  The dynamics of M-site, N-particle Bose-Hubbard systems is described in
quantum phase space constructed in terms of generalized SU(M) coherent states.
These states have a special significance for these systems as they describe
fully condensed states. Based on the differential algebra developed by Gilmore,
we derive an explicit evolution equation for the (generalized) Husimi-(Q)- and
Glauber-Sudarshan-(P)-distributions. Most remarkably, these evolution equations
turn out to be second order differential equations where the second order terms
scale as 1/N with the particle number. For large N the evolution reduces to a
(classical) Liouvillian dynamics. The phase space approach thus provides a
distinguished instrument to explore the mean-field many-particle crossover. In
addition, the thermodynamic Bloch equation is analyzed using similar
techniques.
