Mixing systems

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This article is about ergodic theory. For multicomponent systems see Mixtures

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.


  1. Eberhard Hopf "Complete Transitivity and the Ergodic Principle", PNAS 18 pp. 204-209 (1932)
  2. B. O. Koopman and J. v. Neumann "Dynamical Systems of Continuous Spectra", PNAS 18 pp. 255-263 (1932)
  3. 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))