Modelling the stochastic single particle dynamics of relativistic fermions and bosons using nonlinear drift-diffusion equations

Recently, there is a considerable interest in quantum kinetic models of fermions and bosons given by generalised Boltzmann equations and their corresponding diffusion approximations: nonlinear drift-diffusion equations. In this context, we propose a model that describes the dynamical properties of relativistic fermions and bosons consistent with the Fermi-Dirac and Bose-Einstein statistics. In particular, we discuss the first- and second-order statistics of quantum particle trajectories in energy space.