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

Abstract

We consider one-dimensional infinite chains of harmonic oscillators with stochastic perturbations and long-range interactions which have polynomial decay rate $|x{|}^{-\theta },x\to \infty ,\theta >1$, where $x\in \mathbb{Z}$ is the interaction range. We prove that if $2<\theta \le 3$, then the time evolution of the macroscopic thermal energy distribution is superdiffusive and governed by a fractional diffusion equation with exponent $\frac{3}{7-\theta }$, while if $\theta >3$, then the exponent is $\frac{3}{4}$. The threshold is $\theta =3$ because the derivative of the dispersion relation diverges as $k\to 0$ when $\theta \le 3$.

arXiv comment43 pages

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Cited by in corpus (2)

• Superballistic and superdiffusive scaling limits of stochastic harmonic chains with long-range interactions
• Vlasov limit for a chain of oscillators with Kac potentials