Scaling properties of the conservative bouncer model (standard map) and a dissipative bouncer model
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)Abstract
The process by which cosmic rays accelerate acquiring high energies have been
studied since 1949, when Enrico Fermi proposed a mechanism in which charged particles interact with time dependent magnetic fields. The original model was later modified and, based on different applications, studied by other authors.
The bouncer model, proposed by Pustilnikov, consists of a bouncing particle which, under action of a constant gravitational acceleration $g$, hits an oscillating platform with amplitude $\epsilon$ and angular frequency $ømega$. The bouncer model presents, for some values of the control parameters, the property of unlimited energy growth. After that (1980) Lichenberg et al. showed, by appropriate variables transformations, that the bouncing model is globally equivalent to the Chirikov's standard map and, thus, the origin of the unlimited energy growth of the bouncing model was explained. In this way, the parameters $g$, $\epsilon$ and $ømega$ of the bouncer model are related to the parameter $K$ of the standard map as $K=4ømega^2\epsilon/g$ and the velocity $V$ of the particle is related to the action variable $I$ as $I=2V/g$. Thus, $I^2$ is related to the dimensionless energy of the particle in the bouncer model and, therefore, we describe the scaling relations for the variable $I^2$. The parameter $K$ defines the nonlinearity strength of the standard map and if $K=0$ then the system is integrable. If, however, $K0$ then the system is non integrable and the phase space is characterized by regions of chaotic motion. Thus, the dynamics of the bouncer model presents a transition from integrable to non integrable at $K=0$. If $K$ is enough small then the presence of spanning curves limits the orbits in phase space. However, at $K=K_c0.9716$ the last spanning curve is destroyed and, for $K>K_c$, the energy grows without limits. This phenomena is known as Fermi acceleration. In this way we present the scaling descriptions of the standard map (or bouncer model) in three regimes of the nonlinearity, namely $K0$, $KK_c$ and $K>>K_c$. We present also the scaling descriptions for a dissipative version of the bouncer model. Defining $\alphaın0,1$ as the restitution coefficient, the dissipation is introduced via inelastic collisions. If $\alpha1$ the average energy grows and reaches a saturation regime and, for $\alpha=1$, the non dissipative regime is recovered. In this way, it is observed a transition from bounded to unbounded energy gain at $\alpha=1$. As a consequence of this transition, we can define a new parameter $(1-\alpha)$ and, for the limite $\alpha1$, we can obtain the scaling descriptions of the deviation of the average velocity as function of the transformed time $t\varepsilon^2$, $(1-\alpha)$ and $\varepsilon$, where $\varepsilon=ømega^2\epsilon/g$.