Minimal stochastic model for the Fermi's acceleration

We introduce a simple stochastic system able to generate anomalous diffusion for both position and velocity. The model represents a viable description of the Fermi's acceleration mechanism and it is amenable to analytical treatment through a linear Boltzmann equation. The asymptotic probability distribution functions for velocity and position are explicitly derived. The diffusion process is highly non-Gaussian and the time growth of moments is characterized by only two exponents nu(x) and nu(v). The diffusion process is anomalous (non-Gaussian) but with a defined scaling property, i.e., P(\r\,t) = 1/t(nux)F(x)(\r\/t(nux)) and similarly for velocity.