  Let us consider a projective manifold and $\Omega$ a volume form. We define
the gradient flow associated to the problem of $\Omega$-balanced metrics in the
quantum formalism, the \Omega$-balacing flow. At the limit of the quantization,
we prove that the $\Omega$-balacing flow converges towards a natural flow in
K\"ahler geometry, the $\Omega$-K\"ahler flow. We study the existence of the
$\Omega$-K\"ahler flow and proves its long time existence and convergence
towards the solution to the Calabi problem of prescribing the volume form in a
given K\"ahler class. We derive some natural geometric consequences of our
study.
