  The number-conserving quantum phase space description of the Bose-Hubbard
model is discussed for the illustrative case of two and three modes, as well as
the generalization of the two-mode case to an open quantum system. The
phase-space description based on generalized SU(M) coherent states yields a
Liouvillian flow in the macroscopic limit, which can be efficiently simulated
using Monte Carlo methods even for large systems. We show that this description
clearly goes beyond the common mean-field limit. In particular it resolves
well-known problems where the common mean-field approach fails, like the
description of dynamical instabilities and chaotic dynamics. Moreover, it
provides a valuable tool for a semi-classical approximation of many interesting
quantities, which depend on higher moments of the quantum state and are
therefore not accessible within the common approach. As a prominent example, we
analyse the depletion and heating of the condensate. A comparison to methods
ignoring the fixed particle number shows that in this case artificial number
fluctuations lead to ambiguities and large deviations even for quite simple
examples.
