Mixing (mathematics)

In mathematics, a measure-preserving transformation T is said to be mixing if

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.