Mixing systems: Difference between revisions
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Systems that '''mix''' have a tendency over the course of time to head towards a state of statistical equilibrium. Mixing implies [[Ergodic hypothesis | ergodicity]]. | Systems that '''mix''' have a tendency over the course of time to head towards a state of statistical equilibrium. Mixing implies [[Ergodic hypothesis | ergodicity]]. In 1963 (Ref. 3) Sinai outlined a proof that a system of <math>N</math> [[hard sphere model| hard spheres]] enclosed within a cube which has perfectly reflecting walls | ||
is ergodic and mixing. | |||
==References== | ==References== | ||
#[http://www.pnas.org/content/18/2/204.full.pdf+html Eberhard Hopf "Complete Transitivity and the Ergodic Principle", PNAS '''18''' pp. 204-209 (1932)] | #[http://www.pnas.org/content/18/2/204.full.pdf+html Eberhard Hopf "Complete Transitivity and the Ergodic Principle", PNAS '''18''' pp. 204-209 (1932)] | ||
#[http://www.pnas.org/content/18/3/255.full.pdf+html B. O. Koopman and J. v. Neumann "Dynamical Systems of Continuous Spectra", PNAS '''18''' pp. 255-263 (1932)] | #[http://www.pnas.org/content/18/3/255.full.pdf+html B. O. Koopman and J. v. Neumann "Dynamical Systems of Continuous Spectra", PNAS '''18''' pp. 255-263 (1932)] | ||
#Ya. G. Sinai "On the Foundation of the Ergodic Hypothesis for a Dynamical System of Statistical Mechanics", Doklady Akademii Nauk SSSR '''153''' pp. 1261–1264 (1963) (English version: Soviet Math. Doklady '''4''' pp. 1818-1822 (1963)) | |||
[[category: statistical mechanics]] | [[category: statistical mechanics]] | ||
Revision as of 11:52, 29 July 2008
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Systems that mix have a tendency over the course of time to head towards a state of statistical equilibrium. Mixing implies ergodicity. In 1963 (Ref. 3) Sinai outlined a proof that a system of hard spheres enclosed within a cube which has perfectly reflecting walls is ergodic and mixing.
References
- Eberhard Hopf "Complete Transitivity and the Ergodic Principle", PNAS 18 pp. 204-209 (1932)
- B. O. Koopman and J. v. Neumann "Dynamical Systems of Continuous Spectra", PNAS 18 pp. 255-263 (1932)
- Ya. G. Sinai "On the Foundation of the Ergodic Hypothesis for a Dynamical System of Statistical Mechanics", Doklady Akademii Nauk SSSR 153 pp. 1261–1264 (1963) (English version: Soviet Math. Doklady 4 pp. 1818-1822 (1963))