Fermi acceleration with memory-dependent excitation

Some scaling properties for a classical particle confined to bounce between two walls, where one wall is fixed and the other one moves in time according to a random signal with a memory length are studied. We have considered two different kinds of collisions of the particle with the moving wall namely: (i) elastic and (ii) inelastic. The dynamics of the model is described in terms of a two-dimensional nonlinear mapping. For the case of elastic collisions, we show that the memory of the stochastic signal affects directly the behaviour of the average velocity of the particle. It then exhibits different slopes for the average velocity at different stages of the series with β≅3/4 for a short time, β≅1 for the average stage and β≅1/2 for a long time, as predicted by the Central Limit Theorem, therefore leading to the Fermi acceleration. The situation where inelastic collisions are taken into account yields a more drastic change, particularly suppressing the Fermi acceleration.