Critical Velocity for the Onset of Fast Fermi Acceleration

Abstract

One of the well-known effects in nonlinear science is Fermi acceleration that is the unlimited growth of the velocity of a particle when it collides with oscillating walls. It is rather common for time-dependent billiard-like systems in which a particle moves inside a closed oscillating boundary and elastically collides with it. But if the oscillations are weak, more complex effects may arise in the system. Earlier one of these effects was shown for time-dependent stadium-like billiards. In the case of a weak boundary oscillation the particle’s velocity grows if its initial value is greater than a certain critical value and tends to the limit otherwise. We consider the dynamics of a particle moving between two planar and harmonically corrugated walls and colliding elastically with them. We tried to find a similar effect and revealed that the ensemble-averaged velocity grows rapidly if the initial velocity is larger than some critical value. Otherwise, the ensemble-averaged velocity grows much slower and the velocities of all particles tend to become equal. We numerically estimated the band of the oscillation amplitude in which this effect exists and illustrated the dependence of the behavior of the critical initial velocity on various parameters.