Transport and dynamical properties for a bouncing ball model with regular and stochastic perturbations

Highlights

• Transport and dynamical properties are obtained.

• A bouncing ball model is studied.

• Regular and stochastic perturbations are considered.

Abstract

Some statistical properties related to the diffusion in energy for an ensemble of classical particles in a bouncing ball model are studied. The particles are confined to bounce between two rigid walls. One of them is fixed while the other oscillates. The dynamics is described by a two dimensional nonlinear map for the velocity of the particle and time at the instant of the collision. Two different types of change of momentum are considered: (i) periodic due to a sine function and; (ii) stochastic. For elastic collisions case (i) leads to finite diffusion in energy while (ii) produces unlimited diffusion. On the other hand, inelastic collisions yield either (i) and (ii) to have limited diffusion. Scaling arguments are used to investigate some properties of the transport coefficient in the chaotic low energy region. Scaling exponents are also obtained for both conservative and dissipative case for cases (i) and (ii). We show that the parameter space has complicated structures either in Lyapunov as well as period coordinates. When stochasticity is introduced in the dynamics, we observed the destruction of the parameter space structures.

Introduction

Unlimited diffusion in energy of a classical particle due to collisions with an infinitely heavy and moving wall was called as Fermi acceleration [1]. The original idea was proposed by Enrico Fermi in 1949 as an attempt to explain a possible origin of the high energy of the cosmic rays. Fermi claimed that the unlimited diffusion in energy (which leads to the Fermi acceleration phenomena) was due to interactions of the charged particles with the time varying magnetic fields present in the space. After his original idea was launched, several systems were proposed in the literature trying to model and describe the unlimited diffusion [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]. Among several models, one that we discuss in this paper is often called the Fermi–Ulam model (FUM). It consists of a classical particle, or then an ensemble of non interacting particles, confined to move between two rigid walls [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27]. One of them is moving in time therefore corresponding to the time-varying magnetic fields and the other one is fixed. The functionality of the second one is just to produce a returning mechanism for a further collision with the moving wall.

A modification of the Fermi–Ulam model was suggested some years ago, it was called as the bouncer ball model [28], [29]. It considers a particle falling down, in a constant gravitational field, on a moving platform. Different approaches to the bouncer model have been studied theoretically and experimentally [8], [30], [31], [32]. One example can be cited, which considers a propagating surface wave that travels on the surface of the platform, while the platform remains motionless [32]. This model can be used to describe the transport of particles by propagating surface waves, which is an important problem with numerous applications including powder transport by piezoelectrically excited ultrasonic surface wave [33] and manipulation of bioparticles using travelling wave electrophoresis [32], [34].

For a sufficiently smooth and periodic movement of the moving wall in the FUM, the dynamics of the particle leads to three different types of behaviour (see for instance [9]): (i) regular – characterised by period motion; (ii) quasi-periodic – leading the invariant spanning curves or even curves circling periodic fixed points and; (iii) chaotic dynamics yielding in an unpredictability of the dynamics. The dynamics is often described by a two dimensional, nonlinear and area preserving mapping for the variables velocity of the particle and instant of the collisions with the moving wall. The chaotic sea is not allowed to diffuse with unbounded energy due to the existence of a set of infinitely many invariant spanning curves in the phase space [9]. Indeed, the position of the lowest one defines the law of the behaviour of several observables along the chaotic sea, including average velocity [16], deviation around the average velocity and many others. It is then concluded from the literature that a periodic perturbation of the moving wall leads to a failure to generate unlimited diffusion. This is basically connected to the fact the high energy of a bouncing particle leads to correlation between two successive collisions therefore producing regularity in the dynamics. For stochastic perturbation of the moving wall, the energy of the particle undergoes unlimited diffusion. Such diffusion is however limited when a fractional loss of energy is introduced upon collision with the moving wall [10] via introduction of a restitution coefficient. Other types of dissipation also prevent the unlimited diffusion including a viscous drag force [35], [36].

In this paper we revisit the Fermi–Ulam model seeking to understand and describe some of its dynamical properties considering three regimes of external perturbation: (i) entirely stochastic; (ii) totally periodic and; (iii) low stochasticity. For case (i) we confirm the unlimited diffusion in energy is taking place for elastic collisions but is suppressed when a fractional loss of energy upon collisions is introduced. Then we explore some properties around the steady state particularly focusing on the characterisation of the diffusion coefficient. Our results show it is indeed scaling invariant with respect either to the number of collisions as well as the restitution coefficient. For case (ii) we discuss some of the dynamical properties present in the parameter space of the model including the so called shrimp-like structures obtained as a function of the Lyapunov exponent as well as the period. Finally for case (iii) we explore the influences of the stochasticity in the periodic structures present in the parameter space and how they change as the stochastic parameter rises.

The organisation of this paper is as follows. In Section 2 we discuss the model and the map. Section 3 is devoted to discuss the stochastic model focusing particularly on the diffusion coefficient and its scaling invariance. The deterministic model and parameter space is left for Section 4 while the influences of a partially stochastic dynamics is discussed in Section 5. Conclusions and final remarks are presented in Section 6.