A one-dimensional Fermi accelerator model with moving wall described by a nonlinear van der Pol oscillator

TL;DR

This paper studies a modified one-dimensional Fermi accelerator with a moving wall modeled by a nonlinear van der Pol oscillator, analyzing how particle-wall interactions influence energy gain and phase space dynamics.

Contribution

It introduces a novel model combining Fermi acceleration with a nonlinear van der Pol oscillator and explores the effects of particle mass on energy transfer and phase space structure.

Findings

For negligible particle mass, the system shows Fermi acceleration with velocity diffusion.

Large non-linearity parameter leads to velocity scaling and acceleration.

When particle mass affects the wall, energy absorption suppresses unlimited acceleration, resulting in attractors of various periods.

Abstract

A modification of the one-dimensional Fermi accelerator model is considered in this work. The dynamics of a classical particle of mass $m$, confined to bounce elastically between two rigid walls where one is described by a non-linear van der Pol type oscillator while the other one is fixed, working as a re-injection mechanism of the particle for a next collision, is carefully made by the use of a two-dimensional non-linear mapping. Two cases are considered: (i) the situation where the particle has mass negligible as compared to the mass of the moving wall and does not affect the motion of it; (ii) the case where collisions of the particle does affect the movement of the moving wall. For case (i) the phase space is of mixed type leading us to observe a scaling of the average velocity as a function of the parameter ($\c{hi}$) controlling the non-linearity of the moving wall. For large $\c{hi}$, a diffusion on the velocity is observed leading us to conclude that Fermi acceleration is taking place. On the other hand for case (ii), the motion of the moving wall is affected by collisions with the particle. However due to the properties of the van der Pol oscillation, the moving wall relaxes again to a limit cycle. Such kind of motion absorbs part of the energy of the particle leading to a suppression of the unlimited energy gain as observed in case (i). The phase space shows a set of attractors of different periods whose basin of attraction has a complicate organization.