  We study the quantum many-body dynamics and the entropy production triggered
by an interaction quench in a system of $N=10$ interacting identical bosons in
an external one-dimensional harmonic trap. The multiconfigurational
time-dependent Hartree method for bosons (MCTDHB) is used for solving the
time-dependent Schr\"odinger equation at a high level of accuracy. We consider
many-body entropy measures such as the Shannon information entropy, number of
principal components, and occupation entropy that are computed from the
time-dependent many-body basis set used in MCTDHB. These measures quantify
relevant physical features such as irregular or chaotic dynamics, statistical
relaxation and thermalization. We monitor the entropy measures as a function of
time and assess how they depend on the interaction strength. For larger
interaction strength, the many-body information entropy approaches the value
predicted for the Gaussian orthogonal ensemble of random matrices and implies
statistical relaxation. The basis states of MCTDHB are explicitly
time-dependent and optimized by the variational principle in a way that
minimizes the number of significantly contributing ones. It is therefore a
non-trivial fact that statistical relaxation prevails in MCTDHB computations.
Moreover, we demonstrate a fundamental connection between the production of
entropy, the build-up of correlations and loss of coherence in the system.
Since the coherence and correlations are experimentally accessible, their
present connection to many-body entropies can be scrutinized to detect
statistical relaxation. Our results are the first ones obtained for
thermalization of finite quantum systems using an optimized time-dependent and
genuinely many-body basis set.
