  Recently, there have been significant new insights concerning conditions
under which closed systems equilibrate locally. The question if subsystems
thermalize---if the equilibrium state is independent of the initial state---is
however much harder to answer in general. Here, we consider a setting in which
thermalization can be addressed: A quantum quench under a Hamiltonian whose
spectrum is fixed and basis is drawn from the Haar measure. If the Fourier
transform of the spectral density is small, almost all bases lead to local
equilibration to the thermal state with infinite temperature. This allows us to
show that, under almost all Hamiltonians that are unitarily equivalent to a
local Hamiltonian, it takes an algebraically small time for subsystems to
thermalize.
