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: e.g. mixing paint, mixing drinks, industrial mixing.
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 implies ergodicity: that is, every system that is weakly mixing is also ergodic.
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.The set is understood to be the total space to be filled: the mixing bowl, the smoke-filled room, etc. The measure is understood to define the natural volume of the space and of its subspaces. The collection of subspaces is denoted by, and the size of any given subset is ; the size is its volume. Naively, one could imagine to be the power set of ; this doesn't quite work, as not all subsets of a space have a volume. Thus, conventionally, 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. Given some subset, its map will in general be a deformed version of – 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 must have the same volume as ; the squashing/stretching does not alter the volume of the space, only its distribution. Such a system is "measure-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 with. Worse, a single point has no size. These difficulties can be avoided by working with the inverse map ; it will map any given subset 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 map is the inverse of some map. The proper definition of a volume-preserving map is one for which because describes all the pieces-parts that came from.
One is now interested in studying the time evolution of the system. If a set eventually visits all of over a long period of time, the system is said to be ergodic. If every set behaves in this way, the system is a conservative system, placed in contrast to a dissipative system, where some subsets 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, and not just between some set and. That is, given any two sets, a system is said to be mixing if there is an integer such that, for all and, one has that. Here, denotes set intersection and is the empty set.
The above definition of topological mixing should be enough to provide an informal idea of mixing. However, it made no mention of the volume of 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.
The concept of strong mixing is made in reference to the volume of a pair of sets. Consider, for example, a set 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 that hard-to-reach corner. The question of mixing is then, can, after a long enough period of time, not only penetrate into but also fill with the same proportion as it does elsewhere?
One phrases the definition of strong mixing as the requirement that
The time parameter serves to separate and in time, so that one is mixing while holding the test volume fixed. The product is a bit more subtle. Imagine that the volume is 10% of the total volume, and that the volume of dye will also be 10% of the grand total. If is uniformly distributed, then it is occupying 10% of, which itself is 10% of the total, and so, in the end, after mixing, the part of that is in is 1% of the total volume. That is, 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, even as they use different notation.
The reason for using instead of in the definition is a bit subtle, but it follows from the same reasons why
was used to define the concept of a measure-preserving map. When looking at how much dye got mixed into the corner, one wants to look at where that dye "came from". One must be sure that every place it might have "come from" eventually gets mixed into.
Mixing in dynamical systems
Let 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, one hasFor shifts parametrized by a continuous variable instead of a discrete integer n, the same definition applies, with replaced by with g being the continuous-time parameter.
A dynamical system is said to be weak mixing if one has
In other words, is strong mixing if in the usual sense, weak mixing if
in the Cesàro sense, and ergodic if 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, so one of has zero measure, and the other one has full measure.
Covering families
Given a topological space, such as the unit interval, 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. Next, we define a measure on the Borel sets, then add in all the subsets of measure-zero. This is how we obtain the Lebesgue measure and the Lebesgue measurable sets.In most applications of ergodic theory, the underlying space is almost-everywhere isomorphic to an open subset of some, 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.
A covering family 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. For Lebesgue measure spaces, if is measure-preserving, and for all in a covering family, then is strong mixing.
Proof. Extend the mixing equation from all in the covering family, to all open sets by disjoint union, to all closed sets by taking the complement, to all measurable sets by using the regularity of Lebesgue measure to approximate any set with open and closed sets. Thus, for all measurable.
''L''2 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 is equivalent to the property that, for any function, the sequence converges strongly and in the sense of Cesàro to, i.e.,A dynamical system is weakly mixing if, for any functions and
A dynamical system is strongly mixing if, for any function, the sequence converges weakly to, i.e., for any function
Since the system is assumed to be measure preserving, this last line is equivalent to saying that the covariance, so that the random variables and become orthogonal as grows. Actually, since this works for any function, one can informally see mixing as the property that the random variables and become independent as grows.