  We consider the behavior of an interacting many particle system under slow
external driving -- a many body generalization of the Landau-Zener paradigm. We
find that a conspiracy of interactions and driving leads to physics profoundly
different from that of the single particle limit: for practically all values of
the driving rate the particle distributions in Hilbert space are very broad, a
phenomenon caused by a strong amplification of quantum fluctuations in the
driving process. These fluctuations are 'non-adiabatic' in that even at very
slow driving it is exceedingly difficult to push the center of the distribution
towards the limit of full ground state occupancy. We obtain these results by a
number of complementary theoretical approaches, including diagrammatic
perturbation theory, semiclassical analysis, and exact diagonalization.
