We now imagine that we prepare our system in an initial state for which the expectation value of A ^ ^ is far from its value predicted in a microcanonical ensemble appropriate to the energy scale in question (we assume that our initial state is some superposition of energy eigenstates which are all sufficiently "close" in energy). The eigenstate thermalization hypothesis says that for an arbitrary initial state, the expectation value of A ^ ^ will ultimately evolve in time to its value predicted by a microcanonical ensemble, and thereafter will exhibit only small fluctuations around that value, provided that the following two conditions are met: