caa a22 4500 467891397 CHVBK 20180406152750.0 cr unu---uuuuu 170328e20060801xx s 000 0 eng 10.1007/s10955-005-8088-5 doi (NATIONALLICENCE)springer-10.1007/s10955-005-8088-5 Spohn Herbert Zentrum Mathematik and Physik Department, TU München, Boltzmannstr. 3, D - 85747, Garching, Germany aut The Phonon Boltzmann Equation, Properties and Link to Weakly Anharmonic Lattice Dynamics [Elektronische Daten] [Herbert Spohn] 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 program 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. Springer Science + Business Media, Inc., 2006 Kinetic limit nationallicence Feynman diagrams nationallicence Wigner function nationallicence H-theorem nationallicence transport equations nationallicence Journal of Statistical Physics Kluwer Academic Publishers-Plenum Publishers; http://www.springer-ny.com 124/2-4(2006-08-01), 1041-1104 0022-4715 124:2-4<1041 2006 124 10955 https://doi.org/10.1007/s10955-005-8088-5 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10955-005-8088-5 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Spohn Herbert Zentrum Mathematik and Physik Department, TU München, Boltzmannstr. 3, D - 85747, Garching, Germany aut NATIONALLICENCE 773 0- Journal of Statistical Physics Kluwer Academic Publishers-Plenum Publishers; http://www.springer-ny.com 124/2-4(2006-08-01), 1041-1104 0022-4715 124:2-4<1041 2006 124 10955 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer