  We review the basic concepts of a non-equilibrium kinetic theory of a trapped
bosonic gas. By extending the successful mean-field concept of the
Gross-Pitaevskii equation with the effects of non-local, two particle quantum
correlations, one obtains a renormalized binary interaction and allows for the
dynamic establishment of non-classical many-particle quantum correlations.
These concepts are illustrated by self-consistent numerical calculations of the
first and second order ground state quantum correlations of a harmonically
trapped, quasi one-dimensional bosonic gas. We do find a strong suppression of
the density fluctuations or, in other words, an enhanced number squeezing with
decreasing particle density.
