Strong stochasticity threshold in nonlinear large Hamiltonian systems: Effect on mixing times

Pettini, Marco; Cerruti-Sola, M.

doi:10.1103/physreva.44.975

Cited by 107 publications

(86 citation statements)

References 25 publications

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“…In the first comprehensive numerical comparison of the two systems, (2.1) and (2.9), some physical differences were observed and, qualitatively, understood. In particular, at a given ε, the φ 4 took significantly shorter time to obtain equipartition from long-wavelength mode initial conditions and significantly longer time from short-wavelength modes, than the FPU [64]. The fact that from short wavelengths it was generally a longer process to obtain equipartition was remarked in that early work, but little background theory had been done for these initial conditions.…”

Section: Comparison Of Different Oscillator Chainsmentioning

confidence: 97%

“…Although the relaxation phenomenology depends on the details of the initial conditions, they always group into two families separated by ε c . For example, below ε c the initial excitation of high-frequency normal modes results in a slower relaxation to equipartition with respect to the initial excitation of lowfrequency modes, but this situation is reversed above ε c [64]. We can estimate the ε-scaling of λ to a transition from weak to strong chaos from a random matrix approximation (RMA) for the tangent dynamics, which approximately accounts for the high-energy scaling of λ, whereas the extrapolation of such a scaling law to lower energy increasingly overestimates the actual values of λ [53,64].…”

Section: Random Matrix Approximation In the Strong Mode Overlap Regimementioning

confidence: 98%

“…These results will be discussed further, after presenting additional data from other oscillator chains. An early comparison of the FPU-β chain with the φ 4 chain was made by Pettini and Cerruti-Sola [64]. Both oscillator chains have quartic nonlinear potentials, but the φ 4 quartic term is on-site, while the FPU-β quartic term is between oscillators.…”

Section: Numerical Determination Of T Eq Scaling With εmentioning

confidence: 99%

“…This transition has been called [64] the Strong Stochasticity Threshold (SST), and it has been ascribed to the (smooth) transition between a regime of weak chaos and a regime of strong chaos. Even if the system is prepared far from energy equipartition, corresponding to the crossover energy density ε c there is a transition between slow and fast relaxation to equipartition; the relaxation time approximately constant for energy densities greater than ε c , but steeply growing with decreasing the energy density below this value.…”

Section: Random Matrix Approximation In the Strong Mode Overlap Regimementioning

confidence: 99%

“…Using that method for the FPU equations of motion, the instability boundary was found by Berman and Kolovskii [81] in the so-called "narrow-packet" approximation. Detailed numerics over longer times were obtained for the FPU model by Pettini [64] and for the discretized sine-Gordon equation by Goedde [54], both indicating that, for a given energy, short-wavelength (high-frequency) modes required longer times to reach equipartition than long-wavelength modes. At about the same time it was demonstrated that stable intrinsic localized modes (ILMs) could exist for anharmonic periodic structures [82].…”

Section: Dynamics At Short Wavelengths: Chaotic Breathersmentioning

confidence: 99%

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Abstract. The Fermi-Pasta-Ulam (FPU) nonlinear oscillator chain has proved to be a seminal system for investigating problems in nonlinear dynamics. First proposed as a nonlinear system to elucidate the foundations of statistical mechanics, the initial lack of confirmation of the researchers expectations eventually led to a number of profound insights into the behavior of high-dimensional nonlinear systems. The initial numerical studies, proposed to demonstrate that energy placed in a single mode of the linearized chain would approach equipartition through nonlinear interactions, surprisingly showed recurrences. Although subsequent work showed that the origin of the recurrences is nonlinear resonance, the question of lack of equipartition remained. The attempt to understand the regularity bore fruit in a profound development in nonlinear dynamics: the birth of soliton theory. A parallel development, related to numerical observations that, at higher energies, equipartition among modes could be approached, was the understanding that the transition with increasing energy is due to resonance overlap. Further numerical investigations showed that time-scales were also important, with a transition between faster and slower evolution. This was explained in terms of mode overlap at higher energy and resonance overlap at lower energy. Numerical limitations to observing a very slow approach to equipartition and the problem of connecting high-dimensional Hamiltonian systems to lower dimensional studies of Arnold diffusion, which indicate transitions from exponentially slow diffusion along resonances to power-law diffusion across resonances, have been considered. Most of the work, both numerical and theoretical, started from low frequency (long wavelength) initial conditions. Coincident with developments to understand equipartition was another program to connect a statistical phenomenon to nonlinear dynamics, that of understanding classical heat conduction. The numerical studies were quite different, involving the excitation of a boundary oscillator with chaotic motion, rather than the excitation of

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Section: Comparison Of Different Oscillator Chainsmentioning

confidence: 97%