_156.
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.