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.

Mixing in a ball of colored putty after consecutive iterations of the Smale horseshoe map (i.e. squashing and folding in two)

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 a convective diffusion equation that may involve non-Fickian diffusion as in spinodal decomposition. The convective portion of the governing equation contains fluid motion terms that are governed by the Navier-Stokes equations. When fluid properties such as viscosity depend on composition, the governing equations may be coupled. There may also be temperature effects. 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.

See also

• Miscibility

References

• V.I. Arnold and A. Avez. Ergodic Problems of Classical Mechanics. New York: W.A. Benjamin. 1968.
• J Lebowitz and O. Penrose, Modern ergodic theory. Physics Today, 26, 155-175, February 1973.