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Mathematical description of mixing substances

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 (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).

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. Next, we define a measure 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. 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 .

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.

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 dyadic map is "shift to left in binary". In general, for any , the "shift to left in base " map 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 We can either construct the singly infinite space 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.