Dynamics of infinite classical anharmonic crystals

TL;DR

This paper analyzes the dynamics of infinite classical anharmonic crystals on an unbounded lattice, providing bounds on energy growth and perturbation propagation, extending previous results to a broader class of polynomial potentials.

Contribution

It extends existing results on anharmonic crystal dynamics to cases where the restoring potential degree is less than twice the interaction potential degree minus one.

Findings

Bound on local energy growth over time

Nontrivial bound on perturbation velocity

Extension of known results to broader potential classes

Abstract

We consider an unbounded lattice and at each point of this lattice an anharmonic oscillator, that interacts with its first neighborhoods via a pair potential $V$ and is subjected to a restoring force of potential $U$. We assume that $U$ and $V$ are even nonnegative polynomials of degree $2\sigma_1$ and $2\sigma_2$. We study the time evolution of this system, with a control of the growth in time of the local energy, and we give a nontrivial bound on the velocity of propagation of a perturbation. This is an extension to the case $\sigma_1 < 2\sigma_2-1$ of some already known results obtained for $\sigma_1 \geq 2\sigma_2-1$.