  The manipulation of many-body systems often involves time-dependent forces
that cause unwanted heating. One strategy to suppress heating is to use
time-periodic (Floquet) forces at large driving frequencies. For quantum spin
systems with bounded spectra, it was shown rigorously that the heating rate is
exponentially small in the driving frequency. Recently, the exponential
suppression of heating has also been observed in an experiment with ultracold
atoms, realizing a periodically driven Bose-Hubbard model. This model has an
unbounded spectrum and, hence, is beyond the reach of previous theoretical
approaches. Here, we study this model with two semiclassical approaches valid,
respectively, at large and weak interaction strengths. In both limits, we
compute the heating rates by studying the statistical probability to encounter
a many-body resonance, and obtain a quantitative agreement with the exact
diagonalization of the quantum model. Our approach demonstrates the relevance
of statistical arguments to Floquet perthermalization of interacting many-body
quantum systems.
