Minimal Stochastic Model for 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 ν_x and ν_v. The diffusion process is anomalous (non-Gaussian) but with a defined scaling property, i.e., P(|r|,t)=1/t^ν_xF_x(|r|/t^ν_x) and similarly for velocity.