Noise-Enhanced Trapping in Chaotic Scattering

We show that noise enhances the trapping of trajectories in scattering systems. In fully chaotic systems, the decay rate can decrease with increasing noise due to a generic mismatch between the noiseless escape rate and the value predicted by the Liouville measure of the exit set. In Hamiltonian systems with mixed phase space we show that noise leads to a slower algebraic decay due to trajectories performing a random walk inside Kolmogorov-Arnold-Moser islands. We argue that these noise-enhanced trapping mechanisms exist in most scattering systems and are likely to be dominant for small noise intensities, which is confirmed through a detailed investigation in the Hénon map. Our results can be tested in fluid experiments, affect the fractal Weyl’s law of quantum systems, and modify the estimations of chemical reaction rates based on phase-space transition state theory.

Figure 1

Noise-enhanced trapping for fully chaotic systems. (a) The lifetime $\tau$ for the baker map (1) is shown as a function of the noise intensity $\xi$ for two BCs: periodic (red ▪) and open (blue ●). Horizontal lines: ${\tau }_{\xi =0}=6.06$ (solid) and ${\tau }^{*}=9.49$ (dashed). Inset: $P(t)$ for $\xi =0$, $\xi ={\xi }_m=0.035$ (open BC) and $\xi =1$ (periodic BC). (b) Values of ${\tau }^{*}$, ${\tau }_{\xi =0}$, and ${\tau }_m$ for different positions $x_c$ of the leak (${\Delta }_x=0.05$). Noise-enhanced trapping occurs when ${\tau }^{*}>{\tau }_{\xi =0}$ for periodic BC and when ${\tau }_m>{\tau }_{\xi =0}$ for open BC (blue shading).

Figure 2

Noise tends to uniformly distribute surviving trajectories. Quasi-invariant density ${\rho }_{\xi }$ of map (1) obtained at $t=40$ for (a) $\xi =0$ ($\tau ={\tau }_{\xi =0}$), ${\rho }_{\xi =0}$ distributed along the unstable manifold of the chaotic saddle [2, 10, 11]; (b) $\xi ={\xi }_m$ ($\tau ={\tau }_m$, open BC), ${\rho }_{\xi }$ is smoothed in $I$ and negligible outside $[0,1]\times [0,1]$; (c) $\xi =1$ ($\tau ={\tau }^{*}$, periodic BC), ${\rho }_{\xi =1}={\rho }_{\mu }$ is uniformly distributed. The leak $I$ ($x_c=0.5$, $\Delta =0.05$) and its forward iteration $M(I)$ are indicated.

Figure 3

Random-walk model for trajectories inside KAM islands. (a) Illustrative KAM island [Hénon map (4) with $k=2$]. Inset: Magnification around the period three orbit ▴ showing the intersection of manifolds. (b) Random-walk model with probabilities $p_{\pm }=0.5$ of stepping $\Delta r=\pm 1$, one reflecting ($r=0$, center of the island) and one absorbing boundary ($r=r^{*}\sim {\tilde{r}}^{*}/\xi$, where ${\tilde{r}}^{*}$ is proportional to island width and $\xi$ to step length). (c) $P(t)$ for a simulation of ${10}^5$ walkers started in $r=r^{*}-1$, with $r^{*}=\infty$ (thin red line) and $r^{*}=100$ (thick black line). Dashed line: Scaling $\alpha =0.5$. Inset: Linear-log plot emphasizing the exponential tail (for $r^{*}=100$) starting at $t_e\approx {10}^4=(r^{*}{)}^2\sim ({\tilde{r}}^{*}/\xi {)}^2$.

Figure 4

Noise-enhanced trapping in the fully chaotic ($k=6$) Hénon map (4). (a) Phase space with trapped region, delimited by the solid black line (first intersection of the manifolds $W^{U,S}$ of the fixed point ▪). Escape (inflow) denotes an unbounded forward (backward) invariant set that diverges in forward (backward) iterations [32]. Initial conditions and the region where trajectories are removed were chosen inside these sets [35]. The exit set $I$ is the set of points that exit the trapped region in one iteration [31]. (b) Dependence of $\tau$ on $\xi$ with ${\tau }_{\xi =0}=1.557$ (dashed line) and ${\tau }^{*}=3.239$ (dotted line). Inset: $P(t)$ for $\xi =0$, $\xi ={\xi }_m$, and with ${\tau }^{*}$.

Figure 5

Noise-enhanced trapping in the Hénon map (4) with mixed phase space ($k=2$). (a) $P(t)$ for $\xi =0$ (thick black line) and $\xi ={10}^{-3}$ (thin blue line) [35]. The times $t_b$ and $t_e$ indicate the beginning and end of the mechanism described in Fig. 3. Power-law scalings are shown as a reference (dashed lines). (b) Dependence of $t_b$ and $t_e$ on $\xi$ with fits of the model predictions (with $\beta =0.92$; see text).