Testing Boltzmann's ergodic hypothesis with electron gas models

Kubo proposed a physical approach to proving the validity of Boltzmann's ergodic hypothesis. It is to perform the time averages on dynamical functions, thereby avoiding the difficulties of measure theory inherent in the classical approach. To perform time averaging properly, one must have a general solution for the Heisenberg equation of motion such as by the recurrence relations method. A time averaging carried over with a recurrence relations solution is found to yield an ergodic condition in the form of an infinite product. It is linked to the energy transfer mechanisms, hence to the ergodicity itself. The electron gas models are fertile ground for testing ergodicity by this approach. For several static domains, we have evaluated the infinite product and drawn from them a general physical picture that underlies the ergodic hypothesis.