Detailed Examination of Transport Coefficients in Cubic-Plus-Quartic Oscillator Chains

TL;DR

This paper investigates the transport properties of a 1D cubic-plus-quartic oscillator chain, identifying three frequency regimes and predicting the behavior of bulk viscosity and thermal conductivity using mode-coupling theory, with simulation validation.

Contribution

It provides a detailed analysis of frequency-dependent transport coefficients in a 1D oscillator chain, introducing the concept of the bulk Prandtl number and validating theoretical predictions with simulations.

Findings

Bulk viscosity and thermal conductivity diverge with the same power law at low frequency.

The bulk Prandtl number approaches a constant as frequency approaches zero.

Simulation results agree with mode-coupling theory predictions across parameter ranges.

Abstract

We examine the thermal conductivity and bulk viscosity of a one-dimensional (1D) chain of particles with cubic-plus-quartic interparticle potentials and no on-site potentials. This system is equivalent to the FPU-alpha beta system in a subset of its parameter space. We identify three distinct frequency regimes which we call the hydrodynamic regime, the perturbative regime and the collisionless regime. In the lowest frequency regime (the hydrodynamic regime) heat is transported ballistically by long wavelength sound modes. The model that we use to describe this behaviour predicts that as the frequency goes to zero the frequency dependent bulk viscosity and the frequency dependent thermal conductivity should diverge with the same power law dependence on frequency. Thus, we can define the bulk Prandtl number as the ratio of the bulk viscosity to the thermal conductivity (with suitable prefactors to render it dimensionless). This dimensionless ratio should approach a constant value as frequency goes to zero. We use mode-coupling theory to predict the zero frequency limit. Values of the bulk Prandtl number from simulations are in agreement with these predictions over a wide range of system parameters. In the middle frequency regime, which we call the perturbative regime, heat is transported by sound modes which are damped by four-phonon processes. We call the highest frequency regime the collisionless regime since at these frequencies the observing times are much shorter than the characteristic relaxation times of phonons. The perturbative and collisionless regimes are discussed in detail in the appendices.