Mixing (physics)

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

In physics, there are several distinct notions of mixing, all of which try to capture the common-sense notion of mixing, but using rather disparate formal methods, techniques and definitions. One approach is to focus on mixtures of fluids, in three-dimensional space, described by differential equations suitable for fluids, such as the Navier–Stokes equations. The route to the final mixed state typically proceeds through turbulence created during mixing. A second approach considers the mixing of aggregates, such as rocks and sand, which are lumpy on the small scale; this is commonly seen in mixing in process engineering. A third approach uses the mathematical formalisms of measure theory and measure-preserving dynamical systems, to define mixing abstractly for generic dynamical systems in arbitrary dimensions. For example, by assigning a position and a velocity to each atom in a fluid, the mixing takes place in a $6^{N}$-dimensional space, where $N$ is approximately the Avogadro number and $6=3+3$ is the dimension of the phase space (position plus velocity) of a single atom.

Mixing of fluids

Main article: Mixture

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.

Mixing of aggregates

Main article: Mixing (process engineering)

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).

Dynamical systems

Main article: Mixing (mathematics)

A dynamical system is said to be mixing if the phase space of the system becomes strongly intertwined over time, according to at least one of several formal 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 $\mu$ is the associated measure. Other definitions are possible, including weak mixing and topological mixing.

The above definition of mixing is meant to capture the intuitive notion of physical mixing. A canonical example is the Cuba libre: suppose one is adding rum (the set $A$) to a glass of cola. After stirring the glass (the transformation $T^{k}$ for large $k$), the bottom half of the glass (the set $B$) will contain rum, and it will be in equal proportion as it is elsewhere in the glass. The mixing is uniform: no matter which region $B$ one looks at, some of $A$ will be in that region. The use of $T^{-k}$ in the definition, instead of $T^{k}$, is required because the time evolution transformation $T$ is typically not an injective function (one-to-one). That is, if $T$ maps two points into one, the measure is preserved only by counting the two points in the pre-image, instead of one point in the image.

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

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.