Arnold web

There are a number of open issues associated with statistical descriptions of unimolecular reactions, particularly in many-dimensional systems. One fundamental issue is to find a qualitative criterion for predicting if a reaction in a many-dimensional system is statistical or nonstatistic al. In a recent review article, Toda [17] discussed different aspects of the Arnold web — that is, the network of nonlinear resonances in many-dimensional systems. Toda pointed out the importance of analyzing the qualitative features of the Arnold web— for example, how different resonance zones intersect and how the intersections further overlap with one another. However, as pointed out earlier, even in the case of fully developed global chaos it remains challenging to define a nonlocal reaction separatrix and to calculate the flux crossing the separatrix in a manydimensional phase-space. 

In addition to the NHIMs with saddles, other kinds of NHIMs can also become important when intramolecular vibrational energy redistribution (IVR) comes into play. They are NHIMs with whiskered tori that are created by nonlinear resonances within the potential well [5]. In the network of nonlinear resonances, which is called the Arnold web [16,17], these NHIMs will be connected in an interwoven way with each other and also with NHIMs with saddles. 

In his celebrated article, Arnold actually constructed a model where these movements take place [15]. In this model, orbits move along a nonlinear resonance under the influence of other resonances. From the results of this study, it is found that the dynamics on the network of nonlinear resonances is characteristic for systems of N degrees of freedom with N >3. The network is called the Arnold web [16,17]. 

The Arnold web is supposed to play an important role in intramolecular vibrational energy redistribution (IVR) [1]. However, in order to reveal its role for IVR, the following two problems must be investigated. 

Phase-space structure of Hamiltonian systems with multiple degrees of freedom—in particular, normally hyperbolic invariant manifolds (NHlMs), intersections between their stable and unstable manifolds, and the Arnold web. 

Nonlinear resonances are important factors in reaction processes of systems with many degrees of freedom. The contributions of Konishi and of Honjo and Kaneko discuss this problem. Konishi analyzes, by elaborate numerical calculations, the so-called Arnold diffusion, a slow movement along a single resonance under the influence of other resonances. Here, he casts doubt on the usage of the term diffusion. In other words, Arnold diffusion is a dynamics completely different from random behavior in fully chaotic regions where most of the invariant structures are lost. Hence, understanding Arnold diffusion is essential when we go beyond the conventional statistical theory of reaction dynamics. The contribution of Honjo and Kaneko discusses dynamics on the network of nonlinear resonances (i.e., the Arnold web), and stresses the importance of resonance intersections since they play the role of the hub there. 

This model is clearly incomplete, since it does not account for vague tori [355] and the complex Arnold web [357, 358] structure of a multidimensional phase space with both chaotic and quasi-periodic trajectories. However, Eq. (74) does properly describe that, with non-ergodic dynamics, the lifetime distribution will have an initial component that decays faster than the RRKM prediction as found in the simulations by Bunker [323,324] and the more recent study of HCO dissociation [51]. Additionally, there will be a component to the classical rate, which is slower than /srrkm, for example, in the dissociations of NO2 and O3 this component cannot be described by an expression as simple as the one in Eq. (74). 

A complete model for the non-ergodic classical dynamics of a polyatomic molecule will need to represent the complete Arnold web structure of the phase space. There may be multiple bottlenecks for IVR and vague tori may exist in the vicinity of invariant tori. These complex phase space structures, leading to non-ergodic dynamics, are the origins of the... 

We introduce a simple model to investigate and calculate a diffusion coefficient as a basic quantity describing transport in Section II, and then we visualize resonances to detect the structure of the Arnold web and overlapped resonances in Section III. With the aid of this representation, to clarify the relevance of Arnold diffusion and diffusion induced by resonance overlap to global transport in the phase space, we compute transition diagrams in the frequency space in Section IV. In Section V, we extend the resonance overlap criterion to multidimensional systems to identify the pathway for fast transport, and in Section VI we revisit the diffusion coefficient to ensure fast transport affecting the global diffusion. A brief summary is given in Section VII. 

Froeschle C., Guzzo, M. and Lega, E. (2000) Graphical Evolution of the Arnold Web From Order to chaos. Science, Volume 289, n. 5487. 

Recent experiments make it possible to actually analyze the Arnold web of specific molecules. A typical example is vibrationally highly excited acetylene. These analyses indicate that ergodicity is doubtful for this molecule, which opens the possibility of studying selective IVR, and would even lead to laser control of reactions. As a preliminary investigation toward this direction, we will report our study on the behavior of acetylene under external fields. 

In Section V, we discuss the relationship between IVR and the network of nonlinear resonances (the Arnold web). By studying the web for vibra-tionally highly excited acetylene, we show that a nonstatistical nature of the IVR is explained by a sparse feature of the web, which comes from the selection rules imposed by the symmetry of the molecule. Furthermore, in studying the dynamics of the molecule under external fields, we show that the effects of the external field and the resonances are combined to produce a new behavior that each of the two cannot reveal. We suggest that combining the effects of external fields and resonances would enable us to manipulate reactions. 

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