  We illustrate how dynamical transitions in nonlinear semiclassical models can
be recognized as phase transitions in the corresponding -- inherently linear --
quantum model, where, in a Statistical Mechanics framework, the thermodynamic
limit is realized by letting the particle population go to infinity at fixed
size. We focus on lattice bosons described by the Bose-Hubbard (BH) model and
Discrete Self-Trapping (DST) equations at the quantum and semiclassical level,
respectively.
  After showing that the gaussianity of the quantum ground states is broken at
the phase transition, we evaluate finite populations effects introducing a
suitable scaling hypothesis; we work out the exact value of the critical
exponents and we provide numerical evidences confirming our hypothesis. Our
analytical results rely on a general scheme obtained from a large-population
expansion of the eigenvalue equation of the BH model. In this approach the DST
equations resurface as solutions of the zeroth-order problem.
