An Interdisciplinary Journal

* 2015, Vol.18, No.3, pp.288-302*

As we all know, and Marko Robnik has often emphasized in his work, many problems
in theoretical physics are expressed in the form of Hamiltonian systems. Of these the first
to be extensively studied were low-dimensional, possessing as few as two (or three) degrees
of freedom. In the last 20 years, however, great attention has been devoted to Hamiltonian
systems of high dimensionality. Among these perhaps the most famous are the ones that deal
with the dynamics and statistics of a large number *N* of mass particles connected with nearest
neighbor interactions. At low energies *E*, these typically execute quasiperiodic motions near
some fundamental stable periodic orbits which represent nonlinear continuations of the
*N* normal mode solutions of the corresponding linear system. However, as the energy is
increased, these solutions destabilize causing the motion in their vicinity to drift into chaotic
domains, thus giving rise to important questions concerning the system's behavior in the
thermodynamic limit where *E* and *N* diverge with *E/N = constant*. In this review, we start
by discussing some very effcient techniques for identifying regular from chaotic domains
in multi-degree of freedom Hamiltonian systems. Then we proceed to describe some highly
complex features of the dynamics connected with the presence of unexpected 'hierarchies'
of order and chaos in such systems. In particular, we will describe how these phenomena
are manifested (a) in the form of low-dimensional tori responsible for the lack of energy
equipartiton among normal modes and (b) in the presence of long lived quasi-stationary
states whose weakly chaotic properties are related to Tsallis type and not Boltzmann-Gibbs
thermodynamics. Finally, we will mention some recent results on the effect of long range
interactions on these important dynamical and statistical phenomena.
This paper is based on the lecture delivered by the first author at the Symposium
'Quantum and Classical Chaos: What comes next?' dedicated to Marko Robnik's 60th
birthday, Ljubljana, October 9 - 11 May, 2014.

*Key words: *
classical statistical mechanics, nonlinear dynamics, chaos, entropy

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Last updated: *October 07, 2015*