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

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-α β 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 ω→0 the frequency dependent bulk viscosity, $\hat{\zeta}(\omega)$ , and the frequency dependent thermal conductivity, $\tilde{\kappa}(\omega)$ , should diverge with the same power law dependence on ω. Thus, we can define the bulk Prandtl number, $Pr_{\zeta}=k_{B}\hat{\zeta}(\omega)/(m\hat{\kappa }(\omega))$ , where m is the particle mass and k _B is Boltzmann’s constant. This dimensionless ratio should approach a constant value as ω→0. We use mode-coupling theory to predict the ω→0 limit of Pr _ζ . Values of Pr _ζ obtained 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. This regime is characterized by an intermediate-frequency plateau in the value of $\hat{\kappa}(\omega)$ . We find that the value of $\hat{\kappa}(\omega)$ in this plateau region is proportional to T ⁻² where T is the temperature; this is in agreement with the expected result of a four-phonon Boltzmann-Peierls equation calculation. The Boltzmann-Peierls approach fails, however, to give a nonvanishing bulk viscosity for all FPU-α β chains. 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.