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.
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.
There exists a regime of overlap between the equilibrium and non-equilibrium behavior of a system, the non-equilibrium behavior of weakly perturbed states. When a system is perturbed ever so slightly, its response will be linear in the perturbation, say the current of the conduction electrons in a metal will be proportional to the strength of the applied electric field. This regime is called the linear response regime, and though the system is in a non-equilibrium state all its characteristics can be inferred from the properties of its equilibrium state. In the next chapter we shall go beyond the linear regime by showing how to obtain quantum kinetic equations. The kinetic-equation approach to transport is a general method, and allows in principle nonlinear effects to be considered. However, in many practical situations one is interested only in the linear response of a system to an external force. The linear response limit is a tremendous simplification in comparison with general non-equilibrium conditions, and is the subject matter of this chapter. In particular the linear response of the density and current of an electron gas are discussed. The symmetry properties of response functions, and the fluctuation–dissipation theorem are established. Lastly we demonstrate how correlation functions can be measured in scattering experiments, as illustrated by considering neutron scattering from matter. Needless to say, in measurements of (say) the current in a macroscopic body, far less information in the current correlation function is probed. Linear response In this section we consider the response of an arbitrary property of a system to a general perturbation.
Introduction There are countless situations in physics when one is interested in calculating the response of a system to a small time-dependent perturbation acting on it. With some luck the response can be expanded in a power series of the strength of the perturbation, so that, to first order, it is a linear function of the latter. To compute this function is the objective of the linear response theory (LRT). Linear response theory has many important applications to the study of electronic matter. Virtually all interactions of electrons with experimental probes (electromagnetic fields, beams of particles) can be regarded as small perturbations to the system: if they were not, one would not be probing the system, but the system modified by the probe. Consequently, the results of these experiments can be expressed in terms of linear response functions, which are properties of the unperturbed system. In particular it will turn out that the analytic structure of these functions is entirely determined by the eigenvalues and eigenfunctions of the unperturbed system. Conversely, a measure of the linear response as a function of the frequency of the perturbation enables us to determine the excitation energies of the system. Beside being a cornerstone for the theory of single-particle properties to be developed in Chapter 8, the linear response functions also provide invaluable information in their own right. For example, as we will discuss in Chapter 5, the extent to which an external electrostatic potential is reduced by screening is controlled by the dynamical dielectric function which, in turn, is determined by the density–density response function.
Abstract Linear response theory is an extremely widely used concept in all branches of physics. It simply states that the response to a weak external perturbation is proportional to the perturbation, and therefore all one needs to understand is the proportionality constant. Below we derive the general formula for the linear response of a quantum system exerted by a perturbation. The physical question we ask is thus: given some external perturbation H′, what is the measured consequence for an observable quantity, A.
Linear response theorySuppose that a solid is hit with a hammer.Sound waves will propagate outwards from the point of contact.How is the frequency of the sound wave related to its wave number?How does a light wave propagate through plasma?What happens when a charge impurity is embedded in an electrically neutral medium?Is it screened, and if so how is that screening described quantitatively?If a medium is disturbed by a small amount one might expect its response also to be small.The quantitative formalism for dealing with small disturbances is called linear response theory.The beauty of the theory is that the response of the system can be expressed as a folding of the external source causing the disturbance with a response function that is computed using equilibrium correlation functions not dependent on the strength of the external source.Therefore, details of the internal dynamics of the thermodynamic system can be studied using weak external probes.Other areas of science where linear response theory has proven to be extremely useful are quite extensive, and include x-ray scattering from crystals and molecules, electron scattering from protons and nuclei, and sound waves generated by earthquakes propagating through the earth's interior. Linear response to an external fieldSuppose we apply some external field to our system, which is initially in equilibrium.The goal of linear response theory is to calculate the change in the ensemble average value of any operator Y (x, t) caused by the external field, to first order in that external field.Let
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.
The problem of nonlinear response is considered by employing a general time-evolution equation, and a Green's function which is the transition or conditional probability density for an unperturbed system. Expansion of the Green's function in terms of orthonormal functions enables us to express the distribution function describing the nonlinear behavior by means of matrix products whose elements are composed of correlation functions in the absence of the perturbation. In other words, it is shown how the distribution function induced by a strong perturbation may be calculated by knowing the Green's function without the perturbation. As the special case of the linear response, we have obtained Kubo's relation. The Laplace-transform technique with respect to time is found quite useful in developing the present theory in which the transient effect is also taken into account. As an application of the theory, a new relation valid in the region of the second-order perturbation connecting the transient rise and decay with the stationary alternating perturbations has been obtained.