  We study the validity of the eigenstate thermalization hypothesis (ETH) and
its role for the occurrence of initial-state independent (ISI) equilibration in
closed quantum many-body systems. Using the concept of dynamical typicality, we
present an extensive numerical analysis of energy exchange in integrable and
nonintegrable spin-1/2 systems of large size outside the range of exact
diagonalization. In case of nonintegrable systems, our finite-size scaling
shows that the ETH becomes valid in the thermodynamic limit and can serve as
the underlying mechanism for ISI equilibration. In case of integrable systems,
however, indication of ISI equilibration has been observed despite the
violation of the ETH. We establish a connection between this observation and
the need of choosing a proper parameter within the ETH.
