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.

中文翻译：

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统计力学中的投影算子：一种教学方法

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，允许通过系统的粗粒度程序从微观动力学中推导出宏观运动方程。它作为一种重要的物理研究方法，对非平衡输运方程的一般结构和微观推导的一般程序提供了许多见解。因此，它是统计力学基础和高级课程的重要组成部分。然而，这种方法的可访问介绍——特别是其更高级的形式——是极其罕见的。在本文中，我们对 Mori-Zwanzig 形式主义进行了简单而系统的介绍，让学生以当前研究中使用的形式来理解该方法论。这包括该理论的基本版本和现代版本。此外，我们将形式主义与统计力学和量子力学的更一般方面联系起来。因此，我们解释了如何将这种方法纳入统计力学的讲座课程中，作为对非平衡系统研究的一般介绍。还讨论了应用，特别是自旋弛豫和动态密度泛函理论。