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

Journal of Mathematical Physics

Linear response theory revisited. I. The many-body van Hove limit

K. M. van Vliet

Published Jun 1, 1978• 84 citations
Publisher

Abstract

A critical discussion of linear response theory is given. It is argued that in the formalism as it stands no dissipation is manifest. A physical reinterpretation for the case of a system in weak interaction with a reservoir is given. Mathematically this means that the van Hove limit, as well as the large system limit, is applied to the time-dependent Heisenberg operators of the Kubo formalism. The reduced operators can be put in a very compact form, viz.,BRα(t) =[exp(−Λdt)]Bα, where Bα is a Schrödinger operator and Λd is the Liouville space superoperator corresponding to the transition operator of the master equation. In this form the relaxation character of the transport expressions, and the approach to equilibrium is at once evident. New expressions for the generalized susceptibility and conductivity in this limit are presented. Also, the Onsager relations and other symmetry properties are confirmed.

DOI
10.1063/1.523833

Related papers

Linear response theory revisited. II. The master equation approach

K. M. van Vliet

We consider a system perturbed by an external field and subject to dissipative processes. From the von Neumann equation for such a system in the weak coupling limit we derive an inhomogeneous master equation, i.e., a master equation with dissipative terms and streaming terms, using Zwanzig’s projection operator technique in Liouville space. From this equation the response function, as well as expressions for the generalized conductivity and susceptibility, is obtained. It is shown that for large times only the diagonal part of the density operator is required. The various expressions are found to be in complete harmony with previous results (Part I) obtained via the van Hove limit of the Kubo–Green linear response formulas. In order to account for the properties at quantum frequencies, the evolution of the nondiagonal part in the weak coupling limit is also established. The complete time dependent behavior of the dynamic variables in the van Hove limit is expressed by B (t) =exp[−(Λd−iℒ0) t] B, where Λd is the master operator and L0 the Liouville operator in the interaction picture. The cause of irreversibility is discussed. Finally, the inhomogeneous master equation is employed to obtain as first moment equation a Boltzmann equation with streaming terms, applicable to quantum systems.

Journal of Mathematical Physics•Dec 1, 1979•81 citations

Linear response theory revisited III: One-body response formulas and generalized Boltzmann equations

M. Charbonneau, K. M. van Vliet, Pagiotis Vasilopoulos

The many-body linear response expressions obtained in previous papers [J. Math. Phys. 19, 1345 (1978); 20, 2573 (1979)] are applied to systems of weakly interacting particles. General expressions for the susceptibility and conductivity in such systems are obtained. The diagonal parts depend on the scattering processes, for which we consider interactions with bosons with mass and electron-phonon interaction. For elastic collisions simple closed forms result. For general two-body collisions, the closed expressions are cumbersome, except when the current is due to collisional current through localized states, such as Landau orbits; in that event a generalized Adams-Holstein result is obtained. The nondiagonal electrical conductivity is shown to be of paramount importance for the quantum mechanical Hall effect. We also derive quantum mechanical Boltzmann equations, both for the diagonal occupancy operator 〈nζ〉t and for the nondiagonal operator 〈c+ζ′ cζ\〉t. The total Boltzmann equation is shown to be fully equivalent with the linear response results. Finally, in the last part we derive the Boltzmann equation for the Wigner function of inhomogeneous systems. In the classical limit this yields the usual Boltzmann transport equation. This equation has therefore been obtained by first principles from the von Neumann equation.

Journal of Mathematical Physics•Feb 1, 1982•162 citations

Linear response theory

Helmut Hofmann

Abstract This chapter presents a universal definition of linear response, which at first is studied for the example of the damped oscillator. From the quantal time evolution operator lowest order transition rates are deduced, together with microscopic response functions. Their fundamental properties are studied, in particular, the meaning of their reactive and dissipative parts for energy transfer. Correlation functions are introduced and related to response functions by way of the fluctuation dissipation theorem. For a random matrix model, the latter is shown to also hold true for microcanonical ensembles. Connections to thermal Green functions are established and an extension of response theory to unstable modes is discussed. With the help of the Mori product, relations among the static response, the adiabatic and the isothermal susceptibilities are given. Time dependent irreversible processes are studied by introducing relaxation functions and by following the variation of the density operator and of entropy. Onsager relations are presented and Kubo's expressions for transport coefficients are given.

•Apr 17, 2008•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

Response theory of open quantum systems

C. H. Obcemea

We discuss in this paper the various objections to the Kubo formalism initiated by van Kampen. We motivate the need to embed the system of interest in a larger reservoir to resolve these objections using the van Hove limit. In particular, in the context of spin magnetic systems, we show that it is precisely the spin–lattice coupling together with the above limit which is responsible for the validity of the linearization ansatz.

International Journal of Quantum Chemistry•Mar 1, 1984•1 citation

Quantum transport equations

John Sykes

We consider a system of weakly interacting Fermions which satisfies all those requirements necessary for the derivation of Van Hove’s weak coupling master equation. We show how the Boltzmann equation can be derived with the help of a Factorisation Theorem which we then prove, in the limit of an infinite system. We then use this Factorisation Theorem to show that the electrical conductivity derived from the Kubo formula is equivalent to the conductivity derived from the Boltzmann equation, for the same system of Fermions. Finally, we illustrate how the theory must be modified for a weakly interacting system of electrons and phonons.

•Jan 1, 1964•0 citations
A critical discussion of linear response theory is given. It is argued that in the formalism as it stands no dissipation is manifest. A physical reinterpretation for the case of a system in weak interaction with a reservoir is given. Mathematically this means that the van Hove limit, as well as the large system limit, is applied to the time-dependent Heisenberg operators of the Kubo formalism. The reduced operators can be put in a very compact form, viz.,BRα(t) =[exp(−Λdt)]Bα, where Bα is a Schrödinger operator and Λd is the Liouville space superoperator corresponding to the transition operator of the master equation. In this form the relaxation character of the transport expressions, and the approach to equilibrium is at once evident. New expressions for the generalized susceptibility and conductivity in this limit are presented. Also, the Onsager relations and other symmetry properties are confirmed.
We consider a system perturbed by an external field and subject to dissipative processes. From the von Neumann equation for such a system in the weak coupling limit we derive an inhomogeneous master equation, i.e., a master equation with dissipative terms and streaming terms, using Zwanzig’s projection operator technique in Liouville space. From this equation the response function, as well as expressions for the generalized conductivity and susceptibility, is obtained. It is shown that for large times only the diagonal part of the density operator is required. The various expressions are found to be in complete harmony with previous results (Part I) obtained via the van Hove limit of the Kubo–Green linear response formulas. In order to account for the properties at quantum frequencies, the evolution of the nondiagonal part in the weak coupling limit is also established. The complete time dependent behavior of the dynamic variables in the van Hove limit is expressed by B (t) =exp[−(Λd−iℒ0) t] B, where Λd is the master operator and L0 the Liouville operator in the interaction picture. The cause of irreversibility is discussed. Finally, the inhomogeneous master equation is employed to obtain as first moment equation a Boltzmann equation with streaming terms, applicable to quantum systems.
The many-body linear response expressions obtained in previous papers [J. Math. Phys. 19, 1345 (1978); 20, 2573 (1979)] are applied to systems of weakly interacting particles. General expressions for the susceptibility and conductivity in such systems are obtained. The diagonal parts depend on the scattering processes, for which we consider interactions with bosons with mass and electron-phonon interaction. For elastic collisions simple closed forms result. For general two-body collisions, the closed expressions are cumbersome, except when the current is due to collisional current through localized states, such as Landau orbits; in that event a generalized Adams-Holstein result is obtained. The nondiagonal electrical conductivity is shown to be of paramount importance for the quantum mechanical Hall effect. We also derive quantum mechanical Boltzmann equations, both for the diagonal occupancy operator 〈nζ〉t and for the nondiagonal operator 〈c+ζ′ cζ\〉t. The total Boltzmann equation is shown to be fully equivalent with the linear response results. Finally, in the last part we derive the Boltzmann equation for the Wigner function of inhomogeneous systems. In the classical limit this yields the usual Boltzmann transport equation. This equation has therefore been obtained by first principles from the von Neumann equation.
Abstract This chapter presents a universal definition of linear response, which at first is studied for the example of the damped oscillator. From the quantal time evolution operator lowest order transition rates are deduced, together with microscopic response functions. Their fundamental properties are studied, in particular, the meaning of their reactive and dissipative parts for energy transfer. Correlation functions are introduced and related to response functions by way of the fluctuation dissipation theorem. For a random matrix model, the latter is shown to also hold true for microcanonical ensembles. Connections to thermal Green functions are established and an extension of response theory to unstable modes is discussed. With the help of the Mori product, relations among the static response, the adiabatic and the isothermal susceptibilities are given. Time dependent irreversible processes are studied by introducing relaxation functions and by following the variation of the density operator and of entropy. Onsager relations are presented and Kubo's expressions for transport coefficients are given.
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.
We consider a system of weakly interacting Fermions which satisfies all those requirements necessary for the derivation of Van Hove’s weak coupling master equation. We show how the Boltzmann equation can be derived with the help of a Factorisation Theorem which we then prove, in the limit of an infinite system. We then use this Factorisation Theorem to show that the electrical conductivity derived from the Kubo formula is equivalent to the conductivity derived from the Boltzmann equation, for the same system of Fermions. Finally, we illustrate how the theory must be modified for a weakly interacting system of electrons and phonons.