Transition from normal to ballistic diffusion in a one-dimensional impact system

We characterize a transition from normal to ballistic diffusion in a bouncing ball dynamics. The system is composed of a particle, or an ensemble of noninteracting particles, experiencing elastic collisions with a heavy and periodically moving wall under the influence of a constant gravitational field. The dynamics lead to a mixed phase space where chaotic orbits have a free path to move along the velocity axis, presenting a normal diffusion behavior. Depending on the control parameter, one can observe the presence of featured resonances, known as accelerator modes, that lead to a ballistic growth of velocity. Through statistical and numerical analysis of the velocity of the particle, we are able to characterize a transition between the two regimes, where transport properties were used to characterize the scenario of the ballistic regime. Also, in an analysis of the probability of an orbit to reach an accelerator mode as a function of the velocity, we observe a competition between the normal and ballistic transport in the midrange velocity.

Figure 1

Phase space for the complete dynamics of the bouncing ball model. (a, c) $\epsilon =0.71$; (b, d) $\epsilon =1.71$.

Figure 2

Final velocity after ${10}^8$ collisions as function of the control parameter $\epsilon$, where the several peaks denote the AM. The range of the first period 1 AM is depicted inside the dashed box. Also, if we did not have any AM in the dynamics, a quadratic curve seems to fit well the dependence of the final velocity as a function of $\epsilon$.

Figure 3

(a) The root-mean-square velocity for two regimes of velocity growth defined as regular Fermi acceleration and ballistic Fermi acceleration. (b) The diffusion coefficient, given by Eq. (5) for the same values of the control parameter $\epsilon$ of (a), where the normal and ballistic diffusion behavior is depicted. (c) The dispersion of the mean-square velocity by collision iteration given by Eq. (7). (d) The final plateau of (c) as a function of $\epsilon$.

Figure 4

(a) The behavior of $\rho (v,n)$ for a range where the AM is active for two different holes, where all the decays are exponential. (b) The $\zeta$ exponent for both holes as function of $\epsilon$.

Figure 5

(a) Escape rate $\zeta$ for the 20 different velocity holes for some values of $\epsilon$, where a power law seems to be the best fit. (b) The power law exponent $\gamma$ obtained in (a) as function of $\epsilon$.

Figure 6

(a) The linear coefficient $a$ as function of $n$; (b) a zoom-in of the convergence to $\pi$ region.

Figure 7

(a) The probability of an ensemble of initial conditions to reach the AM as a function of the velocity for some values of $\epsilon$. The first Gaussian-like peak represents a competition between normal and ballistic diffusion, and the maximum of $\rho (A/N)$ denotes the critical velocity for which the AM was finally achieved. (b) The critical velocity $V_c$, obtained in (a), as function of $\epsilon$. The similar peaks scenario with Fig. 5 gives robustness to our probability analysis.