Publication: Detailed Examination of Transport Coefficients in Cubic-Plus-Quartic Oscillator

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Title Detailed Examination of Transport Coefficients in Cubic-Plus-Quartic Oscillator
Authors/Editors* Geoffrey Lee-Dadswell, Bernie Nickel, Chris Gray
Where published* Journal of Statistical Physics
How published* Journal
Year* 2007
Volume -1
Number -1
Publisher Springer
We examine the thermal conductivity and bulk viscosity of a onedimensional (1D) chain of particles with cubic-plus-quartic interparticle potentials and no on-site potentials. This system is equivalent to the FPU-®¯ system in a subset of its parameter space. We predict that in the lowest frequency regime (which we call the hydrodynamic regime) heat is transported ballistically by long wavelength sound modes. The model that we use to describe this behaviour predicts that as ! ! 0 the frequency dependent bulk viscosity, ³(!), and the frequency dependent thermal conductivity, ·(!), should diverge with the same power law dependence on !. Thus, the ratio ³(!)/(¯·(!)) should approach a constant value and we use mode-coupling theory to predict the ! ! 0 value of this ratio. We call this ratio the bulk Prandtl number, Pr³ . We calculate the value of Pr³ using simulations and find that the simulations are in agreement with the predictions. Further, at frequencies higher than the hydrodynamic regime there are indications that there is another regime in which 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 ·(!). We find that the value of ·(!) in this plateau region goes as T−2. This is in agreement with the expected result of a four-phonon Boltzmann-Peierls equation calculation. We call this frequency regime the perturbative regime. We conclude that in weakly anharmonic systems there is a perturbative regime at intermediate frequencies which gives way to a hydrodynamic regime at lower frequencies. In the appendix we summarize results of a recent Boltzmann-Peierls equation calculation.
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