  I show that the dynamics of the weakly interacting bose gas can be described
by a modified time dependent Bogoliubov theory. The novelty of the approach is
to include decoherence steps that gradually transform the entanglement entropy
of the pure state into the von Neumann entropy of a statistical mixture. This
approximation drastically reduces the entanglement that is needed in order to
represent the system's state while becoming exponentially accurate in the mean
field limit. I argue that this scheme can be extended to all quantum systems
whose ground state can be well approximated by a variational wave function. The
upshot is that the dynamics of almost all quantum systems can be reduced to
stochastic classical motion supplemented with small quantum fluctuations.
