Physics

This book begins where elementary books and courses leave off and covers the advances made in statistical mechanics in the past fifty years. The book is divided into three parts. The first part is on general theory which includes a summary of the basic principles of statistical mechanics; a presentation of the physical phenomena covered and the models used to discuss them; theorems on the existence and uniqueness of partition functions; theorems on order; and critical phenomena and scaling theory. The second part is on series and numerical methods which includes derivations of the Mayer and Ree–Hoover expansions of the low density virial equation of state; Groeneveld's theorems; the application to hard spheres and discs; a summary of numerical studies of systems at high density; and the use of high temperature series expansions to estimate critical exponents for magnets. The third part covers exactly solvable models which includes a detailed presentation of the Pfaffian methods of computing the Ising partition function, magnetization, correlation functions, and susceptibility; the star-triangle (Yang–Baxter equation); functional equations and the free energy for the eight-vertex model; and the hard hexagon and chiral Potts models. All needed mathematics is developed in detail and many open questions are discussed. The goal is to guide the reader to the current forefront of research.

The anomaly, which forms the central part of this book, is the failure of classical symmetry to survive the process of quantization and regularization. The study of anomalies is the key to a deeper understanding of quantum field theory and has played an increasingly important role in the theory over the past twenty years. This book presents all the different aspects of the study of anomalies in an accessible and self-contained way. Much emphasis is now being placed on the formulation of the theory using the mathematical ideas of differential geometry and topology. This approach is followed here, and the derivations and calculations are given explicitly. Topics discussed include the relevant ideas from differential geometry and topology and the application of these paths (path integrals, differential forms, homotopy operators, etc.) to the study of anomalies. Chapters are devoted to abelian and nonabelian anomalies, consistent and covariant anomalies, and gravitational anomalies.

This book introduces the full range of activity in the rapidly growing field of nonlinear dynamics. Using a step-by-step introduction to dynamics and geometry in state space as the central focus of understanding nonlinear dynamics, this book includes a thorough treatment of both differential equation models and iterated map models (including a detailed derivation of the famous Feigenbaum numbers). It includes the increasingly important field of pattern formation and a survey of the controversial question of quantum chaos. Important tools such as Lyapunov exponents, fractal dimensions, and correlation dimensions are treated in detail. Several appendices provide a detailed derivation of the Lorenz model from the Navier-Stokes equation, a summary of bifurcation theory, and some simple computer programs to study nonlinear dynamics. Each chapter includes an extensive, annotated bibliography.

This book illustrates the interplay between distinct domains of mathematics. Firstly, this book provides an introduction to hyperbolic geometry, based on the Lorentz group. The Lorentz group plays, in relativistic space-time, a role analogue to the rotations in Euclidean space. The hyperbolic geometry is the geometry of the unit pseudo-sphere. The boundary of the hyperbolic space is defined as the set of light rays. Special attention is given to the geodesic and horocyclic flows. Hyperbolic geometry is presented via special relativity to benefit from physical intuition. Secondly, this book introduces basic notions of stochastic analysis: the Wiener process, Itô's stochastic integral, and calculus. This introduction allows study in linear stochastic differential equations on groups of matrices. In this way the spherical and hyperbolic Brownian motions, diffusions on the stable leaves, and the relativistic diffusion are constructed. Thirdly, quotients of the hyperbolic space under a discrete group of isometries are introduced. In this framework some elements of hyperbolic dynamics are presented, as the ergodicity of the geodesic and horocyclic flows. This book culminates with an analysis of the chaotic behaviour of the geodesic flow, performed using stochastic analysis methods. This main result is known as Sinai's central limit theorem.

Quantum phase transitions describe the violent rearrangement of electrons or atoms as they evolve from well defined excitations in one phase to a completely different set of excitations in another. The book chapters give insights into how a coherent metallic or superconducting state can be driven into an incoherent insulating state by increasing disorder, magnetic field, carrier concentration and inter-electron interactions. They illustrate the primary methods employed to develop a multi-faceted theory of many interacting particle systems. They describe how recent experiments probing the microscopic structure, transport, charge and spin dynamics have yielded guiding insights. What sets this book apart is this strong dialog between experiment and theory, which reveals the recent progress and emergent opportunities to solve some major problems in many body physics. The pedagogical style of the chapters has been set for graduate students starting in this dynamic field.

The investigation of discrete symmetries is a fascinating subject which has been central to the agenda of physics research for fifty years, and has been the target of many experiments, ongoing and in preparation, all over the world. This book approaches the subject from a somewhat less traditional angle: it puts more emphasis on the experimental aspects of the field, trying to provide a wider picture than usual and to convey the intellectual challenge of experimental physics. The book includes the related connection to phenomenology, a purpose for which the precision experiments in this field — often rather elegant and requiring a good amount of ingenuity — are very well suited. The book discusses discrete symmetries (parity, charge conjugation, time reversal, and of course CP symmetry) in microscopic (atomic, nuclear, and particle) physics, and includes a detailed description of some key or representative experiments. The book discusses their principles and challenges more than the historical development. The main past achievements and the most recent developments are both included.

Most of the solid materials we use in everyday life, from plastics to cosmetic gels exist in a non-crystalline, amorphous form: they are glasses. Yet, we are still seeking a fundamental explanation as to what glasses really are, why they form, and what their properties are. This book surveys the most recent theoretical and experimental research dealing with the physics of glassy and disordered materials, from molecular fluids to colloidal glasses, granular media and foams. Chapters present broad and original perspectives on one of the deepest mystery of condensed matter physics, with a particular emphasis on the key role played by dynamic heterogeneities to understand from a unified viewpoint phenomena occurring in scores of disordered materials. The book covers both fundamental aspects, and reviews extensively several recent theoretical developments in the field of the glass transition that have changed our view of the glass transition. It also provides an up-to-date perspective on new experimental tools that have been developed to study with unprecedented resolution the structural relaxation of systems like molecular and polymer liquids, colloidal glasses, foams and granular materials. It also confronts, discusses, compares, and challenges the different perspectives actively promoted by different research groups and communities in a very active area of condensed-matter and statistical physics.

Phase transitions and critical phenomena have consistently been among the principal subjects of active studies in statistical physics. The simple act of transforming one state of matter or phase into another, for instance by changing the temperature, has always captivated the curious mind. This book provides an introductory account on the theory of phase transitions and critical phenomena, a subject now recognized to be indispensable for students and researchers from many fields of physics and related disciplines. The first five chapters are very basic and quintessential, and cover standard topics such as mean-field theories, the renormalization group and scaling, universality, and statistical field theory methods. The remaining chapters develop more advanced concepts, including conformal field theory, the Kosterlitz-Thouless transition, the effects of randomness, percolation, exactly solvable models, series expansions, duality transformations, and numerical techniques. Moreover, a comprehensive series of appendices expand and clarify several issues not developed in the main text. The important role played by symmetry and topology in understanding the competition between phases and the resulting emergent collective behaviour, giving rise to rigidity and soft elementary excitations, is stressed throughout the book. Serious attempts have been directed toward a self-contained modular approach so that the reader does not have to refer to other sources for supplementary information. Accordingly, most of the concepts and calculations are described in detail, sometimes with additional/auxiliary descriptions given in appendices and exercises. The latter are presented as the topics develop with solutions found at the end of the book, thus giving the text a self-learning character.

This book is based on the premise that the entropy concept, a fundamental element of probability theory as logic governs all of the thermal physics, both equilibrium and nonequilibrium. The variational algorithm of J. Willard Gibbs — dating from the 19th century and extended considerably over the following 100 years — is shown to be the governing feature over the entire range of thermal phenomena, such that only the nature of the macroscopic constraints changes. Beginning with a short history of the development of the entropy concept by Rudolph Clausius and his predecessors, along with the formalization of classical thermodynamics by Gibbs, the first part of the book describes the quest to uncover the meaning of thermodynamic entropy, which leads to its relationship probability and information as first envisioned by Ludwig Boltzmann. Recognition of entropy first of all as a fundamental element of probability theory in mid-20th Century led to deep insights into both statistical mechanics and thermodynamics, the details of which are presented here in several chapters. The later chapters extend these ideas to nonequilibrium statistical mechanics in an unambiguous manner, thereby exhibiting the overall unifying role of the entropy.

This book focuses attention on two aspects of equations of motion in general relativity: the motion of extended bodies(stars) and the motion of small black holes. The objective is to offer a guide to prospective researchers into these areas of general relativity and to point out open questions and topics that are ripe for further development. It is over forty years since a text on this subject was published and in that time the research area of equations of motion in general relativity has undergone extraordinary development stimulated by the discovery of the binary neutron star PSR 1913+16 in 1974, which was the first isolated gravitating system found in which general relativity plays a fundamental role in describing theoretically its evolution, and more recently by the advent of kilometre size interferometric gravitational wave detectors which are expected to detect gravitational waves produced by coalescing binary neutron stars. Included in the book are novel topics in general relativistic celestial mechanics: choreographic configurations and the relativistic motion of small black holes.