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 [Formula presented] and [Formula presented]. The diffusion process is anomalous (non-Gaussian) but with a defined scaling property, i.e., [Formula presented] and similarly for velocity.