physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...

, a

dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...

is said to be mixing if the

phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usuall ...

of the system becomes strongly intertwined, according to at least one of several mathematical definitions. For example, a

measure-preserving transformation In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special ca ...

''T'' is said to be strong mixing if :

\lim_ \, \mu(T^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 In mathematics, mixing is an abstract concept originating from physics: the attempt to describe the irreversible thermodynamic process of mixing in the everyday world: mixing paint, mixing drinks, industrial mixing, ''etc''. The concept appear ...

Ergodic mixing of putty ball after repeated Smale horseshoe map

The mathematical definition of mixing is meant to capture the 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 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. A far more detailed, but still informal description of mixing can be found in the article on mixing (mathematics). Every mixing transformation is

ergodic In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies tha ...

, 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).

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. Ergodic theory Statistical mechanics {{statisticalmechanics-stub

Physical mixing

See also

References