# Mori-Zwanzig projection operator formalism for systems with time-dependent Hamiltonians

## TL;DR

This paper extends the Mori-Zwanzig projection operator formalism to handle systems with explicitly time-dependent Hamiltonians, enabling its application to driven systems like soft matter and NMR, with practical approximation techniques.

## Contribution

We derive a generalized Mori-Zwanzig formalism for time-dependent Hamiltonians applicable to classical and quantum systems, including approximation methods for practical use.

## Key findings

- Derived microscopic expressions for relaxation times in time-dependent magnetic fields.
- Established a generalized formalism applicable to non-equilibrium and driven systems.
- Demonstrated the approach with spin relaxation and Bloch equations.

## Abstract

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 a lot of scenarios like, for example, 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.