A general type of fluctuation-dissipation theorem is discussed to show that the physical quantities such as complex susceptibility of magnetic or electric polarization and complex conductivity for electric conduction are rigorously expressed in terms of time-fluctuation of dynamical variables associated with such irreversible processes. This is a generalization of statistical mechanics which affords exact formulation as the basis of calculation of such irreversible quantities from atomistic theory. The general formalism of this statistical-mechanical theory is examined in detail. The response, relaxation, and correlation functions are defined in quantummechanical way and their relations are investigated. The formalism is illustrated by simple examples of magnetic and conduction problems. Certain sum rules are discussed for these examples. Finally it is pointed out that this theory may be looked as a generalization of the Einstein relation.

Statistical-Mechanical Theory of Irreversible Processes : I. General Theory and Simple Applications to Magnetic and Conduction Problems

The linear response theory has given a general proof of the fluctuation-dissipation theorem which states that the linear response of a given system to an external perturbation is expressed in terms of fluctuation properties of the system in thermal equilibrium. This theorem may be represented by a stochastic equation describing the fluctuation, which is a generalization of the familiar Langevin equation in the classical theory of Brownian motion. In this generalized equation the friction force becomes retarded or frequency-dependent and the random force is no more white. They are related to each other by a generalized Nyquist theorem which is in fact another expression of the fluctuation-dissipation theorem. This point of view can be applied to a wide class of irreversible process including collective modes in many-particle systems as has already been shown by Mori. As an illustrative example, the density response problem is briefly discussed.

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The fluctuation-dissipation theorem

1. Introduction 2. Theoretical foundations 3. Propagation and focusing of optical fields 4. Spatial resolution and position accuracy 5. Nanoscale optical microscopy 6. Near-field optical probes 7. Probe-sample distance control 8. Light emission and optical interaction in nanoscale environments 9. Quantum emitters 10. Dipole emission near planar interfaces 11. Photonic crystals and resonators 12. Surface plasmons 13. Forces in confined fields 14. Fluctuation-induced phenomena 15. Theoretical methods in nano-optics Appendices Index.

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Principles of nano-optics

Preface Acknowledgements Notation Part I. Overview and Background Topics: 1. Introduction 2. Overview 3. Theoretical background 4. Periodic solids and electron bands 5. Uniform electron gas and simple metals Part II. Density Functional Theory: 6. Density functional theory: foundations 7. The Kohn-Sham ansatz 8. Functionals for exchange and correlation 9. Solving the Kohn-Sham equations Part III. Important Preliminaries on Atoms: 10. Electronic structure of atoms 11. Pseudopotentials Part IV. Determination of Electronic Structure, The Three Basic Methods: 12. Plane waves and grids: basics 13. Plane waves and grids: full calculations 14. Localized orbitals: tight binding 15. Localized orbitals: full calculations 16. Augmented functions: APW, KKR, MTO 17. Augmented functions: linear methods Part V. Predicting Properties of Matter from Electronic Structure - Recent Developments: 18. Quantum molecular dynamics (QMD) 19. Response functions: photons, magnons ... 20. Excitation spectra and optical properties 21. Wannier functions 22. Polarization, localization and Berry's phases 23. Locality and linear scaling O (N) methods 24. Where to find more Appendixes References Index.

Electronic Structure: Basic Theory and Practical Methods

The Theory of a general quantum system interacting with a linear dissipative system

The formalism of nonlinear irreversible thermodynamics, as recently given by us, is applied to a circuit containing a capacitor and a nonlinear resistor. The solution is compared with those obtained by M. Lax and other investigators. It is shown that the fluctuation-dissipation theorem is rigorous, and that no correction factors need be introduced for nonlinear systems. The dynamical behavior of the microscopic fluctuations, from which the macroscopic motion can be obtained, is also derived. Finally, a specific Markoffian model of a nonlinear $R\ensuremath{-}C$ system, in strong interaction with a temperature reservoir, is shown to be consistent with the general analysis.

Irreversible Thermodynamics of a Nonlinear R − C System

In an earlier paper [Amer. J. Phys. 36, 735 (1968)] of the same title one of us described a particularly simple method by which symmetry considerations can be applied to determine the possible forms of macroscopic laws. In this sequel we clarify the role of time-reversal symmetry, describe more fully the criteria for the occurrence of spontaneous electric and magnetic moments, discuss the application of the method to magnetically ordered systems, and correct a number of errors in the tables.

Crystal symmetry and macroscopic laws. II

The ground state of a ferromagnetic system with an antiferromagnetically coupled impurity is calculated exactly. The spin of the host is arbitrary, as is the ratio of impurity-host exchange to host-host exchange, but the impurity spin is taken to have ${S}_{0}=\frac{1}{2}$. The ground-state energy, wave function, spin defect, and critical field for a metamagnetic transition are computed. Results are compared with the approximate spin-wave method as employed by Ishii, Kanamori, and Nakamura. It is shown that the spin-wave method is ambiguous and can lead to grossly incorrect results if applied in particular ways.

Antiferromagnetically Coupled Impurities in a Heisenberg Ferromagnet

Prigogine has shown that in the steady state in which certain macroscopic affinities ${F}_{1}, {F}_{2}, \ensuremath{\cdots} {F}_{k}$ are fixed and other macroscopic affinities ${F}_{k+1}, {F}_{k+2}, \ensuremath{\cdots} {F}_{r}$ are unconstrained, the values assumed by the unconstrained affinities are such as to minimize the rate of production of entropy. We here show that the complete microscopic density matrix of the system is that which minimizes the rate of entropy production subject to the imposed constraints. All magnetic fields are assumed to be zero.

Herbert B. Callen

Papers

Irreversible Thermodynamics of a Nonlinear R − C System

Crystal symmetry and macroscopic laws. II

Antiferromagnetically Coupled Impurities in a Heisenberg Ferromagnet

Principle of Minimum Entropy Production

Spin structures of europium selenide