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
6N
N
6=3+3
See 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.
See 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).
See 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
\limk → infty\mu(T-kA\capB)=\mu(A) ⋅ \mu(B)
whenever
A
B
\mu
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
Tk
k
B
B
A
T-k
Tk
T
T
Every mixing transformation is ergodic, but there are ergodic transformations which are not mixing.