Mixing (physics)
Encyclopedia
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
whenever A and B are any measurable sets and μ is the associated measure
. Other definitions are possible, including weak mixing and topological mixing.
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.
. 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.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, a dynamical system
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
is said to be mixing if the phase space
Phase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in 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
whenever A and B are any measurable sets and μ is the associated measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
. Other definitions are possible, including weak mixing and topological mixing.
The mathematical definitions of mixing are meant to capture the notion of physical mixing. A canonical example is the Cuba libre
Cuba Libre
The Cuba Libre is a highball made of cola, lime, and white rum. This highball is often referred to as a Rum and Coke in the United States, Canada, the UK and Ireland, where the lime juice is optional.-History:...
: 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
Ergodic theory
Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
, 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 equationsNavier-Stokes equations
In physics, the Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton's second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing viscous...
. 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
Draft lottery (1969)
On December 1, 1969, the Selective Service System of the United States conducted two lotteries to determine the order of call to military service in the Vietnam War for men born between 1944 and 1950...
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.