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

TL;DR

This paper investigates the macroscopic behavior of one-dimensional harmonic chains with long-range interactions, revealing superballistic, ballistic, superdiffusive, and diffusive scaling limits depending on the decay rate of interactions.

Contribution

It establishes the superballistic and superdiffusive scaling limits for harmonic chains with polynomial decay interactions, extending understanding of their macroscopic evolution.

Findings

Superballistic scaling for 1<θ<3

Ballistic scaling for θ>3

Superdiffusive behavior when 2<θ≤4

Abstract

We consider one-dimensional infinite chains of harmonic oscillators with random exchanges of momenta and long-range interaction potentials which have polynomial decay rate $|x|^{-\theta}, x \to \infty, \theta > 1$ where $x \in \mathbb{Z}$ 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 $(y \varepsilon^{-1}, t \varepsilon^{- \frac{\theta - 1}{2}})$ when $1 < \theta < 3$, $(y \varepsilon^{-1}, t \varepsilon^{-1} \sqrt{- \log(\varepsilon^{-1})}^{-1})$ when $\theta = 3$, and ballistic space-time scale $(y \varepsilon^{-1}, t \varepsilon^{-1})$ when $\theta > 3$. Combining our results and the results in [10], we show the existence of two different space-time scales on which the systems evolve. In addition, we prove scaling limits of recentered normal modes of superballistic wave equations, which are analogues of Riemann invariants and capture fluctuations around characteristics. The space-time scale is superdiffusive when $2 < \theta \le 4$ and diffusive when $\theta > 4$.