  Starting from a many-body classical system governed by a trace-form entropy
we derive, in the stochastic quantization picture, a family of non linear and
non-Hermitian Schroedinger equations describing, in the mean filed
approximation, a quantum system of interacting particles. The time evolution of
the main physical observables is analyzed by means of the Ehrenfest equations
showing that, in general, this family of equations takes into account
dissipative and damped effects due to the interaction of the system with the
background. We explore the presence of steady states by means of solitons,
describing conservative solutions. The results are specialized to the case of a
system governed by the Boltzmann-Gibbs entropy.
