On Temporal Evolution of Isolated Dynamical Systems

It is shown that if 〈 A (t) 〉 is the expectation value of a real observable A at time t, then for a finite closed dynamical system 〈 A (t) 〉 can converge to a limit as t→ ∞ only if 〈 A (t) 〉 is altogether independent of t. This conclusion is unaffected by time-smoothing or coarse-graining. On the other hand, for an infinite system whose energy levels form a purely continuous spectrum, 〈 A (t) 〉 tends to a limit as t→ ∞ under very general conditions. This conclusion does not depend on the introduction of either time-smoothing or coarse-graining.