Effective Stochastic Model for Chaos in the Fermi–Pasta–Ulam–Tsingou Chain

Goldfriend, Tomer

doi:10.1007/s10955-023-03080-z

Cited by 3 publications

(4 citation statements)

References 39 publications

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“…[34], at E = 1/6). V nm , and of Toda truncations T j , with the integrable Toda dynamics, using as an easy tool the Lyapunov exponents (for a recent extensive study of Lyapunov exponents in truncated Toda and other FPU models, see [13,26]). Consider any initial datum z in the phase space, let F t (z) be its evolution at time t, and for any tangent vector u in z, let DF t z u be the evolved tangent vector.…”

Section: Bmentioning

confidence: 99%

“…while times O(ε −5/2 ) look necessary to observe full diffusion [12]. An intermediate time scale also exists, namely the Lyapunov time (the inverse of the maximal Lyapunov exponent), see [13,15,26]; however, on such a time scale the diffusion of actions is not affected, since chaos turns out to be tangent to tori. So, if the observation time is sufficiently large, the FPU paradox disappears, and in general, pre-thermalization scenarios turn out to be a matter of interplay between observation time and dynamical time-scales of the given physical system.…”

Section: Introductionmentioning

confidence: 99%

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Understanding the FPU state in FPU–like models

Benettin¹,

Ponno

2021

Mathematics in Engineering

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We prove that the common Mie-Lennard-Jones (MLJ) molecular potentials, appropriately normalized via an affine transformation, converge, in the limit of hard-core repulsion, to the Toda exponential potential. Correspondingly, any Fermi-Pasta-Ulam (FPU)-like Hamiltonian, with MLJ-type interparticle potential, turns out to be 1/n-close to the Toda integrable Hamiltonian, n being the exponent ruling repulsion in the MLJ potential. This means that the dynamics of chains of particles interacting through typical molecular potentials, is close to integrable in an unexpected sense. Theoretical results are accompanied by a numerical illustration; numerics shows, in particular, that even the very standard 12-6 MLJ potential is closer to integrability than the FPU potentials which are more commonly used in the literature.

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Section: Bmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Understanding the FPU state in FPU–like models

Benettin¹,

Ponno

2021

Mathematics in Engineering

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“…Since dealing with genuine non-linear forces is a formidable task, it is convenient to adopt an intermediate 'mesoscopic' approach whereby nonintegrable interactions are replaced by a random noise. This idea has proved to be useful, for instance, to study the Lyapunov time-scale [32,33] and the onset of diffusive dynamics in almost-integrable systems [34]. Also, such approach reproduces many features of nonlinear lattices in steady nonequilibrium states [35,36].…”

Section: Introductionmentioning

confidence: 99%

“…Also, such approach reproduces many features of nonlinear lattices in steady nonequilibrium states [35,36]. This semplification may overlook some of the subtle correlations of the deterministic dynamics but can provide also quantitative results, as for instance the scaling law of the Lyapunov exponent with the energy density for the Fermi-Pasta-Ulam-Tsingou model [33]. In this spirit, we focus on a specific model: an harmonic network with momentum and energy conserving noise that we studied in [37].…”

Section: Introductionmentioning

confidence: 99%

Large-deviations approach to thermalization: the case of harmonic chains with conservative noise

Lepri

2024

J. Stat. Mech.

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We investigate the possibility of characterizing the different thermalization pathways through a large-deviation approach. Specifically, we consider clean, disordered and quasi-periodic harmonic chains under energy and momentum-conserving noise. For their associated master equations, describing the dynamics of normal modes energies, we compute the fluctuations of activity and dynamical entropy in the corresponding biased ensembles. First-order dynamical phase transition are found that originates from different activity regions in action space. At the transitions, the steady-state in the biased ensembles changes from extended to localized, yielding a kind of condensation in normal-modes space. For the disordered and quasi-periodic models, we argue that the phase-diagram has a critical point at a finite value of the disorder or potential strength.

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On the Role of the Integrable Toda Model in One-Dimensional Molecular Dynamics

2023

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We prove that the common Mie–Lennard-Jones (MLJ) molecular potentials, appropriately normalized via an affine transformation, converge, in the limit of hard-core repulsion, to the Toda exponential potential. Correspondingly, any Fermi–Pasta–Ulam (FPU)-like Hamiltonian, with MLJ-type interparticle potential, turns out to be 1/n-close to the Toda integrable Hamiltonian, n being the exponent ruling repulsion in the MLJ potential. This means that the dynamics of chains of particles interacting through typical molecular potentials, is close to integrable in an unexpected sense. Theoretical results are accompanied by a numerical illustration; numerics shows, in particular, that even the very standard 12–6 MLJ potential is closer to integrability than the FPU potentials which are more commonly used in the literature.

show abstract

Effective Stochastic Model for Chaos in the Fermi–Pasta–Ulam–Tsingou Chain

Cited by 3 publications

References 39 publications

Understanding the FPU state in FPU–like models

Understanding the FPU state in FPU–like models

Large-deviations approach to thermalization: the case of harmonic chains with conservative noise

On the Role of the Integrable Toda Model in One-Dimensional Molecular Dynamics

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