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

Suda, Hayate

doi:10.1214/20-aihp1133

Cited by 2 publications

(3 citation statements)

References 20 publications

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“…Note that the ballistic transport of the phononic terms with diffusive fluctuations and 3/4-superdiffusion of thermal energy agree with the theoretical prediction of [20]. We also note that for the polynomial decay model 3/4-superdiffusion of thermal energy is proved in [21] when θ > 3. Now we generalise the above argument to the case 2 < θ 4.…”

Section: Fluctuations Of Normal Modessupporting

confidence: 82%

“…For the polynomial decay model, in [21] we show that at mechanical equilibrium the thermal energy also evolves on a superdiffusive space-time scale and the time evolution law of the macroscopic thermal energy T(y, t) is given by a fractional diffusion equation and the exponent of the fractional Laplacian depends on θ > 2:…”

Section: Two Different Space-time Scales Without Stochastic Noisementioning

confidence: 94%

“…Theorem (Theorem 1 in [21]). Under suitable initial conditions and γ > 0, the following weak convergence holds for any test function…”

Section: Two Different Space-time Scales Without Stochastic Noisementioning

confidence: 99%

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Superballistic and superdiffusive scaling limits of stochastic harmonic chains with long-range interactions

Suda

2022

Nonlinearity

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We consider one-dimensional infinite chains of harmonic oscillators with random exchanges of momenta and long-range interaction potentials which have polynomial decay rate |x|−θ , x → ∞, θ > 1 where x ∈ Z is the interaction range. The dynamics conserve total momentum, total length and total energy. We prove that the systems evolve macroscopically on superballistic space-time scale ( y ε − 1 , t ε − θ − 1 2 ) when 1 < θ < 3, ( y ε − 1 , t ε − 1 − log ( ε − 1 ) − 1 ) when θ = 3, and hyperbolic space-time scale (yɛ −1, tɛ −1) when θ > 3. Combining our results and the results in (Suda 2021 Ann. Inst. Henri Poincare B 57 2268–2314), we show the existence of two different space-time scales on which the systems evolve. In addition, we prove the fluctuations of the normal modes of the superballistic wave equation, which are analogues of the Riemann invariants and capture fluctuations along the characteristics. For the normal modes, the space-time scale is superdiffusive when 2 < θ ⩽ 4 and diffusive when θ > 4.

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Section: Fluctuations Of Normal Modessupporting

confidence: 82%

Section: Two Different Space-time Scales Without Stochastic Noisementioning

confidence: 94%

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Superballistic and superdiffusive scaling limits of stochastic harmonic chains with long-range interactions

Suda

2022

Nonlinearity

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Non-Fourier heat transport in nanosystems

et al. 2023

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Energy transfer in small nano-sized systems can be very different from that in their macroscopic counterparts due to reduced dimensionality, interaction with surfaces, disorder, and large fluctuations. Those ingredients may induce non-diffusive heat transfer that requires to be taken into account on small scales. We provide an overview of the recent advances in this field from the points of view of nonequilibrium statistical mechanics and atomistic simulations. We summarize the underlying basic properties leading to violations of the standard diffusive picture of heat transport and its universal features, with some historical perspective. We complete this scenario by illustrating also the effects of long-range interaction and integrability on non-diffusive transport. Then we discuss how all of these features can be exploited for thermal management, rectification and to improve the efficiency of energy conversion. We conclude with a review on recent achievements in atomistic simulations of anomalous heat transport in single polymers, nanotubes and two-dimensional materials. A short account of the existing experimental literature is also given.

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

Cited by 2 publications

References 20 publications

Superballistic and superdiffusive scaling limits of stochastic harmonic chains with long-range interactions

Superballistic and superdiffusive scaling limits of stochastic harmonic chains with long-range interactions

Non-Fourier heat transport in nanosystems

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