Thermal fluctuations in quantized chaotic systems

We consider a quantum system with N degrees of freedom which is classically chaotic. When N is large, and both and the quantum energy uncertainty are small, quantum chaos theory can be used to demonstrate the following results: (i) given a generic observable A, the infinite time average of the quantum expectation value is independent of all aspects of the initial state other than the total energy, and equal to an appropriate thermal average of A; (ii) the time variations of are too small to represent thermal fluctuations; (iii) however, the time variations of can be consistently interpreted as thermal fluctuations, even though these same time variations would be called quantum fluctuations when N is small.