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Canadian Journal of Physics

Remark on memory function approach and linear response theory

T. H. Cheung, E. Tong, K. P. Wang

Published Jul 1, 1970• 2 citations
Publisher

Abstract

It is generally understood (Kubo) that the linearized version of the memory function approach of Mori and Zwanzig is equivalent, through the application of the fluctuation–dissipation theorem, to the corresponding linear response theory of correlation function calculations developed in the work of Kadanoff and Martin. We give here a simple example of spin diffusion to show this explicitly, which reflects the general connection between these two approaches.

DOI
10.1139/p70-206

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It is generally understood (Kubo) that the linearized version of the memory function approach of Mori and Zwanzig is equivalent, through the application of the fluctuation–dissipation theorem, to the corresponding linear response theory of correlation function calculations developed in the work of Kadanoff and Martin. We give here a simple example of spin diffusion to show this explicitly, which reflects the general connection between these two approaches.
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Abstract Linear response theory is a general formalism applicable to any physical system slightly departing from equilibrium. It allows us to express the linear response functions and the generalized susceptibilities in terms of the equilibrium correlation functions of the relevant dynamical variables. For instance, in the case of the response to an applied external field, one of the two relevant dynamical variables represents the physical quantity conjugate to the field whereas the other represents the physical quantity whose out-of-equilibrium average is measured. The computation of the linear response functions relies on a first-order perturbation expansion of the density operator (or of the phase space distribution function) of the system coupled to the field. The expressions thus obtained constitute the Kubo formulas of linear response theory.
An introduction to the linear response theory will allow us to define the concepts of susceptibility, fluctuation spectra, autocorrelation function... and to connect these quantities through the fluctuation-dissipation theorem. We will then determine the origin of the scattering of light and calculate the Raman and Brillouin spectra within a (classical) macroscopic theory [1]. These calculations will reveal selection rules that will be applied to few examples taken from crystalline solids, liquids, and glasses [2].
We have given in two previous chapters a first introduction to non-equilibrium phenomena. The present chapter is devoted to a presentation of more general approaches, in which time dependence will be made explicit, whereas in practice we had to limit ourselves to stationary situations in Chapters 6 and 8. In the first part of the chapter, we examine the relaxation toward equilibrium of a system that has been brought out of equilibrium by an external perturbation. The main result is that, for small deviations from equilibrium, this relaxation is described by equilibrium time correlation functions, called Kubo (or relaxation) functions: this result is also known as ‘Onsager's regression law’. The Kubo functions turn out to be basic objects of non-equilibrium statistical mechanics. First they allow one to compute the dynamical susceptibilities, which describe the response of the system to an external time dependent perturbation: the dynamical susceptibilities are, within a multiplicative constant, the time derivatives of Kubo functions. A second crucial property is that transport coefficients can be expressed in terms of time integrals of Kubo functions. As we limit ourselves to small deviations from equilibrium, our theory is restricted to a linear approximation and is known as linear response theory . The classical version of linear reponse is somewhat simpler than the quantum one, and will be described first in Section 9.1. We shall turn to the quantum theory in Section 9.2, where one of our main results will be the proof of the fluctuationdissipation theorem.
We derive the general Kubo formula in a form that solely utilizes the time evolution of displacement operators. The derivation is based on the decomposition of the linear response function into its time-symmetric and time-antisymmetric parts. We relate this form to the well-known fluctuation-dissipation formula and discuss theoretical and numerical aspects of it. The approach is illustrated with an analytical example for magnetic resonance as well as a numerical example where we analyze the electrical conductivity tensor and the Chern insulating state of the disordered Haldane model. We introduce a highly efficient time-domain approach that describes the quantum dynamics of the resistivity of this model with an at least 1000-fold better performance in comparison to existing time-evolution schemes.