  We consider many-body quantum systems that exhibit quantum chaos, in the
sense that the observables of interest act on energy eigenstates like banded
random matrices. We study the time-dependent expectation values of these
observables, assuming that the system is in a definite (but arbitrary) pure
quantum state. We induce a probability distribution for the expectation values
by treating the zero of time as a uniformly distributed random variable. We
show explicitly that if an observable has a nonequilibrium expectation value at
some particular moment, then it is overwhelmingly likely to move towards
equilibrium, both forwards and backwards in time. For deviations from
equilibrium that are not much larger than a typical quantum or thermal
fluctuation, we find that the time dependence of the move towards equilibrium
is given by the Kubo correlation function, in agreement with Onsager's
postulate. These results are independent of the details of the system's quantum
state.
