Anomalous Energy Transport in FPU- $$\beta $$ β Chain

Mellet, Antoine; Merino-Aceituno, Sara

doi:10.1007/s10955-015-1273-2

Cited by 15 publications

(8 citation statements)

References 33 publications

(104 reference statements)

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“…As we saw in the previous section, NFH provides some understanding of the emergence of Lévy-walk behaviour, which seems to capture several aspects of anomalous transport. However, providing a completely rigorous microscopic derivation of the Lévy-walk picture in a Hamiltonian model has been difficult, though there have been some attempts [79]. While generic non-linear Hamiltonian models are difficult to analyze, analytical results have been obtained for harmonic chains whose Hamiltonian dynamics is perturbed by stochastic noise that breaks integrability of the system [9,30,52].…”

Section: Stochastic Models: Exact Results On Fractional Equation mentioning

confidence: 99%

Anomalous Heat Transport in One Dimensional Systems: A Description Using Non-local Fractional-Type Diffusion Equation

Dhar

Kundu

2019

Front. Phys.

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It has been observed in many numerical simulations, experiments and from various theoretical treatments that heat transport in one-dimensional systems of interacting particles cannot be described by the phenomenological Fourier's law. The picture that has emerged from studies over the last few years is that Fourier's law gets replaced by a spatially non-local linear equation wherein the current at a point gets contributions from the temperature gradients in other parts of the system. Correspondingly the usual heat diffusion equation gets replaced by a non-local fractional-type diffusion equation. In this review, we describe the various theoretical approaches which lead to this framework and also discuss recent progress on this problem.

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Section: Stochastic Models: Exact Results On Fractional Equation mentioning

confidence: 99%

Anomalous Heat Transport in One Dimensional Systems: A Description Using Non-local Fractional-Type Diffusion Equation

Dhar

Kundu

2019

Front. Phys.

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“…Note that they did not prove whether c α,γ 0 = C α,γ 0 or not. In general, the two-step limit for an anharmonic chain does not coincide with the direct limit of that [18,19].…”

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confidence: 97%

A family of fractional diffusion equations derived from stochastic harmonic chains with long-range interactions

Suda¹

2019

Preprint

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We consider one-dimensional infinite chains of harmonic oscillators with stochastic perturbations and long-range interactions which have polynomial decay rate x −θ , x → ∞, θ > 1, where x ∈ Z is the interaction range. We prove that if 2 < θ ≤ 3, then the time evolution of the macroscopic thermal energy distribution is superdiffusive and governed by a fractional diffusion equation with exponent 3 7−θ , while if θ > 3, then the exponent is 3 4 . The threshold is θ = 3 because the derivative of the dispersion relation diverges as k → 0 when θ ≤ 3.

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“…The case with degenerate collision frequency we are considering has been studied in [2]. It does not correspond to a particles dynamics physical situation but rather to the modelization of some chains of oscillators, see [13,22].…”

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confidence: 99%

Numerical Schemes for Kinetic Equations in the Anomalous Diffusion Limit. Part II: Degenerate Collision Frequency

Crouseilles¹,

Hivert²,

Lemou³

2016

SIAM J. Sci. Comput.

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Abstract. In this work, which is the continuation of [9], we propose numerical schemes for linear kinetic equation which are able to deal with the fractional diffusion limit. When the collision frequency degenerates for small velocities it is known that for an appropriate time scale, the small mean free path limit leads to an anomalous diffusion equation. From a numerical point of view, this degeneracy gives rise to an additional stiffness that must be treated in a suitable way to avoid a prohibitive computational cost. Our aim is therefore to construct a class of numerical schemes which are able to undertake these stiffness. This means that the numerical schemes are able to capture the effect of small velocities in the small mean free path limit with a fixed set of numerical parameters. Various numerical tests are performed to illustrate the efficiency of our methods in this context.

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Anomalous Energy Transport in FPU- $$\beta $$ β Chain

Cited by 15 publications

References 33 publications

Anomalous Heat Transport in One Dimensional Systems: A Description Using Non-local Fractional-Type Diffusion Equation

Anomalous Heat Transport in One Dimensional Systems: A Description Using Non-local Fractional-Type Diffusion Equation

A family of fractional diffusion equations derived from stochastic harmonic chains with long-range interactions

Numerical Schemes for Kinetic Equations in the Anomalous Diffusion Limit. Part II: Degenerate Collision Frequency

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