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arXiv:1903.00583 · Physical review. E

Mori-Zwanzig projection operator formalism for far-from-equilibrium systems with time-dependent Hamiltonians

Michael te Vrugt, Raphael Wittkowski

Published Jun 17, 2019• 52 citations
arXivHTMLPDF

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 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.

DOI
10.1103/physreve.99.062118

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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.
This chapter analyzes the memory functions in detail. The tool is the projection operator technique introduced by R. Zwanzig and extended by H. Mori. With the help of this technique, the dispersion relations can be made into a powerful and versatile instrument as our later applications, particularly to systems of broken symmetry, should amply demonstrate. However, the projector formalism can be applied to quantum systems as well. In order not to be encumbered by incidentals, the chapter considers the formally simplest case: fluctuations of a single classical dynamical variable. It looks at the spin dynamics, to see how all of the formalism can be used. The chapter discusses the magnetization correlation function and the equilibrium-averaged autocorrelation function. The formal developments were presented for a classical system because such objects as, say, the Liouville operator are a little more familiar classically.
Overview Projection operator technique is simple in purpose. It allows us to separate the dynamical evolution of a quantity into two parts: known and unknown, the latter to be evaluated. However, this technique serves multiple purposes in TDSM. It is used to derive a reduced equation of motion for the relaxation of a dynamical variable. It is also used to derive expression for an experimentally observed TCF in terms of theoretically known TCFs. This technique was developed by Zwanzig, who was inspired by Kubo’s approach of “reduction in degrees of freedom” in an interacting many-body system. In the projection operator method, we divide the time evolution of a dynamical variable A ( t ) into two parts: (i) one through a relevant subspace that is closely related to A and varies on a similar time scale, and (ii) an orthogonal subspace, which could be fast and therefore can be averaged out to obtain a time-dependent friction kernel. The idea then is that the first contribution is known while the second may be described accurately by projecting it onto the relevant subspace [that is, (i)]. In the derivation of the reduced equation of motion, one starts with the Liouville equation and projects out irrelevant degrees of freedom to obtain, after specific algebraic manipulations, an equation of motion of the dynamical variable. The final relaxation equation has the form of the Langevin equation with microscopic expressions for the terms involved (frequency term, the friction, and the random force term). This final equation of motion is often termed relaxation equation or memory function equation. It finds wide use in the study of time-dependent statistical mechanics. In the present chapter we derive the memory function equation and discuss a few applications.
In a previous paper we showed how time-dependent projection operators may be employed to enable the Nakajima-Zwanzig projection-operator technique for deriving exact master equations to deal efficiently with two or more coupled classical or quantum systems, neither of which is reservoir like. We considered in detail the case where the relevant part of the classical-system probability-density function (PDF) or quantumsystem density operator (DO) is a product of the PDF's or DO's for the separate subsystems, and we applied the techniques developed to problems in quantum optics and in the kinetic theory of dilute nonideal gases. In this paper we make the time-dependent projection-operator approach useful for a greater variety of systems by allowing the relevant part of the PDF or DO to include correlations between two of the interacting subsystems. This extension allows us to describe well the dynamics of strongly interacting systems in a low degree of approximation while avoiding the use of infinite resummations. We derive exact generalized master equations in this manner for the same two cases as in our earlier work, namely a classical gas of NNN molecules interacting via a two-body potential and a quantum-optical system of NNN two-level atoms interacting with an electromagnetic field. In the former case, the relevant part of the PDF contains two-body correlations, and we obtain two exact coupled master equations for the singlet and doublet PDF's F1(t){F}_{1}(t)F1​(t) and F2(t){F}_{2}(t)F2​(t). In the latter case, the relevant part of the DO contains atom-field correlations, and the result is three exact coupled equations for the single-atom DO ${\ensuremath{\rho}}_{1}(t)$, the field DO R(t)R(t)R(t), and the DO for one atom plus the field $\ensuremath{\chi}(t)$. From the exact equations we derive approximate equations by making simplifying assumptions. In the case of the gas we carry out a straightforward density expansion of the ${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{F}}_{2}$ master equation and obtain a set of kinetic equations for dense gases, which have been derived previously by Frieman, Goldman, and Dorfman and which describe well effects due to the finite mean free path. In the quantum-optical case we derive kinetic equations for a single-mode laser by treating the $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\chi}}$ equation in the Born-Markoff approximation. The equations describe gain saturation and other effects on the field dynamics which are of infinite order in the atom-field coupling constant. It is shown that these equations reduce to previously derived laser master equations if atom-atom correlations can be neglected. Finally, we mention possible generalizations of the time-dependent projection-operator approach used in this paper and briefly discuss its application to other problems, including the kinetics of liquids, collisional line broadening, superradiance, and amplified spontaneous emission.
We consider the relationship of the dual Lanczos transformation method, introduced in paper I, to the projection operator methods of Mori and Zwanzig for treating relaxation processes. It is shown that Mori’s memory function formalism and ‘‘generalized’’ Langevin description of relaxation processes represents a special case in the reversible limit of the ‘‘global’’ dual Lanczos transformation method. Unlike Mori’s memory function formalism, the present results are not limited to the description of autocorrelation functions for reversible systems. Our results apply to auto- and cross-correlation functions, as well as, the ensemble average of observables for both reversible and irreversible systems. In fact, we are able to construct a continued fraction representation of both auto- and cross-correlation functions. Introducing a ‘‘bath’’ dual Lanczos transformation scheme, we show that the first bath dual Lanczos projection gives rise to the Zwanzig projection operator method for constructing an equation of motion for a reduced distribution function. Additional bath dual Lanczos projections allow us to go beyond the usual results obtained in applications of the Zwanzig method.
In this thesis we discuss Mori Projected Time Dynamics in a quantum mechanical system. As a precursor to calculating the time derivative of a mixed state of the system we examine the derivation of the Mori-Zwanzig formalism and different ways of calculating the time dynamics of various quantum systems. We consider the exact calculation of the time derivative of a mixed state. We then calculate the same time evolution using Mori Theory and compare the two results. From the general calculation of the Mori Equation we were able to perform a series of simple tests to compare Mori Theory to the known result. We discovered that in each of the three simple cases the Mori Equation and the direct calculation of the derivative give the same result, but in the more complicated situations the two calculations differed. This result leads us to believe that the Mori Equation is an accurate way of calculating the derivative of a mechanical variable in a quantum system.