  We investigate finite size effects in quantum quenches on the basis of simple
energetic arguments. Distinguishing between the low-energy part of the
excitation spectrum, below a microscopic energy-scale, and the high-energy
regime enables one to define a crossover number of particles that is shown to
diverge in the small quench limit. Another crossover number is proposed based
on the fidelity between the initial and final ground-states. Both criteria can
be computed using ground-state techniques that work for larger system sizes
than full spectrum diagonalization. As examples, two models are studied: one
with free bosons in an harmonic trap which frequency is quenched, and the
one-dimensional Bose-Hubbard model, that is known to be non-integrable and for
which recent studies have uncovered remarkable non-equilibrium behaviors. The
diagonal weights of the time-averaged density-matrix are computed and
observables obtained from this diagonal ensemble are compared with the ones
from statistical ensembles. It is argued that the "thermalized" regime of the
Bose-Hubbard model, previously observed in the small quench regime, experiences
strong finite size effects that render difficult a thorough comparison with
statistical ensembles. In addition, we show that the non-thermalized regime,
emerging on finite size systems and for large interaction quenches, is not
related to the existence of an equilibrium quantum critical point but to the
high energy structure of the energy spectrum in the atomic limit. Its features
are reminiscent of the quench from the non-interacting limit to the atomic
limit.
