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Quantum to macroscale and linear response

Sandip Tiwari

Published Sep 24, 2020• 0 citations
Publisher

Abstract

Abstract This chapter focuses on the properties associated with linear response. Reversibility holds in linear transformations. Schrödinger and Maxwell equations are linear, yet the world is irreversible, with time marching forward and dissipation quite ubiquitous. The connections between the quantum and microscopic scale, which are reversible and non-deterministic, to the macroscale, where irreversibility and determinism abounds, arise through interactions where both linear and nonlinear responses can appear. Causality’s implication in linear response is illustrated through a toy example and a quantum-statistical view of response. Linear response theory—using Green’s functions—is applied to develop dispersion relationships and dielectric function. The tie-in between real and imaginary parts is illustrated as one example of the Kramers-Kronig relationship, and the linear response of a damped oscillator and the Lorentz model, together with the oscillating electron model, employed to illustrate the dielectric function implications.

DOI
10.1093/oso/9780198759867.003.0014

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International Journal of Quantum Chemistry•May 1, 1986•43 citations
Abstract This chapter focuses on the properties associated with linear response. Reversibility holds in linear transformations. Schrödinger and Maxwell equations are linear, yet the world is irreversible, with time marching forward and dissipation quite ubiquitous. The connections between the quantum and microscopic scale, which are reversible and non-deterministic, to the macroscale, where irreversibility and determinism abounds, arise through interactions where both linear and nonlinear responses can appear. Causality’s implication in linear response is illustrated through a toy example and a quantum-statistical view of response. Linear response theory—using Green’s functions—is applied to develop dispersion relationships and dielectric function. The tie-in between real and imaginary parts is illustrated as one example of the Kramers-Kronig relationship, and the linear response of a damped oscillator and the Lorentz model, together with the oscillating electron model, employed to illustrate the dielectric function implications.
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.
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.
The overview of the principles of quantum statistical mechanics are given, emphasizing the fundamental differences with respect to classical statistical mechanics, as well as the analogies prevailing for the formulation of the properties. A functional time-reversal symmetry relation is presented, allowing the deduction of response theory. The Kubo formula is obtained for the linear response properties and the fluctuation–dissipation theorem is established. For weakly coupled systems, the quantum master equation and the corresponding stochastic Schrödinger equation are deduced. The slippage of initial conditions is discussed in relation to the positivity of the reduced statistical operator. The results are illustrated with the spin-boson model.
Abstract The aim of our work was to find an unambiguous connection between irreversible macroscopic dynamics and reversible microdynamics that makes it possible to manifest irreversibility on a submacroscopic level without the use of coarse graining arguments. On this level the state of a physical system can be described by a field amplitude Ψ and the time evolution of this system is determined by a field equation for Ψ. For conservative systems, this field equation is formally identical with the linear Schrödinger equation, which can be constructed with the help of the classic Hamiltonian function for the corresponding problem. Regarding irreversible phenomena like damping due to a frictional force, for those dissipative systems no classic Hamiltonian function exists. Therefore the corresponding field equation cannot be obtained in the usual way. Nevertheless, also for dissipative systems it is possible to obtain a field equation in an unambiguous way using only Newton's form of classic mechanics. The result of our method is a nonlinear Schrödinger‐type field equation with a logarithmic nonlinearity. We discuss in more detail the properties of our logarithmic nonlinearity that corresponds to a macroscopic frictional force in a unique way. A figurative interpretation in terms of environment and interaction can be given. From a more formal point of view, the compatibility of our nonlinear operator with principles known from the theory of linear operators is investigated. One of the surprising results is the fact that although our nonlinear Hamiltonian HNL is “Hermitean” in the usual sense, in contrast to the linear theory an operator exp(i · HNL) is not unitary. Furthermore, in our theory the time‐derivative of the mean value of an operator is not only essentially determined by (the mean value of) its commutator with the Hamiltonian. There also occurs an additional term that causes irreversibility of the evolution and is connected with the feature of our theory that (in general) time derivative and construction of the mean value are no longer commuting operations. This fact shows some similarity with coarse graining theories, but in our theory the reason can be traced back unambiguously to an irreversible physical phenomenon.