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Journal of Mathematical Physics

Linear response theory revisited. II. The master equation approach

K. M. van Vliet

Published Dec 1, 1979• 81 citations
Publisher

Abstract

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.

DOI
10.1063/1.524020

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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.
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.
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.
A linear response theory for open quantum systems is formulated by means of the time-local and time-nonlocal quantum master equations, where a relevant quantum system interacts with a thermal reservoir as well as with an external classical field. A linear response function that characterizes how a relaxation process deviates from its intrinsic process by a weak external field is obtained by extracting the linear terms with respect to the external field from the quantum master equation. It consists of four parts. One represents the linear response of a quantum system when system-reservoir correlation at an initial time and correlation between reservoir states at different times are neglected. The others are correction terms due to these effects. The linear response function is compared with the Kubo formula in the usual linear response theory. To investigate the properties of the linear response of an open quantum system, an exactly solvable model for a stochastic dephasing of a two-level system is examined. Furthermore, the method for deriving the linear response function is applied for calculating two-time correlation functions of open quantum systems. It is shown that the quantum regression theorem is not valid for open quantum systems unless their reduced time evolution is Markovian.
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.
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 A generalized master equation is derived to describe intramolecular rearrangement processes. It is an inhomogeneous equation, including memory effects. The derivation is based on the Liouville space formalism. Because chemically relevant information is contained in the off-diagonal elements of the density matrix, a non-diagonal coarse-graining projector is used. All necessary assumptions are stated explicitly. By making further approximations, the master equation can be reduced to an inhomogeneous von Neumann equation with an effective Liouville operator the imaginary part of which is responsible for relaxation-like coarse-grained solutions. All neglected terms are given in closed form. The character of the solutions of the master equation is discussed in "coordinate-free" manner, i.e. without referring to the underlying Hilbert space.