Universality of One-Dimensional Heat Conductivity

TL;DR

This paper analytically demonstrates that one-dimensional oscillator chains exhibit a universal divergence in heat conductivity proportional to system size to the power of one-third, applicable to both fluids and stiff systems like nanotubes.

Contribution

It provides an analytical proof of the universal N^{1/3} divergence of heat conductivity in 1D systems, extending understanding beyond numerical simulations.

Findings

Heat conductivity diverges as N^{1/3} in 1D oscillator chains.

Hydrodynamic equations for 1D crystals predict the same scaling as fluids.

The results are relevant for stiff systems such as nanotubes.

Abstract

We show analytically that the heat conductivity of oscillator chains diverges with system size N as N^{1/3}, which is the same as for one-dimensional fluids. For long cylinders, we use the hydrodynamic equations for a crystal in one dimension. This is appropriate for stiff systems such as nanotubes, where the eventual crossover to a fluid only sets in at unrealistically large N. Despite the extra equation compared to a fluid, the scaling of the heat conductivity is unchanged. For strictly one-dimensional chains, we show that the dynamic equations are those of a fluid at all length scales even if the static order extends to very large N. The discrepancy between our results and numerical simulations on Fermi-Pasta-Ulam chains is discussed.