The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics

For low density gases the validity of the Boltzmann transport equation is well established. The central object is the one-particle distribution function, $f$, which in the Boltzmann-Grad limit satisfies the Boltzmann equation. Grad and, much refined, Cercignani argue for the existence of this limit on the basis of the BBGKY hierarchy for hard spheres. At least for a short kinetic time span, the argument can be made mathematically precise following the seminal work of Lanford. In this article a corresponding programme is undertaken for weakly nonlinear, both discrete and continuum, wave equations. Our working example is the harmonic lattice with a weakly nonquadratic on-site potential. We argue that the role of the Boltzmann $f$-function is taken over by the Wigner function, which is a very convenient device to filter the slow degrees of freedom. The Wigner function, so to speak, labels locally the covariances of dynamically almost stationary measures. One route to the phonon Boltzmann equation is a Gaussian decoupling, which is based on the fact that the purely harmonic dynamics has very good mixing properties. As a further approach the expansion in terms of Feynman diagrams is outlined. Both methods are extended to the quantized version of the weakly nonlinear wave equation. The resulting phonon Boltzmann equation has been hardly studied on a rigorous level. As one novel contribution we establish that the spatially homogeneous stationary solutions are precisely the thermal Wigner functions. For three phonon processes such a result requires extra conditions on the dispersion law. We also outline the reasoning leading to Fourier's law for heat conduction.Comment: special issue on "Kinetic Theory", Journal of Statistical Physics, improved versio

Heat transport by lattice and spin excitations in the spin chain compounds SrCuO_2 and Sr_2CuO_3

We present the results of measurements of the thermal conductivity of the quasi one-dimensional spin S=1/2 chain compound SrCuO_2 in the temperature range between 0.4 and 300 K along the directions parallel and perpendicular to the chains. An anomalously enhanced thermal conductivity is observed along the chains. The analysis of the present data and a comparison with analogous recent results for Sr_2CuO_3 and other similar materials demonstrates that this behavior is generic for cuprates with copper-oxygen chains and strong intrachain interactions. The observed anomalies are attributed to the one-dimensional energy transport by spin excitations (spinons), limited by the interaction between spin and lattice excitations. The energy transport along the spin chains has a non-diffusive character, in agreement with theoretical predictions for integrable models.Comment: 12 pages (RevTeX), 8 figure

Impacts of Atomistic Coating on Thermal Conductivity of Germanium Nanowires

By using non-equilibrium molecular dynamics simulations, we demonstrated that thermal conductivity of Germanium nanowires can be reduced more than 25% at room temperature by atomistic coating. There is a critical coating thickness beyond which thermal conductivity of the coated nanowire is larger than that of the host nanowire. The diameter dependent critical coating thickness and minimum thermal conductivity are explored. Moreover, we found that interface roughness can induce further reduction of thermal conductivity in coated nanowires. From the vibrational eigen-mode analysis, it is found that coating induces localization for low frequency phonons, while interface roughness localizes the high frequency phonons. Our results provide an available approach to tune thermal conductivity of nanowires by atomic layer coating.Comment: 24 pages, 5 figure

The Cohen-Macaulay Property and F-Rationality in Certain Rings of Invariants

AbstractConsider the action of a group G ≤ Sn that permutes the n variables in a polynomial ring k[X1,...,Xn] over a field k. Two related properties, the Cohen-Macaulay property and F-rationality, are studied in the ring of invariants, and the following results are obtained. (1) The invariant ring k[X1,...,Xn]Cn produced by cyclic permutation of the variables is shown not to be Cohen-Macaulay in characteristics dividing n for n > 4. This completes the analysis of the characteristics in which this invariant ring is Cohen-Macaulay. (2) The non-F-rational locus of k[X1,...,Xn]An is found to have positive dimension for certain n and k, although this ring possesses many of the properties of F-rational rings