  The problem of how complex quantum systems eventually come to rest lies at
the heart of statistical mechanics. The maximum entropy principle put forward
in 1957 by E. T. Jaynes suggests what quantum states one should expect in
equilibrium but does not hint as to how closed quantum many-body systems
dynamically equilibrate. A number of theoretical and numerical studies
accumulate evidence that under specific conditions quantum many-body models can
relax to a situation that locally or with respect to certain observables
appears as if the entire system had relaxed to a maximum entropy state. In this
work, we report the experimental observation of the non-equilibrium dynamics of
a density wave of ultracold bosonic atoms in an optical lattice in the regime
of strong correlations. Using an optical superlattice, we are able to prepare
the system in a well-known initial state with high fidelity. We then follow the
dynamical evolution of the system in terms of quasi-local densities, currents,
and coherences. Numerical studies based on the time-dependent density-matrix
renormalization group method are in an excellent quantitative agreement with
the experimental data. For very long times, all three local observables show a
fast relaxation to equilibrium values compatible with those expected for a
global maximum entropy state. We find this relaxation of the quasi-local
densities and currents to initially follow a power-law with an exponent being
significantly larger than for free or hardcore bosons. For intermediate times
the system fulfills the promise of being a dynamical quantum simulator, in that
the controlled dynamics runs for longer times than present classical algorithms
based on matrix product states can efficiently keep track of.
