Mixing (mathematics)
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In mathematics, a measure-preserving transformation T is said to be mixing if
- <math> \lim_{k\rightarrow\infty} \, \mu(T^k(A) \cap B) = \mu(A) \cdot \mu(B) <math>
whenever A and B are any measurable sets and μ is the associated measure.
This definition is meant to capture the notion of physical mixing. Suppose that a glass initially contains 20% rum and 80% cola in separate regions. After stirring the glass, any region of the glass contains approximately 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.