In
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 r ...
, a
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
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
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 cas ...
''T'' is said to be
strong mixing if
:
whenever ''A'' and ''B'' are any measurable sets and μ is the associated
measure
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Mea ...
. Other definitions are possible, including
weak mixing
In mathematics, mixing is an abstract concept originating from physics: the attempt to describe the irreversible thermodynamic process of mixing (physics), mixing in the everyday world: mixing paint, mixing drinks, Mixing (process engineering), in ...
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 appea ...
.
![Ergodic mixing of putty ball after repeated Smale horseshoe map](https://upload.wikimedia.org/wikipedia/commons/4/4f/Ergodic_mixing_of_putty_ball_after_repeated_Smale_horseshoe_map.jpg)
The mathematical definition of mixing is meant to capture the notion of physical mixing. A canonical example is the
Cuba libre
Rum and Coke, or the Cuba libre ( , ; literally "Free Cuba"), is a highball cocktail consisting of cola, rum, and in many recipes lime juice on ice. Traditionally, the cola ingredient is Coca-Cola ("Coke") and the alcohol is a light rum such ...
: 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)
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 appears ...
.
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
In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
. 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).
See also
*
Miscibility
Miscibility () is the property of two substances to mix in all proportions (that is, to fully dissolve in each other at any concentration), forming a homogeneous mixture (a solution). The term is most often applied to liquids but also appl ...
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
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