An Interdisciplinary Journal

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

Complex Aspects in Hamiltonian Dynamics and Statistics
Tassos Bountis and Helen Christodoulid

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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