Weak and strong chaos in Fermi–Pasta–Ulam models and beyond

Pettini, Marco; Casetti, Lapo; Cerruti-Sola, M.; Franzosi, Roberto; Cohen, E. G. D.

doi:10.1063/1.1849131

Cited by 46 publications

(62 citation statements)

References 71 publications

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“…For the FPU model, the main goal is to understand the transition to global ergodicity. Mostly related to the general problem of weak chaos in Hamiltonian systems with many degrees of freedom are the results of papers [18][19][20][21]. There, a transition between weak and strong chaos was found marking a stochasticity threshold between slow and fast relaxation to equilibrium.…”

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confidence: 99%

Strong and Weak Chaos in Weakly Nonintegrable Many-Body Hamiltonian Systems

et al. 2011

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We study properties of chaos in generic one-dimensional nonlinear Hamiltonian lattices comprised of weakly coupled nonlinear oscillators by numerical simulations of continuous-time systems and symplectic maps. For small coupling, the measure of chaos is found to be proportional to the coupling strength and lattice length, with the typical maximal Lyapunov exponent being proportional to the square root of coupling. This strong chaos appears as a result of triplet resonances between nearby modes. In addition to strong chaos we observe a weakly chaotic component having much smaller Lyapunov exponent, the measure of which drops approximately as a square of the coupling strength down to smallest couplings we were able to reach. We argue that this weak chaos is linked to the regime of fast Arnold diffusion discussed by Chirikov and Vecheslavov. In disordered lattices of large size we find a subdiffusive spreading of initially localized wave packets over larger and larger number of modes. The relations between the exponent of this spreading and the exponent in the dependence of the fast Arnold diffusion on coupling strength are analyzed. We also trace parallels between the slow spreading of chaos and deterministic rheology.

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Strong and Weak Chaos in Weakly Nonintegrable Many-Body Hamiltonian Systems

et al. 2011

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“…Notice that since the minimum of S(λ δ ) occurs for a small λ δ value, S(0) is quadratically small and this implies that a direct estimate is rather problematic. To test our prediction in a different system, we have also studied a Hamiltonian model, namely a chain of Fermi-Pasta-Ulam (FPU) oscillators [34], again with periodic boundary conditions,…”

Section: Convergence Towards the Covariant Lyapunov Vectorsmentioning

confidence: 99%

Covariant Lyapunov vectors

Ginelli¹,

Chaté²,

Livi³

et al. 2013

J. Phys. A: Math. Theor.

140

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Abstract. The recent years have witnessed a growing interest for covariant Lyapunov vectors (CLVs) which span local intrinsic directions in the phase space of chaotic systems. Here we review the basic results of ergodic theory, with a specific reference to the implications of Oseledets' theorem for the properties of the CLVs. We then present a detailed description of a "dynamical" algorithm to compute the CLVs and show that it generically converges exponentially in time. We also discuss its numerical performance and compare it with other algorithms presented in literature. We finally illustrate how CLVs can be used to quantify deviations from hyperbolicity with reference to a dissipative system (a chain of Hénon maps) and a Hamiltonian model (a Fermi-Pasta-Ulam chain).

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“…A long debate then followed [3,4,5,6,7] concerning the questions (still unanswered) whether the phenomenon persists in the "thermodynamic limit" (i.e., when the number N of particles and the energy E both grow to infinity with a finite value of the specific energy ǫ = E/N ), and whether it can be interpreted in a metastability perspective [8,9]. Another still open problem is whether the phenomenon persists when the dimensions are increased, passing from a chain of particles to a 2-or a 3-dimensional lattice [10].…”

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Fermi-Pasta-Ulam phenomenon for generic initial data

et al. 2007

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The well known FPU phenomenon (lack of attainment of equipartition of the mode-energies at low energies, for some exceptional initial data) suggests that the FPU model does not have the mixing property at low energies. We give numerical indications that this is actually the case. This we show by computing orbits for sets of initial data of full measure, sampled out from the microcanonical ensemble by standard Montecarlo techniques. Mixing is tested by looking at the decay of the autocorrelations of the mode-energies, and it is found that the high-frequency modes have autocorrelations that tend instead to positive values. Indications are given that such a nonmixing property survives in the thermodynamic limit. It is left as an open problem whether mixing obtains within time-scales much longer than the presently available ones. PACS numbers: Valid PACS appear hereBy the "standard" FPU phenomenon we mean the celebrated one observed for the first time in the year 1955 [1]. Namely, in numerical integrations of the equations of motion for a chain of particles coupled by weakly nonlinear springs, equilibrium is not attained within the available computational time, and a kind of anomalous pseudoequilibrium does instead show up. This is observed at low energies, for initial data very far from equilibrium. The quantities studied were the energies E j (t) of the normal modes of the linearized system, and their time-averages E j (t) were found to relax each to a different value rather than to a common one, against the equipartition principle (see especially the last figure of the original FPU report).It was later found by Izrailev and Chirikov [2] that the phenomenon disappears, i.e., energy equipartition is quickly attained, if energy is large enough. A long debate then followed [3,4,5,6,7] concerning the questions (still unanswered) whether the phenomenon persists in the "thermodynamic limit" (i.e., when the number N of particles and the energy E both grow to infinity with a finite value of the specific energy ǫ = E/N ), and whether it can be interpreted in a metastability perspective [8,9]. Another still open problem is whether the phenomenon persists when the dimensions are increased, passing from a chain of particles to a 2-or a 3-dimensional lattice [10].In the present letter we address a further problem, namely whether some analog of the FPU phenomenon occurs if generic initial data are taken rather than some very special ones (see also [11,12,13]). More generally, we would like to look at the problem of the approach to equilibrium from the viewpoint of ergodic theory, in which one considers in principle all initial data, weighted * Electronic address: carati@mat.unimi.it † Electronic address: galgani@mat.unimi.it ‡ Electronic address: giorgilli@mat.unimi.it § Electronic address: paleari@mat.unimi.itwith an invariant measure such as the microcanonical one. Now, in ergodic theory it is well known that an approach to equilibrium is guaranteed if a system is proven to be mixing. Let us recall this. For a function f on p...

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Weak and strong chaos in Fermi–Pasta–Ulam models and beyond

Cited by 46 publications

References 71 publications

Strong and Weak Chaos in Weakly Nonintegrable Many-Body Hamiltonian Systems

Strong and Weak Chaos in Weakly Nonintegrable Many-Body Hamiltonian Systems

Covariant Lyapunov vectors

Fermi-Pasta-Ulam phenomenon for generic initial data

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