The ergodicity requirement is that the ensemble average coincide with the time average. A sufficient condition for ergodicity is that the time evolution of the system is a mixing. (See also ergodic hypothesis.) Not all systems are ergodic. For instance, it is unknown at this time whether classical mechanical flows on a constant energy surface are ergodic in general.

Physically, when a system fails to be ergodic, we may infer that there is more macroscopically discoverable information available about the microscopic state of the system than what we first thought. In turn this may be used to create a better-conditioned ensemble.

— Wikipedia on Statistical ensemble (mathematical physics)

2013.08.22 Thursday ACHK