Te Vrugt Michael, Wittkowski Raphael
Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, 48149 Münster, Germany.
Center for Soft Nanoscience, Westfälische Wilhelms-Universität Münster, 48149 Münster, Germany.
Phys Rev E. 2019 Jun;99(6-1):062118. doi: 10.1103/PhysRevE.99.062118.
The Mori-Zwanzig projection operator formalism is a powerful method for the derivation of mesoscopic and macroscopic theories based on known microscopic equations of motion. It has applications in a large number of areas including fluid mechanics, solid-state theory, spin relaxation theory, and particle physics. In its present form, however, the formalism cannot be directly applied to systems with time-dependent Hamiltonians. Such systems are relevant in many scenarios such as driven soft matter or nuclear magnetic resonance. In this article we derive a generalization of the present Mori-Zwanzig formalism that is able to treat also time-dependent Hamiltonians. The extended formalism can be applied to classical and quantum systems, close to and far from thermodynamic equilibrium, and even in the case of explicitly-time-dependent observables. Moreover, we develop a variety of approximation techniques that enhance the practical applicability of our formalism. Generalizations and approximations are developed for both equations of motion and correlation functions. Our formalism is demonstrated for the important case of spin relaxation in a time-dependent external magnetic field. The Bloch equations are derived together with microscopic expressions for the relaxation times.
森-茨万齐格投影算符形式体系是一种基于已知微观运动方程推导介观和宏观理论的强大方法。它在包括流体力学、固态理论、自旋弛豫理论和粒子物理学在内的大量领域都有应用。然而，就其目前的形式而言，该形式体系不能直接应用于哈密顿量随时间变化的系统。这类系统在许多情形中都很重要，比如驱动软物质或核磁共振。在本文中，我们推导了当前森-茨万齐格形式体系的一种推广形式，它也能够处理哈密顿量随时间变化的情况。扩展后的形式体系可应用于经典和量子系统，无论是接近还是远离热力学平衡的情况，甚至在可观测量明确随时间变化的情形下也适用。此外，我们还开发了多种近似技术，以增强我们形式体系的实际适用性。针对运动方程和关联函数都进行了推广和近似。我们的形式体系在随时间变化的外磁场中自旋弛豫的重要情形下得到了验证。推导了布洛赫方程以及弛豫时间的微观表达式。
