November 2021 A family of fractional diffusion equations derived from stochastic harmonic chains with long-range interactions
Hayate Suda
Author Affiliations +
Ann. Inst. H. Poincaré Probab. Statist. 57(4): 2268-2314 (November 2021). DOI: 10.1214/20-AIHP1133


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 xZ is the interaction range. We prove that if 2<θ3, then the time evolution of the macroscopic thermal energy distribution is superdiffusive and governed by a fractional diffusion equation with exponent 37θ, while if θ>3, then the exponent is 34. The threshold is θ=3 because the derivative of the dispersion relation diverges as k0 when θ3.

Nous considérons une chaîne infinie unidimensionnelle d’oscillateurs harmoniques avec des perturbations stochastiques et une interaction à longue portée avec une décroissance polynomiale d’ordre |x|θ, x, θ>1, où x est la longueur d’interaction. Nous montrons que si 2<θ3, l’évolution en temps de la distribution d’énergie thermique macroscopique est sur diffusive et régie par une équation de diffusion fractionnaire avec exposant 37θ, alors que si θ>3, l’exposant est 34. Le seuil est θ=3 car la dérivée de la relation de dispersion diverge lorsque k0 quand θ3.


We are grateful to Keiji Saito for insightful discussions. HS was supported by JSPS KAKENHI Grant Number JP19J11268 and the Program for Leading Graduate Schools, MEXT, Japan.


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Hayate Suda. "A family of fractional diffusion equations derived from stochastic harmonic chains with long-range interactions." Ann. Inst. H. Poincaré Probab. Statist. 57 (4) 2268 - 2314, November 2021.


Received: 14 December 2019; Revised: 8 October 2020; Accepted: 20 November 2020; Published: November 2021
First available in Project Euclid: 20 October 2021

MathSciNet: MR4328564
zbMATH: 1487.60192
Digital Object Identifier: 10.1214/20-AIHP1133

Primary: 60K35
Secondary: 74A25 , 82C22

Keywords: Fractional diffusion equation , harmonic chain , long-range interaction , Wigner distribution

Rights: Copyright © 2021 Association des Publications de l’Institut Henri Poincaré


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Vol.57 • No. 4 • November 2021
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