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Mixing (physics)

In physics, a dynamical system is said to be mixing if the phase space of the system becomes strongly intertwined, according to at least one of several mathematical definitions. For example, a measure-preserving transformation T is said to be strong mixing if

$\lim_{k\rightarrow\infty} \, \mu(T^k(A) \cap B) = \mu(A) \cdot \mu(B)$

whenever A and B are any measurable sets and μ is the associated measure. Other definitions are possible, including weak mixing and topological mixing.

The mathematical definitions of mixing are meant to capture the notion of physical mixing. A canonical example is the Cuba libre: suppose that a glass initially contains 20% rum (the set A) and 80% cola (the set B) in separate regions. After stirring the glass, any region of the glass contains approximately 20% rum. Furthermore, the stirred mixture is in a certain sense inseparable: no matter where one looks, or how small a region one looks at, one will find 80% cola and 20% rum.

Every mixing transformation is ergodic, but there are ergodic transformations which are not mixing.

Physical mixing

The mixing of gases or liquids is a complex physical process, governed by the Navier-Stokes equations. It is not clear that fluid mixing processes are mixing in the mathematical sense.

Small rigid objects (such as rocks) are sometimes mixed in a rotating drum or tumbler. The 1969 Selective Service draft lottery was carried out by mixing plastic capsules which contained a slip of paper (marked with a day of the year), resulting in a detectable bias towards later days of the year.

References

• V.I. Arnold and A. Avez. Ergodic Problems of Classical Mechanics. New York: W.A. Benjamin. 1968.