Universal scaling for recovery of Fourier's law in low-dimensional solids under momentum conservation

TL;DR

This paper uses dynamic renormalization group analysis to explain the disappearance of anomalous heat conduction in low-dimensional momentum-conserving systems, revealing a universal scaling law confirmed by numerical experiments.

Contribution

It provides a theoretical explanation for the recovery of Fourier's law in low-dimensional solids and predicts a universal scaling law for heat conductivity crossover.

Findings

RG flow destabilizes the inviscid fixed point for nonzero elastic-wave speeds

Universal scaling law describes heat conductivity crossover

Numerical experiments confirm the predicted scaling in FPU-β lattices

Abstract

Dynamic renormalization group (RG) of fluctuating viscoelastic equations is investigated to clarify the cause for numerically reported disappearance of anomalous heat conduction (recovery of Fourier's law) in low-dimensional momentum-conserving systems. RG flow is obtained explicitly for simplified two model cases: a one-dimensional continuous medium under low pressure and incompressible viscoelastic medium of arbitrary dimensions. Analyses of these clarify that the inviscid fixed point of contributing the anomalous heat conduction becomes unstable under the RG flow of nonzero elastic-wave speeds. The dynamic RG analysis further predicts a universal scaling of describing the crossover between the growth and saturation of observed heat conductivity, which is confirmed through the numerical experiments of Fermi-Pasta-Ulam $\beta$ (FPU-$\beta$) lattices.