Simple model with critical velocity for the onset of Fermi acceleration

Abstract

We consider the billiard system of a particle moving between two walls one of which is plain and fixed and the other one is harmonically corrugated (like in Tennyson-Lieberman-Lichtenberg system) and oscillates harmonically. The collisions of the particle and the wall are suggested to be elastic. Recently we have shown that if the oscillation amplitude is small the two types of Fermi acceleration can occur - fast and slow. We assume the corrugation to be zero and the amplitude of the oscillations to be weak so some significant simplifications of the system are justified. It results in the 3D map which is differ from classical Ulam map. We analyze an ensemble-averaged velocity and conclude that Fermi acceleration starts from the critical value of the initial velocity which is similar to the border between fast and slow Fermi acceleration regimes in original system. We calculate numerically the Jacobian of different trajectories of the system and reveal that the regions of conservative (with the Jacobian very close to zero) and dissipative dynamics coexist in the phase space of the system. The regions of conservative dynamics correspond to fast Fermi acceleration in the original system and the regions of dissipative dynamics corresponds to slow Fermi acceleration.

Speaker

Dmitry Lubchenko
Saratov State University, Institute of Physics
Russia

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