  We analytically describe the decay to equilibrium of generic observables of a
non-integrable system after a perturbation in the form of a random matrix. We
further obtain an analytic form for the time-averaged fluctuations of an
observable in terms of the rate of decay to equilibrium. Our result shows the
emergence of a Fluctuation-Dissipation theorem corresponding to a classical
Brownian process, specifically, the Ornstein-Uhlenbeck process. Our predictions
can be tested in quantum simulation experiments, thus helping to bridge the gap
between theoretical and experimental research in quantum thermalization. We
test our analytic results by exact numerical experiments in a spin-chain. We
argue that our Fluctuation-Dissipation relation can be used to measure the
density of states involved in the non-equilibrium dynamics of an isolated
quantum system.
