By Tassos Bountis

Advent -- Hamiltonian structures of Few levels of Freedom -- neighborhood and worldwide balance of movement -- general Modes, Symmetries and balance -- effective signs of Ordered and Chaotic movement -- FPU Recurrences and the Transition from vulnerable to powerful Chaos -- Localization and Diffusion in Nonlinear One-Dimensional Lattices -- The Statistical Mechanics of Quasi-stationary States -- Conclusions, Open difficulties and destiny Outlook

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In Chap. 8, we turn to the investigation of the type of statistics that characterizes the dynamics in “weakly chaotic” regimes, where slow diffusion processes are observed. Perhaps not surprisingly, we find that, as t grows, the probability distribution functions (pdfs) associated with sums of chaotic variables in these domains do not quickly tend to a Gaussian at equilibrium, as expected by BG statistical mechanics. Rather, they go through a sequence of quasi-stationary states (QSS), which are well-approximated by a family of q-Gaussian functions and share some remarkable properties in many examples of multi-dimensional Hamiltonian systems.

If, on the other hand, these manifolds do join smoothly, this suggests that the system has N analytic, single-valued integrals and is hence completely integrable in the sense of the LA theorem, as in the case of the simple pendulum. 2 The Case of N D 2 Degrees of Freedom Following the above discussion, it would be natural to extend our study to Hamiltonian systems of two dof, joining at first two harmonic oscillators, as shown in Fig. 5 and applying the approach of Sect. 1. We furthermore assume that our oscillators have equal masses m1 D m2 D m and spring constants k1 D k2 D k and impose fixed boundary conditions to their endpoints, as shown in Fig.

1 respectively. 1 D 3 is an irrational number. Hence, the orbits produced by these solutions in the 4dimensional phase space are never closed (periodic). Unlike the orbit shown in Fig. 6, they never pass by the same point, covering eventually uniformly the two-dimensional torus of Fig. 6 specified by the values of E1 and E2 . Thus, in the spirit of the LA theorem we have verified for a Hamiltonian system of two coupled harmonic oscillators that two global, single-valued, analytic integrals are necessary and sufficient to completely integrate the equations of motion by quadratures and moreover that the general solutions lie on two-dimensional tori, since the motion on the four-dimensional phase space of the problem is bounded about an elliptic fixed point at the origin qi D pi D 0, i D 1; 2.