Kinetic theory of phonons in weakly anharmonic particle chains

TL;DR

This paper develops a detailed kinetic theory framework for phonons in classical particle chains, enabling comparison of thermal transport properties between conducting and anomalously conducting chains, with explicit derivations and predictions.

Contribution

It provides a streamlined derivation of phonon Boltzmann collision operators and detailed analysis of thermal conductivity in anharmonic chains, advancing understanding of phonon transport.

Findings

Derived explicit solutions for collisional constraints.

Predicted thermal conductivities for specific chain models.

Connected entropy and Boltzmann H-theorem in phonon systems.

Abstract

The aim of this review is to develop the kinetic theory of phonons in classical particle chains to a point which allows comparing the kinetic theory of normally conducting chains, with an anharmonic pinning potential, to the kinetic theory of the anomalously conducting FPU chains. In addition to reviewing existing results from the literature, we present a streamlined derivation of the phonon Boltzmann collision operators using Wick polynomials, as well as details about the estimates which are needed to study the effect of the collision operator. This includes explicit solutions of the collisional constraints, both with and without harmonic pinning. We also recall in detail the derivation of the Green-Kubo formula for thermal conductivity in these systems, and the relation between entropy and the Boltzmann H-theorem for the phonon Boltzmann equations. The focus is in systems which are spatially translation invariant perturbations of thermal equilibrium states. We apply the results to obtain detailed predictions from kinetic theory for the Green-Kubo correlation functions, and hence the thermal conductivities, of the chain with a quartic pinning potential as well as the standard FPU-beta chain.