统计力学中的投影算子：教学方法
Projection operators in statistical mechanics: a pedagogical approach
摘要：
Mori-Zwanzig投影算子形式主义是非平衡统计力学的核心工具之一，它允许通过系统的粗粒化过程从微观动力学导出宏观运动方程。它是物理研究中的一种重要方法，并对非平衡输运方程的一般结构和微观推导的一般过程提供了许多见解。因此，它是统计力学基础课程和高级课程的一个有价值的组成部分。然而，对这种方法的易懂介绍-特别是其更高级的形式-极为罕见。在这篇文章中，我们给出了一个简单而系统的介绍，Mori-Zwanzig形式主义，让学生理解的方法，它是在当前的研究中使用的形式。这包括基本和现代版本的理论。此外，我们将形式主义与统计力学和量子力学的更一般的方面联系起来。因此，我们解释了这种方法如何可以纳入统计力学的讲座课程，作为一种方式，给一般介绍非平衡系统的研究。应用，特别是自旋弛豫和动态密度泛函理论，也进行了讨论。
英文摘要：
The Mori-Zwanzig projection operator formalism is one of the central tools of nonequilibrium statistical mechanics, allowing to derive macroscopic equations of motion from the microscopic dynamics through a systematic coarse-graining procedure. It is important as a method in physical research and gives many insights into the general structure of nonequilibrium transport equations and the general procedure of microscopic derivations. Therefore, it is a valuable ingredient of basic and advanced courses in statistical mechanics. However, accessible introductions to this method - in particular in its more advanced forms - are extremely rare. In this article, we give a simple and systematic introduction to the Mori-Zwanzig formalism, which allows students to understand the methodology in the form it is used in current research. This includes both basic and modern versions of the theory. Moreover, we relate the formalism to more general aspects of statistical mechanics and quantum mechanics. Thereby, we explain how this method can be incorporated into a lecture course on statistical mechanics as a way to give a general introduction to the study of nonequilibrium systems. Applications, in particular to spin relaxation and dynamical density functional theory, are also discussed.
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