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The Journal of Chemical Physics

Dual Lanczos vector space. II. Lanczos transformations and contracted descriptions of relaxation processes

William A. Wassam

Published Apr 1, 1985• 19 citations
Publisher

Abstract

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.

DOI
10.1063/1.448941

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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.
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.
Abstract Explaining the emergence of stochastic irreversible macroscopic dynamics from time-reversible deterministic microscopic dynamics is one of the key problems in philosophy of physics. The Mori-Zwanzig (MZ) projection operator formalism, which is one of the most important methods of modern nonequilibrium statistical mechanics, allows for a systematic derivation of irreversible transport equations from reversible microdynamics and thus provides a useful framework for understanding this issue. However, discussions of the MZ formalism in philosophy of physics tend to focus on simple variants rather than on the more sophisticated ones used in modern physical research. In this work, I will close this gap by studying the problems of probability and irreversibility using the example of Grabert’s time-dependent projection operator formalism. This allows to better understand how general proposals for understanding probability in statistical mechanics, namely (a) quantum approaches and (b) almost-objective probabilities, can be accomodated in the MZ formalism. Moreover, I will provide a detailed physical analysis, based on the MZ formalism, of various proposals from the philosophical literature, such as (a) Robertson’s theory of justifying coarse-graining via autonomous macrodynamics, (b) Myrvold’s problem of explaining autonomous macrodynamics, and (c) Wallace’s simple dynamical conjecture.
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.