Dynamical properties of a particle in a classical time-dependent potential well

Abstract

We study numerically the dynamical behavior of a classical particle inside a box potential that contains a square well which depth varies in time. Two cases of time dependence are investigated: periodic and stochastic. The periodic case is similar to the one-dimensional Fermi accelerator model, in the sense that KAM curves like islands surrounded by an ergodic sea are observed for low energy and invariant spanning curves appear for high energies. The ergodic sea, limited by the first spanning curve, is characterized by a positive Lyapunov exponent. This exponent and the position of the lower spanning curve depend sensitively on the control parameter values. In the stochastic case, the particle can reach unbounded kinetic energies. We obtain the average kinetic energy as function of time and of the iteration number. We also show for both cases that the distributions of the time spent by the particle inside the well and the number of successive reflections have a power law tail.