Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Cambridge University Press eBooks

Linear response theory

Joseph I. Kapusta, Charles Gale

Published Jul 13, 2023• 0 citations
Publisher

Abstract

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

DOI
10.1017/9781009401968.007

Related papers

Linear response theory

Gabriele F. Giuliani, Giovanni Vignale

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.

Cambridge University Press eBooks•Mar 31, 2005•1 citation

General linear response theory

No¨elle Pottier

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.

•Sep 17, 2009•0 citations

Linear response theory

Jørgen Rammer

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.

Cambridge University Press eBooks•Jul 19, 2007•1 citation

Linear response of an interacting electron liquid

Gabriele F. Giuliani, Giovanni Vignale

Introduction and guide to the chapter Linear response functions contain a wealth of information about the physical properties of a many-body system. In the case of the electron liquid, for example, the density–density response function provides a unified framework for the understanding of different phenomena such as static screening, effective interaction, collective modes, electron energy loss spectra (inelastic scattering of electrons), and Raman spectra (inelastic scattering of photons). The spin–spin response function provides the corresponding information for the spin density fluctuations as probed, for example, by cross-polarized Raman scattering experiments, in which the incident and scattered photon have perpendicular polarizations, or by spin-flip electron energy loss spectroscopy, in which the incoming and outgoing electrons have opposite spin orientations. As we have seen, the Hartree–Fock approximation describes each electron as an independent particle moving in a self-consistent field generated by all the other electrons. There is no correlation between this self-consistent field and the instantaneous position of the electron. Reality is however quite different: whenever an electron moves, it acts on the surrounding electrons, causing a collective disturbance which eventually feeds back on its own motion. In order to study these effects we need, first of all, to learn how the electron liquid as a whole responds to disturbances caused by the charge, the spin, and the current of a single electron. The linear response theory of Chapter 3 provides the natural framework for this description. Strictly speaking, the linear response functions describe the readjustments of the electronic density, spin, or current, in response to externally controlled fields.

Cambridge University Press eBooks•Mar 31, 2005•1 citation

Origin of the quantum operator formalism and its connection with linear response theory

Ana Marı́a Cetto, Luis de la Peña

Abstract Linear response theory is concerned with the way in which a physical system reacts to a small change in the applied forces. Here we show that quantum mechanics in the Heisenberg representation can be understood as a linear response theory. To this effect, we first address the question of the physical origin of the quantum operator formalism by considering the interaction of a bound electron with the radiation field, including the zero-point component, following the approach of stochastic electrodynamics. Once the electron has reached a stationary state, it responds linearly and resonantly to a set of modes of the driving radiation field. Such a response can lead the system to a new stationary state. Identifying a one-to-one relationship between the response variables and the corresponding operators results in the ( x,p ) commutator as the Poisson bracket of these variables with respect to the driving field amplitudes. To account for the order of the response variables, which is reflected in the non-commutativity of the operators, we introduce the concept of ordered covariance. The results obtained allow to establish a natural contact with linear response theory at the fundamental quantum level.

arXiv:2504.03144•The European Physical Journal Special Topics•Jul 16, 2025•1 citation

Response and Correlation in Neutral Systems

David Pines, Philippe Nozières

This chapter introduces the reader to the proper language in which to express with precision results of both theory and experiment concerning the response of a many-particle system to macroscopic and microscopic external probes. It shows why the Landau theory is rigorous in the long wavelength limit: as we pointed out earlier, the Landau theory is equivalent to neglecting multipair excitations in the calculation of the response function. The chapter suggests that the Landau theory succeeds because in the limit of small q and ω, the contribution of multipair excitations is usually negligible. By studying the structure of the linear response function, one may describe explicitly the transition from the collisionless high-frequency regime (where the system response is essentially microscopic), to the low-frequency hydrodynamic regime (where the response is governed by macroscopic average laws). The chapter considers a simple physical example of probe-liquid interaction, the inelastic scattering of an external particle beam by a neutral quantum liquid.

CRC Press eBooks•Mar 9, 2018•0 citations
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
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 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 and guide to the chapter Linear response functions contain a wealth of information about the physical properties of a many-body system. In the case of the electron liquid, for example, the density–density response function provides a unified framework for the understanding of different phenomena such as static screening, effective interaction, collective modes, electron energy loss spectra (inelastic scattering of electrons), and Raman spectra (inelastic scattering of photons). The spin–spin response function provides the corresponding information for the spin density fluctuations as probed, for example, by cross-polarized Raman scattering experiments, in which the incident and scattered photon have perpendicular polarizations, or by spin-flip electron energy loss spectroscopy, in which the incoming and outgoing electrons have opposite spin orientations. As we have seen, the Hartree–Fock approximation describes each electron as an independent particle moving in a self-consistent field generated by all the other electrons. There is no correlation between this self-consistent field and the instantaneous position of the electron. Reality is however quite different: whenever an electron moves, it acts on the surrounding electrons, causing a collective disturbance which eventually feeds back on its own motion. In order to study these effects we need, first of all, to learn how the electron liquid as a whole responds to disturbances caused by the charge, the spin, and the current of a single electron. The linear response theory of Chapter 3 provides the natural framework for this description. Strictly speaking, the linear response functions describe the readjustments of the electronic density, spin, or current, in response to externally controlled fields.
Abstract Linear response theory is concerned with the way in which a physical system reacts to a small change in the applied forces. Here we show that quantum mechanics in the Heisenberg representation can be understood as a linear response theory. To this effect, we first address the question of the physical origin of the quantum operator formalism by considering the interaction of a bound electron with the radiation field, including the zero-point component, following the approach of stochastic electrodynamics. Once the electron has reached a stationary state, it responds linearly and resonantly to a set of modes of the driving radiation field. Such a response can lead the system to a new stationary state. Identifying a one-to-one relationship between the response variables and the corresponding operators results in the ( x,p ) commutator as the Poisson bracket of these variables with respect to the driving field amplitudes. To account for the order of the response variables, which is reflected in the non-commutativity of the operators, we introduce the concept of ordered covariance. The results obtained allow to establish a natural contact with linear response theory at the fundamental quantum level.
This chapter introduces the reader to the proper language in which to express with precision results of both theory and experiment concerning the response of a many-particle system to macroscopic and microscopic external probes. It shows why the Landau theory is rigorous in the long wavelength limit: as we pointed out earlier, the Landau theory is equivalent to neglecting multipair excitations in the calculation of the response function. The chapter suggests that the Landau theory succeeds because in the limit of small q and ω, the contribution of multipair excitations is usually negligible. By studying the structure of the linear response function, one may describe explicitly the transition from the collisionless high-frequency regime (where the system response is essentially microscopic), to the low-frequency hydrodynamic regime (where the response is governed by macroscopic average laws). The chapter considers a simple physical example of probe-liquid interaction, the inelastic scattering of an external particle beam by a neutral quantum liquid.