Random fluctuation leads to forbidden escape of particles

A great number of physical processes are described within the context of Hamiltonian scattering. Previous studies have rather been focused on trajectories starting outside invariant structures, since the ones starting inside are expected to stay trapped there forever. This is true though only for the deterministic case. We show however that, under finitely small random fluctuations of the field, trajectories starting inside Kolmogorov-Arnold-Moser (KAM) islands escape within finite time. The nonhyperbolic dynamics gains then hyperbolic characteristics due to the effect of the random perturbed field. As a consequence, trajectories which are started inside KAM curves escape with hyperboliclike time decay distribution, and the fractal dimension of a set of particles that remain in the scattering region approaches that for hyperbolic systems. We show a universal quadratic power law relating the exponential decay to the amplitude of noise. We present a random walk model to relate this distribution to the amplitude of noise, and investigate these phenomena with a numerical study applying random maps.

Figure 1

(Color online) Simplified representation of the orbits on the torus. In (a), it is represented a single torus and its center. In (b), it is represented the effected of the perturbation. The difference between the dashed and the continuous line is due to the effect of the noise, that dislocate the center of the torus from $O_1$ to $O_2$. In (c) we show what would be expected for four iterations. Each continuous line represents the orbit of a particle on the torus for each iteration, hence different values of perturbations at each iteration. The center is expected to move around $O_1$ at each interaction and for long enough time, the distribution of centers would fill densely the area within the dashed circle.

Figure 2

Phase space for the map 2, for $\lambda =6.0$. The inset shows a blowup of the region $\left(x,y\right)∊\left[2.05,2.20\right]\times \left[0.44,0.47\right]$.

Figure 3

KAM structures under the perturbed dynamics for different values of $\xi$. Figure 3a, $\xi =0$, Fig. 3b, $\xi =0.0004$, and Fig. 3c, $\xi =0.002$.

Figure 4

(Color online) Probability distribution of escaping time $t$ from the region $\mathit{W}=\left\{\left|x\right|<5.0,and\left|y\right|<5.0\right\}$ for different values of $\xi$. Initial conditions were randomly chosen in the line $x∊\left[2.05,2.07\right],y=0.465$ inside nest KAM structures. For each value of $\xi$, we present the exponent $\gamma$ that best fits the exponential region of the probability distribution. Figure 4a shows all used values of $\xi$, Fig. 4b, shows small values of $t$.

Figure 5

Different values of exponent $\gamma$ as a function of $\xi$ from Fig. 4, as well as the power law that best fits the distribution. In the inset, we consider only values of $\xi <0.02$.

Figure 6

(Color online) Estimated fractal dimension of the $T$, for different values of $\xi$. We notice that, when the amplitude of the perturbation is decreased, the dynamics approach the nonhyperbolic limit, and so does the estimated values of fractal dimension. The inset shows the value of $\beta$ as a function of $\xi$.