Nekhoroshev

In addition to computer simulations, what drives the research in this direction is elaborated perturbation theories developed almost simultaneously. In particular, the Kolmogorov-Arnold-Moser (KAM) theorem, which has shown the existence of invariant tori under a small perturbation to completely inte-grable systems, and the Nekhoroshev theorem, which has proved exponentially long-time stability of trajectories close to completely integrable ones, are landmarks in this field. Although a lot of works have been done, there still remain unsolved important questions, and the Hamiltonian system is being studied as one of important branches in the theory of dynamical systems [3-5]. 

The perturbation strength for which the Nekhoroshev s theorem holds is also so small that it cannot be applied to realistic physical and chemical situations. Indeed it was shown that the range of perturbation strength is much smaller than the situation where the power spectrum density of observables exhibits a continuous one [24]. This means that, in its rigorous sense, the Nekhoroshev s theorem can only be applied to sufficiently weak perturbed systems. For the same reason as mentioned above, Nekhoroshev s theorem is nevertheless a key guiding principle to sticky or stagnant motions in nearly integrable Hamiltonian systems. 

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. 

It should be noted that the Nekhoroshev s theory is not limited to two-dimensional systems, but rather it holds in general dimensions. Therefore, in the system with many degrees of freedom, the Nekhoroshev s theorem may explain sticky motions. This is also the case with the KAM theory. As given before, the statement of the KAM theorem is not limited to the Hamiltonian with few degrees of freedom. In Section IV, we will discuss to what extend these perturbation theories have capability to predict the slow motions in manydimensional systems. 

Bounded motions on KAM tori, or Nekhoroshev-type long-time stability could, however, hardly explain such variety of time scales, because the trajectory on a KAM torus is confined on /V-dimensional subspaces in... 

A brief sketch of how the Nekhoroshev-type perturbation technique is available in their proof is as follows. First, by the scale change of canonical variables,... 

Following the same prescription as the one used by Nekhoroshev—that is, the reduction of the Hamiltonian into a normal form via a near to identity canonical transformation—one obtains the form... 

In this way, the above statement may be acceptable not only in a qualitative level, but the most important implication of the mathematical statement is that the character of freezing has been shown to be quantitatively the same as adiabaticity predicted by the Nekhoroshev estimate in the nearly integrable system. This solves, at least partially, question (iii) posed in the previous section, because the model Hamiltonian [Eq. (1)] just describes the situation where not all the degrees of freedom do not necessarily show adiabaticity but only a limited number of variables, just the energies of two subsystems in this case, are almost frozen. It is true that there may be, in principle, many other possibilities and the proposed one is not a unique way as for the division of phase space into lower-dimensional subspaces, but the separation induced by the internal structure of molecules is the most natural and plausible candidate. 

One explanation for anomalous diffusion in Hamiltonian dynamics is the presence of self-similar invariant sets or hierarchical structures formed in phase space that play the role of partial barriers. They slow down the normal diffusion. A different explanation for intermittent behavior is given by the existence of deformed and approximate adiabatic invariants in phase space. They are shown in terms of elaborated perturbation theories such as the KAM and Nekhoroshev theorems. 

M. Guzzo and G. Benettin, A Spectral Formulation of the Nekhoroshev Theorem and Its Relevance for Numerical and Experimental Analysis, preprint. 

Incidentally the singular factor in (15) reminds one of the Nekhoroshev inequality [24-28]... 

We can expect that the stagnant motions of which statistical properties are discussed in the previous section reflect a universal structure of the interface between torus and chaos. Here we discuss a conjecture concerning the universality in the stagnant layer based on the Nekhoroshev theorem, which proves the onset of a new time scale accompanied by the collapse of tori [9]. 

Theorem[Nekhoroshev]. Consider nearly integrable systems with n degrees of freedom ... 

Figure 3. (a) The Nekhoroshev plot by T ociexp[p b, for the induction time of discrete... 

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