A Note on Arnold Diffusion

The existence of Arnold diffusion is irrelevant to the properties of separatrix manifolds, which still mediate the transport of chaotic trajectories within the regions of phase space they control. However, if Arnold diffusion is present in a given multidimensional system, the possibility exists for chaotic motion initially trapped between two nonreactive (trapped) KAM layers to eventually become reactive. This would presumably manifest itself as an apparent bottleneck to the rate of population decay, as chaotic trajectories slowly leak out from the region occupied by regular KAM surfaces into the portion of phase space more directly accessible to the hypercylinders. However, transport via the Arnold diffusion mechanism typically manifests itself on time scales much larger than those that we observe in numerical simulations (Arnold diffusion usually occurs on the order of thousands of mappings, or vibrational periods), and so it seems improbable that this effect would be observed in a typical reaction dynamics simulation. It would be interesting to characterize the effect of Arnold diffusion in realistic molecular models. 

There remain a number of interesting challenges in addition to the application of nonlinear dynamics to larger systems. One of these is the development of a better understanding of the significance of these developments to quantum 

Elementary Principles in Statistical Mechanics, Yale University Press, New 

Tolman, The Principles of Statistical Mechanics, Dover, New York, 1938. H. Eyring, D. Henderson, B. J. Stover, and E. M. Eyring, Statistical Mechanics and Dynamics, Wiley, New York, 1982. 

Allen and D. J. Tildesley, Computer Simulation ( Liquids, 2nd edit., Oxford University Press, Oxford, 1989. 

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