Asymptotic localization of energy in non-disordered oscillator chains

TL;DR

This paper rigorously analyzes how energy localizes in weakly coupled one-dimensional classical oscillator chains, showing that thermal conductivity diminishes faster than any power law as coupling strength approaches zero, especially at high temperatures.

Contribution

It provides a rigorous proof that the thermal conductivity in certain oscillator chains decays faster than any power law in the weak coupling limit, revealing non-perturbative behavior.

Findings

Thermal conductivity decays faster than any power law in coupling strength.

Conductivity vanishes rapidly at high temperatures.

Energy localization becomes dominant as coupling approaches zero.

Abstract

We study two popular one-dimensional chains of classical anharmonic oscillators: the rotor chain and a version of the discrete non-linear Schr\"odinger chain. We assume that the interaction between neighboring oscillators, controlled by the parameter $\epsilon >0$, is small. We rigorously establish that the thermal conductivity of the chains has a non-perturbative origin, with respect to the coupling constant $\epsilon$, and we provide strong evidence that it decays faster than any power law in $\epsilon$ as $\epsilon \rightarrow 0$. The weak coupling regime also translates into a high temperature regime, suggesting that the conductivity vanishes faster than any power of the inverse temperature.