Freezing of the optical--branch energy in a diatomic FPU chain

TL;DR

This paper demonstrates that in a diatomic FPU chain with a large mass ratio, the energy in the optical branch remains nearly constant over long times at low temperatures, indicating a form of energy freezing.

Contribution

It proves that for small temperatures and large mass ratios, the optical branch energy is approximately conserved over long times in a diatomic FPU chain.

Findings

Energy exchange between optical and acoustic modes is minimal at low temperatures.

Optical branch energy remains nearly constant for times proportional to eta^{S/2}.

Results hold uniformly in the thermodynamic limit.

Abstract

We study the exchange of energy between the modes of the optical branch and those of the acoustic one in a diatomic chain of particles, with masses $m_1$ and $m_2$. We prove that, at small temperature and provided $m_1\gg m_2$, for the majority of the initial data the energy of each branch is approximately constant for times of order $\beta^{S/2}$, where $S=\lfloor \sqrt{m_1/m_2}/2\rfloor$ and $\beta$ is the inverse temperature. The result is uniform in the thermodynamic limit.