Complex aspects in hamiltonian dynamics and statistics
Bibliographic record
Abstract
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 efficient 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.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.004 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".