  We analyze the correspondence of many-particle and mean-field dynamics for a
Bose-Einstein condensate in an optical lattice. Representing many-particle
quantum states by a classical phase space ensemble instead of one single
mean-field trajectory and taking into account the quantization of the density
by a modified integer Gross-Pitaevskii equation, it is possible to simulate the
superfluid to Mott insulator transition and other phenomena purely classically.
This approach can be easily extended to higher particle numbers and
multidimensional lattices. Moreover it provides an excellent tool to classify
true quantum features and to analyze the mean-field -- many particle
correspondence.
