Weak and strong chaos in Fermi-Pasta-Ulam models and beyond
Year: 2005
Authors: Pettini M., Casetti L., Cerruti-Sola M., Franzosi R., Cohen E. G. D.
Autors Affiliation: INAF, Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125 Firenze, Italy; Dipartimento di Fisica, Universią di Firenze, via G. Sansone 1, I-50019 Sesto Fiorentino (FI), Italy; Dipartimento di Fisica, Universitą di Pisa, via Buonarroti 2, I-56127 Pisa, Italy; Rockefeller University, 1230 York Avenue, New York, NY 10021-6399, United States; Ctro. Interdipartimentale S.D.C., Universitą di Firenze, Unitą di Ricerca di Firenze, via G. Sansone 1, I-50019 Sesto Fiorentino (FI), Italy; Ist. Nazionale di Fisica Nucleare, Sezione di Firenze, via G. Sansone 1, I-50019 Sesto Fiorentino (FI), Italy
Abstract: We briefly review some of the most relevant results that our group obtained in the past, while investigating the dynamics of the Fermi-Pasta-Ulam (FPU) models. The first result is the numerical evidence of the existence of two different kinds of transitions in the dynamics of the FPU models: (i) A stochasticity threshold (ST), characterized by a value of the energy per degree of freedom below which the overwhelming majority of the phase space trajectories are regular (vanishing Lyapunov exponents). It tends to vanish as the number N of degrees of freedom is increased. (ii) A strong stochasticity threshold (SST), characterized by a value of the energy per degree of freedom at which a crossover appears between two different power laws of the energy dependence of the largest Lyapunov exponent, which phenomenologically corresponds to the transition between weak and strong chaotic regimes. It is stable with N. The second result is the development of a Riemannian geometric theory to explain the origin of Hamiltonian chaos. Starting this theory has been motivated by the inadequacy of the approach based on homoclinic intersections to explain the origin of chaos in systems of arbitrarily large N, or arbitrarily far from quasi-integrability, or displaying a transition between weak and strong chaos. Finally, the third result stems from the search for the transition between weak and strong chaos in systems other than FPU. Actually, we found that a very sharp SST appears as the dynamical counterpart of a thermodynamic phase transition, which in turn has led, in the light of the Riemannian theory of chaos, to the development of a topological theory of phase transitions. (C) 2005 American Institute of Physics.
Journal/Review: CHAOS
Volume: 15 (1) Pages from: 015106-1 to: 015106-13
More Information: We thank D. K. Campbell, P. Rosenau, and G. Zaslavsky for their kind invitation to contribute to this special issue commemorating the 50th anniversary of the fundamental work by E. Fermi, J. Pasta, and S. Ulam. E.G.D.C. gratefully acknowledges support from the Office of Basic Energy Sciences of the US Department of Energy under Grant No. DE-FG02-88-ER13847.KeyWords: Choas; Differential Geometry; Dynamical SystemsDOI: 10.1063/1.1849131ImpactFactor: 1.760Citations: 43data from “WEB OF SCIENCE” (of Thomson Reuters) are update at: 2024-11-10References taken from IsiWeb of Knowledge: (subscribers only)Connecting to view paper tab on IsiWeb: Click hereConnecting to view citations from IsiWeb: Click here
