Mixing (physics)

Mixing (physics)

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

 \lim_{k\rightarrow\infty} \, \mu(T^k(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.

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

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.

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), resulting in a detectable bias towards later days of the year[citation needed].

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.

Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

  • Mixing time — may refer to: Blend time, the time to achieve a predefined level of homogeneity of a flow tracer in a mixing vessel Mixing (mathematics), an abstract concept originating from physics used to attempt to describe the irreversible thermodynamic… …   Wikipedia

  • Physics beyond the Standard Model — Beyond the Standard Model Standard Model …   Wikipedia

  • 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, etc. The concept appears in ergodic theory the… …   Wikipedia

  • physics — /fiz iks/, n. (used with a sing. v.) the science that deals with matter, energy, motion, and force. [1580 90; see PHYSIC, ICS] * * * I Science that deals with the structure of matter and the interactions between the fundamental constituents of… …   Universalium

  • PHYSICS — The material presented in this entry emphasizes those contributions which were important in arriving at verified present day scientific results, rather than those that may have appeared important at the time. Unavoidably it will overlap in parts… …   Encyclopedia of Judaism

  • Mixing ratio — In chemistry and physics, the dimensionless mixing ratio is defined as the abundance of one component of a mixture relative to that of all other components. The term can refer either to mole ratio or mass ratio.[1] Mole ratio In atmospheric… …   Wikipedia

  • Technicolor (physics) — Beyond the Standard Model Standard Model …   Wikipedia

  • Tribimaximal mixing — cite journal | author = P.F. Harrison, D. H. Perkins and W. G. Scott | title = Tribimaximal mixing and the neutrino oscillation data | journal = Physics Letters B | volume = 530 | year = 2002 | pages = 167 | url = http://arxiv.org/pdf/hep… …   Wikipedia

  • Trimaximal mixing — cite journal | author = P.F. Harrison, D. H. Perkins and W. G. Scott | title = A Redetermination of the Neutrino Mass Squared Difference in Tri Maximal Mixing with Terrestrial Matter Effects | journal = Physics Letters B | volume = 458 | year =… …   Wikipedia

  • Entropy of mixing — The entropy of mixing is the change in the configuration entropy, an extensive thermodynamic quantity, when two different chemical substances or components are mixed. This entropy change must be positive since there is more uncertainty about the… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”