/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

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

This review summarizes recent developments in the theory of spin glasses, as well as pertinent experimental data. The most characteristic properties of spin glass systems are described, and related phenomena in other glassy systems (dielectric and orientational glasses) are mentioned. The Edwards-Anderson model of spin glasses and its treatment within the replica method and mean-field theory are outlined, and concepts such as "frustration," "broken replica symmetry," "broken ergodicity," etc., are discussed. The dynamic approach to describing the spin glass transition is emphasized. Monte Carlo simulations of spin glasses and the insight gained by them are described. Other topics discussed include site-disorder models, phenomenological theories for the frozen phase and its excitations, phase diagrams in which spin glass order and ferromagnetism or antiferromagnetism compete, the Ne\'el model of superparamagnetism and related approaches, and possible connections between spin glasses and other topics in the theory of disordered condensed-matter systems.

Spin glasses: Experimental facts, theoretical concepts, and open questions

The Renormalization group and the epsilon expansion

N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

New method for the calculation of the pair correlation function. I

The asymptotic time behavior of the velocity autocorrelation function and of the kinetic parts of the correlation functions for the shear viscosity and the heat conductivity is derived. The results are expressed in terms of the transport coefficients and the specific heats and are valid for all densities.

Asymptotic Time Behavior of Correlation Functions

1. Progress and Problems in Real-Space Renormalization.- 1.1 Introduction.- 1.2 Review of Real-Space Renormalization.- 1.3 New Renormalization Methods.- 1.3.1 Bond-Moving and Variational Methods.- 1.3.2 Monte Carlo Renormalization.- 1.3.3 Exact Differential Transformations.- 1.3.4 Phenomenological Renormalization.- 1.4 New Applications.- 1.4.1 Adsorbed Systems.- 1.4.2 Applications to Quantum Systems.- 1.4.3 Percolation and Polymers.- 1.4.4 Dynamic Real-Space Renormalization.- 1.4.5 The Kosterlitz-Thouless Transition.- 1.4.6 Field-Theoretical Applications.- 1.5 Fundamental Problems.- 1.5.1 Choice of the Weight Function.- 1.5.2 Griffiths-Pearce Peculiarities.- 1.6 Exact Differential Real-Space Renormalization.- 1.6.1 The Two-Dimensional Ising Model.- 1.6.2 Discussion.- 1.7 Phenomenological Renormalization.- 1.7.1 Description of the Method.- 1.7.2 Applications.- 1.8 Concluding Remarks.- References.- 2. Bond-Moving and Variational Methods in Real-Space Renormalization.- 2.1 Introduction.- 2.2 Variational Principles.- 2.2.1 Lower-Bound Property of Bond-Moving Approximations.- 2.2.2 Upper-Bound Property of the First-Order Cumulant Approximation.- 2.3 The Migdal-Kadanoff Transformation.- 2.3.1 Application to the Ising Model with Nearest-Neighbor Interactions.- 2.3.2 Inclusion of a Magnetic Field.- 2.3.3 The Bond-Moving Prescription of EMERY and SWENDSEN.- 2.3.4 Inconsistent Scaling of the Correlation Function.- 2.3.5 Relation to Exactly Soluble Hierarchical Models.- 2.3.6 Applications.- 2.3.7 Modifications of the Migdal-Kadanoff Procedure.- 2.4 Variational Transformations.- 2.4.1 The Kadanoff'Lower-Bound Variational Transformation.- 2.4.2 The Kadanoff Criterion for the Optimal Variational Parameter.- 2.4.3 Problems with the Lower-Bound Variational Transformation.- 2.4.4 Determination of an Optimal Sequence of Variational Parameters.- 2.4.5 Applications of the Lower-Bound Variational Transformation.- 2.4.6 Other Variational Methods.- 2.5 Conclusion.- References.- 3. Monte Carlo Renormalization.- 3.1 Introduction.- 3.2 Basic Notation and Renormalization-Group Formalism.- 3.3 Large-Cell Monte Carlo Renormalization Group.- 3.4 MCRG.- 3.4.1 Calculation of Critical Exponents.- 3.4.2 Calculation of Renormalized Coupling Constants.- 3.5 MCRG Calculations for Specific Systems.- 3.6 Other Approaches to the Monte Carlo Renormalization Group.- 3.7 Conclusions.- References.- 4. The Real-Space Dynamic Renormalization Group.- 4.1 Introduction.- 4.2 Dynamic Problem of Interest.- 4.3 RSDRG - Formal Development.- 4.4 Implementation of the RSDRG Using Perturbation Theory.- 4.4.1 General Development.- 4.4.2 Expansion for H and D?.- 4.4.3 Solution to the Zeroth-Order Problem.- 4.4.4 Renormalization to First Order.- 4.4.5 Recursion Relations for the Correlation Functions.- 4.5 Determination of Parameters.- 4.5.1 General Comments.- 4.5.2 The Parameters K0 and KR0.- 4.5.3 The Dynamic Parameters ?0 and ?.- 4.6 Results.- 4.7 Discussion.- References.- 5. Renormalization for Quantum Systems.- 5.1 Background.- 5.2 Application of the Niemeijer-van Leeuwen Renormalization Group Method to Quantum Lattice Models.- 5.3 The Block Method.- 5.3.1 Principles.- 5.3.2 Applications.- a) The Ising Model in a Transverse Field in One Dimension.- b) The Free Fermion Model in One Dimension.- 5.3.3 Extensions of the Method.- a) Extension to Large Blocks.- b) Extension by Increasing the Number nL of Levels Retained.- c) Other Extensions.- 5.4 Applications of the Block Method.- 5.4.1 Spin Systems.- a) The Spin 1/2 Ising Model in a Transverse Field (ITF).- b) The XY Heisenberg Spin 1/2 Chain.- c) The XY Model in a Z Field for d = 2, 3.- d) The Spin 1 XY Model with an Anisotropy Field for d = 1.- 5.4.2 Fermion Systems.- a) The d = 1 Hubbard Model.- b) Interacting Fermions in d = 1.- c) One-Dimensional Model of f and d Electrons with Hybridization V and fd Interaction Ufd.- 5.4.3 Spin Fermion Systems: The Kondo Lattice in d = 1.- 5.4.4 Quantum Versions of Classical Statistical Mechanics in 1 + 1 Dimension.- a) The 0(n) Model.- b) The P(q) Potts Model.- c) Tricritical Point for Ising Systems in 1 + 1 Dimensions.- 5.4.5 Applications to Field Theory.- a) The Thirring Model in One-Space and One-Time Dimension.- b) The U(1) Goldstone Model in Two Dimensions.- c) Lattice Gauge Theories.- 5.5 Discussion.- 5.5.1 When is the BRG More Suitable?.- 5.5.2 How to Control the Method?.- a) The Division of the Lattice into Blocks.- b) Which Levels to Retain for the Truncated Basis?.- 5.5.3 What Has Been Done and What Are the Difficulties Encountered?.- a) Quantum Properties at T = 0.- b) Quantum Properties at T ? 0.- c) Difficulties.- 5.5.4 Comparison Between Different Methods.- a) The Real-Space RG Methods for Classical Systems.- b) Finite-Size Scaling Methods.- 5.6 What to Do Next?.- 5.6.1 Improvement of the Method.- 5.6.2 Applications.- References.- 6. Application of the Real-Space Renormalization to Adsorbed Systems.- 6.1 Introduction.- 6.2 The Sublattice Method.- 6.3 The Prefacing Method and Introduction of Vacancies.- 6.4 The Potts Model.- 6.5 Further Applications of the Vacancy.- 6.6 Summary.- References.- 7. Position-Space Renormalization Group for Models of Linear Polymers, Branched Polymers, and Gels.- 7.1 Three Physical Systems.- 7.1.1 Linear Polymers.- 7.1.2 Branched Polymers.- 7.1.3 Gels.- 7.2 Three Mathematical Models.- 7.2.1 Percolation.- 7.2.2 Self-Avoiding Walks.- 7.2.3 Lattice Animals.- 7.3 Position-Space Renormalization Group Treatment.- 7.3.1 Percolation.- a) Basic Approach.- b) Extensions.- 7.3.2 Self-Avoiding Walks.- a) Basic Approach.- b) Extensions.- 7.3.3 Lattice Animals.- a) Basic Approach.- b) Extensions.- 7.4 Other Approaches.- 7.4.1 Percolation.- 7.4.2 Self-Avoiding Walks.- 7.4.3 Lattice Animals.- 7.5 Concluding Remarks and Outlook.- References.

Real-space renormalization

Wilson theory for 2-dimensional Ising spin systems

The asymptotic time behavior ($\ensuremath{\sim}c{t}^{\ensuremath{-}\frac{d}{2}}$, where $d$ is the dimensionality of the system) of the velocity autocorrelation function and of the kinetic parts of the correlation functions for the shear viscosity and the heat conductivity is derived on the basis of a local equilibrium assumption and the linearized Navier-Stokes equations. The coefficients $c$ are expressed in terms of the transport coefficients and thermodynamic quantities. The physical mechanism responsible for the long-time tail is indicated, and the connections between the present work and investigations based on molecular dynamics and on kinetic theory are discussed.

J. M. J. van Leeuwen

Papers

New method for the calculation of the pair correlation function. I

Asymptotic Time Behavior of Correlation Functions

Real-space renormalization

Wilson theory for 2-dimensional Ising spin systems

Asymptotic Time Behavior of Correlation Functions. I. Kinetic Terms