  Even after almost a century, the foundations of quantum statistical mechanics
are still not completely understood. In this work, we provide a precise account
on these foundations for a class of systems of paradigmatic importance that
appear frequently as mean-field models in condensed matter physics, namely
non-interacting lattice models of fermions (with straightforward extension to
bosons). We demonstrate that already the translation invariance of the
Hamiltonian governing the dynamics and a finite correlation length of the
possibly non-Gaussian initial state provide sufficient structure to make
mathematically precise statements about the equilibration of the system towards
a generalized Gibbs ensemble, even for highly non-translation invariant initial
states far from ground states of non-interacting models. Whenever these are
given, the system will equilibrate rapidly according to a power-law in time as
long as there are no long-wavelength dislocations in the initial second moments
that would render the system resilient to relaxation. Our proof technique is
rooted in the machinery of Kusmin-Landau bounds. Subsequently, we numerically
illustrate our analytical findings by discussing quench scenarios with an
initial state corresponding to an Anderson insulator observing power-law
equilibration. We discuss the implications of the results for the understanding
of current quantum simulators, both in how one can understand the behaviour of
equilibration in time, as well as concerning perspectives for realizing
distinct instances of generalized Gibbs ensembles in optical lattice-based
architectures.
