  We investigate dynamical equilibration of expectation values in closed
quantum systems for realistic non-equilibrium initial states. Thereby we find
that the corresponding long time expectation values depend on the initial
expectation values if eigenstate thermalization is violated. An analytical
expression for the deviation from the expected ensemble value is derived for
small displacements from equilibrium. Additional numerics for magnetization and
energy equilibration in an asymmetric anisotropic spin-1/2-ladder demonstrate
that the analytical predictions persist beyond the limits of the theory. The
results suggest eigenstate thermalization as physically necessary condition for
initial state independent equilibration.
