Noise-Enhanced Trapping in Chaotic Scattering

Abstract

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.PACS numbers: 05.40.Ca,05.40.Fb,05.60.Cd,82.20Db The responses of nonlinear dynamical systems to small uncorrelated random perturbations (noise) can be surprising and seemingly contradictory. Noise usually destroys fine structures of deterministic dynamics, e.g. it fattens fractals [1-3], but it can also combine constructively with the nonlinearities and increase the order of the system [4]. In chaotic Hamiltonian systems, investigations focused on the effect of noise on anomalous transport [5,6] and, very recently, on chaotic scattering [7][8][9].Chaotic scattering is a basic process of Hamiltonian dynamics [2,10,11], with fundamental applications in classical [12] and quantum [2,13] systems, and recent applications ranging from plankton populations [3] to blood flows [14] and even the origin of life [15]. Chemical (dissociative) reactions of simple molecules are also scattering processes where chaos is essential in the microcanonical phase-space formulation of transition state theory (TST) [16]. Scattered trajectories perform transiently chaotic motion while trapped by fine structures of the phase space, such as fractal nonattracting sets and chains of Kolmogorov-Arnold-Moser (KAM) islands [2,10]. Noise destroys the small scales of these structures [7], modifies the temporal decay of trajectories in time from algebraic to exponential [8,9] with an exponent that increases with noise [8], and create otherwise forbidden escape paths [9]. All these effects weaken the deterministic trapping.In this Letter we show that noise also plays a constructive role in chaotic scattering, enhancing the trapping of trajectories. First we introduce and scrutinize two different mechanisms responsible for this surprising effect, arguing that they exist in very general circumstances. The first mechanism acts in fully chaotic systems and reduces the escape rate of particles by blurring the natural measure of the system. The second mechanism acts on mixed-phase-space systems and enhances trapping by throwing trajectories inside KAM islands. We confirm the generality of these mechanisms through simulations in the conservative Hénon map, and we explore the implications ...

Recurrence-time statistics in non-Hamiltonian volume-preserving maps and flows

“…This occurs just for finite times and we expect an exponential decay for larger times (not shown), as nicely discussed in [5] for the case of noise. Amazingly, all the dynamics shown in Figs.…”

“…It was observed that the RTS is capable of describing the relevant aspects of the dynamics in complex systems. In this context, we mention that the RTS is able to describe universal algebraic decays in Hamiltonian systems [2][3][4], including random walk penetration of the Kolmogorov-Arnold-Moser (KAM) islands [5,6], biased random walk to escape from KAM island [7], DNA sequence [8], synchronization of oscillator [9], generalized bifurcation diagram of conservative systems [10], fine structure of resonance islands [11], transient chaos in systems with leaks [12], among others.…”

Diffusion for ensembles of standard maps

“…The map (1) was evolved in the presence of noise with the distribution (4). The variance σ of the noise was set to a small value, namely 10 −5 and was kept constant, while the variance of K, σ K , was varied and the diffusion coefficient was calculated by (9) and (14). The corresponding numerical calculations were done for 200 different values of K in the intervalK min = 5 andK max = 30.…”

“…2 the diffusion coefficient is presented for moderate σ K =1. Numerical and analytical calculations are shown for the two types and compared with (6) where K is replaced bȳ K. Depletion or blurring of the accelerator modes is clearly shown for type II processes, such that for this value of σ K the diffusion is well described by (14). RRW [12] found that the peak of the accelerator mode is obtained by summation Fourier paths of very high order (Fig 9 in [12]).…”

“…Sometimes it is referred to as regularization of the diffusion process. On the other hand it was found that noise may enhance trapping [14]. This motivates calculation of various scenarios and classification of "noisy" phenomena.…”