Fermi-Pasta-Ulam $\beta$ lattice: Peierls equation and anomalous heat conductivity

TL;DR

This paper analyzes the Peierls equation for the Fermi-Pasta-Ulam β lattice, deriving decay rates and showing how these lead to divergent heat conductivity, with implications for modifying the model to achieve finite conductivity.

Contribution

It provides an explicit form of the linearized collision operator and estimates the decay rate of normal mode energy, offering new insights into heat transport in the FPU β lattice.

Findings

Decay rate proportional to k^{5/3}

Long-time current correlation decays as t^{-3/5}

Heat conductivity diverges without modifications

Abstract

The Peierls equation is considered for the Fermi-Pasta-Ulam $\beta$ lattice. Explicit form of the linearized collision operator is obtained. Using this form the decay rate of the normal mode energy as a function of wave vector $k$ is estimated to be proportional to $k^{5/3}$. This leads to the $t^{-3/5}$ long time behavior of the current correlation function, and, therefore, to the divergent coefficient of heat conductivity. These results are in good agreement with the results of recent computer simulations. Compared to the results obtained though the mode coupling theory our estimations give the same $k$ dependence of the decay rate but a different temperature dependence. Using our estimations we argue that adding a harmonic on-site potential to the Fermi-Pasta-Ulam $\beta$ lattice may lead to finite heat conductivity in this model.