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 $\hbar$ and the quantum energy uncertainty $\Delta E$ are small, quantum chaos theory can be used to demonstrate the following results: (1) given a generic observable $A$, the infinite time average $\overline A$ of the quantum expectation value $<A(t)>$ is independent of all aspects of the initial state other than the total energy, and equal to an appropriate thermal average of $A$; (2) the time variations of $<A(t)> - \overline A$ are too small to represent thermal fluctuations; (3) however, the time variations of $<A^2(t)> - <A(t)>^2$ can be consistently interpreted as thermal fluctuations, even though these same time variations would be called quantum fluctuations when $N$ is small.