声子Boltzmann方程、性质及其与弱非谐晶格动力学的联系
The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics
摘要：
对于低密度气体，玻尔兹曼输运方程的有效性是很好的建立。中心目标是单粒子分布函数，$f$，它在玻尔兹曼-格拉德极限下满足玻尔兹曼方程。格拉德和更精确的Cercignani根据硬球的BBGKY等级论证了这种极限的存在。至少在很短的动力学时间跨度内，根据兰福德的开创性工作，这个论点在数学上是精确的。在这篇文章中，相应的程序进行弱非线性，离散和连续，波动方程。我们的工作的例子是调和晶格与弱非二次的在位势。我们认为，玻尔兹曼$f$ -函数的作用是由维格纳函数，这是一个非常方便的设备，以过滤缓慢的自由度。可以说，维格纳函数局部地标记了动态几乎平稳测度的协方差。声子玻尔兹曼方程的一个途径是高斯解耦，这是基于纯谐波动力学具有非常好的混合特性的事实。作为进一步的方法的扩展方面的费曼图概述。这两种方法都推广到了弱非线性波方程的量子化形式。由此产生的声子玻尔兹曼方程几乎没有在严格的水平上研究。作为一个新的贡献，我们建立的空间均匀的固定解决方案正是热维格纳函数。对于三声子过程，这样的结果需要色散定律的额外条件。我们还概述了推理导致傅立叶定律的热传导。
英文摘要：
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.
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