場(chǎng)的統(tǒng)計(jì)物理學(xué)

前言

　　Many scientists and non-scientists are familiar with fractals， abstract self-similarentities which resemble the shapes of clouds or mountain landscapes. Fewer arefamiliar with the concepts of scale-invariance and universality which underliethe ubiquity of these shapes. Such properties may emerge from the collectivebehavior of simple underlying constituents， and are studied through statisticalfield theories constructed easily on the basis of symmetries. This book demon-strates how such theories are formulated， and studied by innovative methodssuch as the renormalization group.　　The material covered is directly based on my lectures for the second semesterof a graduate course on statistical mechanics， which I have been teaching onand off at MIT since 1988. The first semester introduces the student to thebasic concepts and tools of statistical physics， and the corresponding materialis presented in a companion volume. The second semester deals with moreadvanced applications - mostly collective phenomena， phase transitions， andthe renormalization group， and familiarity with basic concepts is assumed. Theprimary audience is physics graduate students with a theoretical bent， but alsoincludes postdoctoral researchers and enterprising undergraduates. Since thematerial is comparatively new， there are fewer textbooks available in this area，although a few have started to appear in the last few years. Starting with theproblem of phase transitions， the book illustrates how appropriate statisticalfield theories can be constructed on the basis of symmetries. Perturbation the-ory， renormalization group， exact solutions， and other tools are then employedto demonstrate the emergence of scale invariance and universality. The finaltwo chapters deal with non-equilibrium dynamics of interfaces， and directedpaths in random media， closely related to the research of the author.　　An essential part of learning the material is doing problems; and in teachingthe course I developed a large number of problems （and solutions） that havebeen integrated into the text. Following each chapter there are two sets ofproblems： solutionsto the first set are included at the end of the book， and areintended to introduce additional topics and to reinforce technical tools. Thereare no solutions provided for a second set of problems which can be used inassignments.

內(nèi)容概要

Introduction、The Landau-Ginzburg Hamiltonian、Saddle point approximation, and mean-field theory、Continuous symmetry breaking and Goldstone modes、Discrete symmetry breaking and domain walls、Fluctuations、Scattering and fluctuations、Correlation functions and susceptibilities、Lower critical dimension、Comparison to experiments、Gaussian integrals等等。

書籍目錄

Preface1 Collective behavior, from particles to fields  1.1 Introduction  1.2 Phonons and elasticity  1.3 Phase transitions  1.4 Critical behavior  Problems2 Statistical fields  2.1 Introduction  2.2 The Landau-Ginzburg Hamiltonian  2.3 Saddle point approximation, and mean-field theory  2.4 Continuous symmetry breaking and Goldstone modes  2.5 Discrete symmetry breaking and domain walls  Problems3 Fluctuations  3.1 Scattering and fluctuations  3.2 Correlation functions and susceptibilities  3.3 Lower critical dimension  3.4 Comparison to experiments  3.5 Gaussian integrals  3.6 Fluctuation corrections to the saddle point  3.7 The Ginzburg criterion  Problems4 The scaling hypothesis  4.1 The homogeneity assumption  4.2 Divergence of the correlation length  4.3 Critical correlation functions and self-similarity  4.4 The renormalization group （conceptual）  4.5 The renormalization group （formal）  4.6 The Gaussian model （direct solution）  4.7 The Gaussian model （renormalization group）  Problems5 Perturbative renormalizafion group  5.1 Expectation values in the Gaussian model  5.2 Expectation values in perturbation theory  5.3 Diagrammatic representation of perturbation theory  5.4 Susceptibility  5.5 Perturbative RG （first order）  5.6 Perturbative RG （second order）  5.7 The e-expansion  5.8 Irrelevance of other interactions  5.9 Comments on the e-expansion  Problems6 Lattice systems  6.1 Models and methods  6.2 Transfer matrices  6.3 Position space RG in one dimension  6.4 The Niemeijer-van Leeuwen cumulant approximation  6.5 The Migdal-Kadanoff bond moving approximation  6.6 Monte Carlo simulations  Problems7 Series expansions  7.1 Low-temperature expansions  7.2 High-temperature expansions  7.3 Exact solution of the one-dimensional Ising model  7.4 Self-duality in the two-dimensional Ising model  7.5 Dual of the three-dimensional Ising model  7.6 Summing over phantom loops  7.7 Exact free energy of the square lattice Ising model  7.8 Critical behavior of the two-dimensional Ising model  Problems8 Beyond spin waves  8.1 The nonlinear tr model  8.2 Topological defects in the XY model  8.3 Renormalization group for the Coulomb gas  8.4 Two-dimensional solids  8.5 Two-dimensional melting  Problems9 Dissipative dynamics  9.1 Brownian motion of a particle  9.2 Equilibrium dynamics of a field  9.3 Dynamics of a conserved field  9.4 Generic scale invariance in equilibrium systems  9.5 Non-equilibrium dynamics of open systems  9.6 Dynamics of a growing surface10 Directed paths in random media  10.1 Introduction  10.2 High-T expansions for the random-bond Ising model  10.3 The one-dimensional chain  10.4 Directed paths and the transfer matrix  10.5 Moments of the correlation function  10.6 The probability distribution in two dimensions  10.7 Higher dimensions  10.8 Random signs  10.9 Other realizations of DPRM  10.10 Quantum interference of strongly localized electrons  10.11 The locator expansion and forward scattering paths  10.12 Magnetic field response  10.13 Unitary propagation  10.14 Unitary averagesSolutions to selected problems  Chapter 1  Chapter 2  Chapter 3  Chapter 4  Chapter 5  Chapter 6  Chapter 7  Chapter 8  Index

章節(jié)摘錄

　　The underlying microscopic Hamiltonian for the interactions of particles isusually quite complicated, making an ab initio particulate approach to the prob-lem intractable. However, there are many common features in the macroscopicbehavior of many such systems that can still be fruitfully studied by the meth-ods of statistical mechanics. Although the interactions between constituentsare quite specific at the microscopic scale, one may hope that averaging oversufficiently many particles leads to a,simpler description. （In the same sensethat the central limit theorem ensures that the sum over many random variableshas a simple Ganssian probability distribution function.） This expectation isindeed justified in many cases where the collective behavior of the interact-ing system becomes more simple at long wavelengths and long times. （Thisis sometimes called the hydrodynamic limit by analogy to the Navier-Stokesequations for a fluid of particles.） The averaged variables appropriate to theselength and time scales are no longer the discrete set of particle degrees offreedom, but slowly varying continuous fields. For example, the velocity fieldthat appears in the Navier-Stokes equations is quite distinct from the velocitiesof the individual particles in the fluid. Hence the appropriate method for thestudy of collective behavior in interacting systems is the statistical mechanicsof fields.

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