# Mixing systems

*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.

## References[edit]

- 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))