Quantum statistical ergodic and H -theorems for incompletely specified systems

Abstract

In incompletely specified situations the passage from quantum mechanics to statistical mechanics requires an averaging process. Typically, one has to average a scalar product of a fixed unit vector α and a unit vector β of random direction over a probability distribu­tion. A mathematically elementary class of such averaging processes is considered. It is used to define the concept of weak uniformity of the distribution of the end point of β over the unit sphere in a many-dimensional unitary space. In the case of strict uniformity , von Neumann’s method of averaging results. The distributions contain a parameter in terms of which a condition for probable ergodicity can be formulated. It expresses a restriction which bears most explicitly on the averaging process, and less explicitly on the Hamiltonian of the system. One finds that, while strict uniformity and large phase cells are together sufficient for probable ergodicity, each condition by itself can occur when ergodicity is not over­-whelmingly probable. The averaging processes over macro-observers, initial states and Hamiltonians occur as special cases.