We introduce a simple stochastic system able to generate anomalous diffusion both for 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 (PDF) 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 properties i.e. $P(|{ \bf x}|,t) = 1/t^{ \nu_x}F_x(|{ \bf x}|/t^{ \nu_x})$ and similarly for velocity.
Categories: cond-mat.stat-mech cond-mat.dis-nn nlin.CD