An introductory review of the central ideas in the modern theory of dynamic critical phenomena is followed by a more detailed account of recent developments in the field. The concepts of the conventional theory, mode-coupling, scaling, universality, and the renormalization group are introduced and are illustrated in the context of a simple example---the phase separation of a symmetric binary fluid. The renormalization group is then developed in some detail, and applied to a variety of systems. The main dynamic universality classes are identified and characterized. It is found that the mode-coupling and renormalization group theories successfully explain available experimental data at the critical point of pure fluids, and binary mixtures, and at many magnetic phase transitions, but that a number of discrepancies exist with data at the superfluid transition of $^{4}\mathrm{He}$.

Theory of Dynamic Critical Phenomena

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

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

We apply the influence-functional method of Feynman and Vernon to the study of Brownian motion at arbitrary temperature. By choosing a specific model for the dissipative interaction of the system of interest with its environment, we are able to evaluate the influence functional in closed form and express it in terms of a few parameters such as the phenomenological viscosity coefficient. We show that in the limit h→0 the results obtained from the influence functional formalism reduce to the classical Fokker-Planck equation. In the case of a simple harmonic oscillator with arbitrarily strong damping and at arbitrary temperature, we obtain an explicit expression for the time evolution of the complete density matrix ϱ(x, x′, t) when the system starts in a particular kind of pure state. We compare our results with those of other approaches to the problem of dissipation in quantum mechanics.

Path integral approach to quantum Brownian motion

Our aim is to propose a theory which goes beyond the classical formulation of thermodynamics. This is achieved by enlarging the space of basic independent variables, through the introduction of non-equilibrium variables, such as the dissipative fluxes appearing in the balance equations. The next step is to find evolution equations for the dissipative fluxes. Whereas the evolution equations for the classical variables are given by the usual balance laws, no general criteria exist concerning the evolution equations of the dissipative fluxes, with the exception of the restrictions imposed on them by the second law of thermodynamics.

https://iopscience.iop.org/article/10.1088/0034-4885/51/8/002/pdf

Extended irreversible thermodynamics

Transport, Collective Motion, and Brownian Motion

A continued-fraction expansion of the Laplace transform of the time-correlation functions is obtained, which enables us to express the generalized susceptibilities and the transport coefficients in terms of the static correlation functions of a set of quantities. This expansion has a different feature from the moment and cumulant expansions, and has a convenient form to introduce the long-time approximation as well as the short-time approximation. Its ap­ plication to the anomalous relaxation and transport phenomena near the second-order phase transition points is discussed An expansion formula is also obtained for the time evolution of dynamical quantities in order to describe the various modes of motion involved according to their characteristic time constants. These two expansions are closely related to the time-correlation function formalism of irreversible processes, and allow us to have physical intuition in calculating dissipative properties.

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A Continued-Fraction Representation of the Time-Correlation Functions

The dynamical behavior of ferromagnetic spins is studied on the basis of the statistical mechanics of irreversible processes. A macroscopic equation determining the change in time of an inhomogeneous magnetization is derived with explicit expressions for the frequency spectrum and damping constant. With the use of the general expressions thus obtained, the following problems are discussed on the basis of the Heisenberg model of ferromagnetic spins: (1) the pair correlation of spins, (2) the magnetic scattering cross section of neutrons, (3) the frequency spectrum and the clamping constant 'Of spin waves, (4) the damping of the longitudinal spin component above and below the Curie point. In the low temperature limit, a straightforward reduction of our expressions leads to the spin wave frequency and damping equivalent to Dyson's theory of spin wave interactions. A general expression is given for van Hove's parameters describing the asymptotic behavior of the spin pair correlation at large distances. It is shown that the longitudinal spin damping in the low temperature limit exhibits a variety of k dependences, depending upon the relative magnitudes of the spin wave energy Dk2, the effective magnetic field H coming from other than the exchange interaction and the thermal energy keT; in particular, for !JtJ.eH<t:Dk2<t,kBT, the longitudinal damping is proportional to k2, namely, the change in time of the longitudinal spin density obeys the diffusion equation. In the vicinity of the Curie point and in the paramagnetic region, the longitudinal clamping is shown to obey the· diffusion equation. The diffusion constant thus obtained vanishes at the Curie point and is in good agreement with the values observed by Ericson and J acrot for iron above the Curie point.

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Theory of Dynamical Behaviors of Ferromagnetic Spins

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Antiferromagnetic Resonance Absorption

A quantum-statistical theory of frequency spectra and damping constants of the col­ lective motion of interacting particles at finite temperatures is presented, and used to clarify several problems. The formulation is based on the explicit recognition of the fact that a set of collective variables , properly chosen describe the collective motion in such a way that the average values of the collective variables at a time determine their average values thereafter. The frequency distribution of the density fluctuations in fluids is thus analyzed, and molecular expressions for the intensities and widths of the spectral lines are obtained. The widths are written in terms of generalized transport coefficients which depend upon the wavelength of the collective oscillation or the frequency spectrum. The expressions are valid even in the case in which the hydrodynamical description is inapplicable, and turn out to be useful for describing the sound attenuation in liquid helium II at low temperatures and the inelastic scattering of neutrons by liquids. The transport coefficients of fluids are formulated in terms of equilibrium fluctuations from a new point of view without the use of the local equilibrium ensemble. The results for the shear viscosity and thermal conductivity agree with those obtained by the author previously. A new term, however, is found to be added to the dynamical flux determining the bulk viscosity. This term arises as a result of subtracting a pressure fluctuation as­ sociated with the fluctuation of the mass and energy densities to define a random force.

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Hazime Mori

Papers

Transport, Collective Motion, and Brownian Motion

A Continued-Fraction Representation of the Time-Correlation Functions

Theory of Dynamical Behaviors of Ferromagnetic Spins

Antiferromagnetic Resonance Absorption

Collective Motion of Particles at Finite Temperatures