Arnold diffusion

This is strictly true only for two-dimensional area-preserving maps in dimensions iV > 4, chaotic orbits may leak through KAM surfaces by a process called Arnold diffusion (see [licht83j). 

The diffusion coefficient for a gas can be experimentally measured in an Arnold diffusion cell. The device is shown in Figure 3.6 consisting of a narrow tube partially filled with pure liquid A. The system is maintained under constant pressure and gas B flows across the open end of the tube. Component A vaporizes and diffuses into the gas phase, hence the rate of vaporization can be physically measured. Develop a general steady-state expression to describe the diffusion of one gas through a second stagnant gas. Assume that the gas has negligible solubility in liquid A and is also chemically inert in A. 

Gaspard and Rice [35] also proposed a four-dimensional map in order to smdy Hamiltonian systems with Arnold diffusion. The model Hamiltonian is a free rotor in a Morse-like kicking field and takes the following form ... 

Here we review some recent results concerning quantum transport in classically chaotic systems, including new results on suppressed quantum transport through cantori, quantum suppression of Arnold diffusion, and faster-than-classical quantum anomalous diffusion. 

Given the important role of Arnold diffusion in understanding chaotic transport in many-dimensional systems, it is quite surprising that a smdy of the quantization effect on Arnold diffusion was not carried out until very recently [94-96]. In particular, Izrailev and co-workers are the first to carefully examine quantum manifestations of Arnold diffusion in a well-studied model system. The model system is comprised of two coupled quartic oscillators, one of which driven by a two-frequency field. Its Hamiltonian is given by... 

Figure 49 shows the energy variance (denoted by and in unit of [/lo(coi + co2) /4]) versus time (denoted by N and in units of T), for three different initial states—that is, below, above, and within the chaotic layer that is responsible for Arnold diffusion. Clearly, the quantum transport depends strongly on the location of the initial quantum state. In particular, with the initial state below or above the chaotic layer (curve 1 or curve 2), quickly saturates whereas with the initial state inside the chaotic layer, after a transient period A keeps increasing for a long time in a more or less linear fashion. The average linear rate of increase of A gives the quantum diffusion coefficient. 

To further demonstrate that the quantum transport of curve 3 in Fig. 49 is intrinsically related to Arnold diffusion, Izrailev and co-workers compared the p-dependence of the quantum diffusion coefficient to that of the Arnold diffusion coefficient. This comparison is shown in Fig. 50. It is seen that the quantum result resembles the classical result. That is, roughly speaking, in either case the logarithm of the diffusion coefficient decreases linearly with increasing 1 / y/p. This makes it clear that quantum manifestations of Arnold diffusion are indeed observed. 

Figure 50 also shows that quantum effects strongly suppress Arnold diffusion (note the logarithmic scale). The suppression effect for very small p is... 

Figure 49. The energy variance (in units of [So(wi + >>2) /4]) versus the time variable N (in units of T). The system parameters are chosen as //p = 0.01, So 1-77 x 10 , lOn/coi = 12kI(A2 = T = 150. The three curves correspond to three initial states, i.e., below, above, or within the separatrix associated the Arnold diffusion. [From V. Ya Demikhovskii, F. M. Izrailev, and A. 1. Malyshev, Phys. Rev. Lett. 88, 154101 (2002).]...

Figure 50. Quantum and classical Arnold diffusion constants versus l/y jl. [From V. Ya Demikhovskii, F. M. Izrailev, and A. I. Malyshev, Phys. Rev. Lett. 88, 154101 (2002).]...

It remains to examine whether or not the results of Izrailev and co-workers are general. In particular, since the Gaspard-Rice four-dimensional mapping model introduced above can mimic the Arnold diffusion in unimolecular predissociation, the corresponding quantum dynamics is of considerable interest. 

The first is that the diffusive dynamics on the web (the Arnold diffusion) is very slow. Indeed, its time scale is so long that Arnold diffusion would be irrelevant in IVR. The second is that a recent study shows that diffusive behavior across nonlinear resonances is much more prominent than that along resonances [28,33,34]. Then, movement along resonances would be surpassed by movement across resonances. 

In order for Arnold diffusion to take place, we need a transverse intersection of the stable and unstable manifolds. In other words, there must be a (xo,ao) that satisfies the following conditions ... 

Also note that the inequality Eq. (92) implies that there would be no Arnold diffusion for H = 0. To the contrary, Xia proved that there still exists a diffusive movement even for H = 0. He called it pseudo-Arnold diffusion [35]. 

Z. Xia, Arnold Diffusion and Instabilities in Hamiltonian Dynamics, preprint, see the web site . 

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. 

Here we should mention the importance of dimensionality of phase space. In two-dimensional phase space, KAM curves can encircle the two-dimensional regions and confine the orbits surrounded by them. However, in the case of the system with more than two dimensions, KAM curves do not serve as the barrier of phase space. Likewise, the partial barriers do not form bottlenecks. The possibility of the Arnold diffusion may be taken into account in more than two dimensions, but the Arnold diffusion is usually discussed instead in relation with the Nekhoroshev-type argument, not considered as a consequence of partial barriers discussed here. 

One of the phenomena that is unique in systems with many degrees of freedom is that the state of the system can undergo large changes regardless of the existence of invariant sets that are remnants of periodic/quasi-periodic orbits of the unperturbed system [1], Arnold showed this phenomenon for a specific model (shown in Section IE) and calculated the upper bound of the rate of change in the action variable by linear perturbation theory. This phenomenon is sometimes called Arnold diffusion [2]. 

Arnold diffusion is important not just in abstract dynamical system theory, but also in realistic systems such as astrodynamics and chemical physics [3,4]. 

One of the important features of Arnold diffusion is that it is supposed to be quite slow. This is shown in the original estimate of Arnold [1], Since Arnold diffusion is an essential process for relaxation in nearly integrable (i.e., weakly coupled) systems with many degrees of freedom, most relaxations can proceed in this very slow time scale. The slow relaxation is acceptable when the system we are concerned is the solar system or some other celestial object. But, does Arnold diffusion really occur in molecular systems in the course of relaxation or reaction Do the systems really show slow relaxation or not What is the phase-space structure and how does the system move in there Answers to all these questions are still unknown. 

Although the term Arnold diffusion is quite popular, it is unfortunately not well understood, although it is about 40 years since its discovery. For example, the original estimate is given as an inequality, and the rate of the actual motion is not known. Second, the range of validity (or the observability) is not known. Moreover, although it is sometimes called Arnold diffusion whether it is diffusive or not is not understood. 

Thus we really need to clarify what Arnold diffusion really is. 

The first answer was given by Arnold [1] by using a method now called the Melnikov-Arnold integral for a specific model, which we will describe in the next section. The mechanism is now called Arnold diffusion and has since been a subject of great interest [2-4, 8-19]. 

Arnold diffusion is a motion along a resonance. One can observe motion across the resonance by using a frequency map [20], Also, motion at the crossing points of resonances are interesting [21-23],... 

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