Stochastic quantization of an interacting classical particle system

Starting from a many-body classical system governed by a trace-form entropy we derive, in the stochastic quantization picture, a family of nonlinear and non-Hermitian Schrodinger equations describing, in the mean field approximation, a quantum system of interacting particles. The time evolution of the main physical observables is analysed 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.