This article traces the development and study of phase transitions from late last century to the present day. We begin with a brief historical sketch and a description of the statistical mechanics of phase transitions. Particular attention is given to the modern era which began in 1944 with Onsager's celebrated solution of the two-dimensional Ising model. Points of development since Onsager, which are highlighted in this article, include the study of critical exponents, the scaling hypothesis, realisation of the universality of critical exponents and the recent renormalization group approach to critical phenomena. The basic idea of the renormalization group method, rather than the detailed application of the recipe to particular cases, is stressed and is discussed critically in some detail.
This article traces the development and study of phase transitions from late last century to the present day. We begin with a brief historical sketch and a description of the statistical mechanics of phase transitions. Particular attention is given to the modern era which began in 1944 with Onsager's celebrated solution of the two-dimensional Ising model. Points of development since Onsager, which are highlighted in this article, include the study of critical exponents, the scaling hypothesis, realisation of the universality of critical exponents and the recent renormalization group approach to critical phenomena. The basic idea of the renormalization group method, rather than the detailed application of the recipe to particular cases, is stressed and is discussed critically in some detail.
These lectures describe the physics of phase transitions and critical phenomena. After introducing the thermodynamics of phase transitions, these general considerations are made explicit by presenting statistical mechanical calculations on a lattice gas model in the mean-field approximation. Then the Landau theory of phase transitions is presented. The form of free energy in the Landau theory is motivated by an explicit mean field calculation for the Ising model. This is followed by a discussion of spatial correlations and the breakdown of the mean field theory due to increase of spatial fluctuations near the critical point. This calculation also brings out the role of dimension of the system for the critical behaviour. The basic ideas of Renormalization Group (RG) are then introduced and illustrated by a simple RG calculation for the one-dimensional Ising model. Real-space RG calculations are then applied to the two-dimensional Ising model on the triangular lattice, which, in fact, brings forth several key features of the RG technique. A discussion of more general aspects of the RG formalism, such as scaling for spatial correlation functions, relevant and irrelevant couplings, universality, upper critical dimension, etc. are introduced.
Abstract The successful calculation of critical exponents for continuous phase transitions is one of the main achievements of theoretical physics over the last quarter-century. This was achieved through the use of scaling and field-theoretic techniques which have since become standard equipment in many areas of physics, especially quantum field theory. This book provides a thorough introduction to these techniques. Continuous phase transitions are introduced, then the necessary statistical mechanics is summarized, followed by standard models, some exact solutions and techniques for numerical simulations. The real-space renormalization group and mean-field theory are then explained and illustrated. The final chapters cover the Landau-Ginzburg model, from physical motivation, through diagrammatic perturbation theory and renormalization to the renormalization group and the calculation of critical exponents above and below the critical temperature.
This text provides a thoroughly modern graduate-level introduction to the theory of critical behaviour. Beginning with a brief review of phase transitions in simple systems and of mean field theory, the text then goes on to introduce the core ideas of the renormalization group. Following chapters cover phase diagrams, fixed points, cross-over behaviour, finite-size scaling, perturbative renormalization methods, low-dimensional systems, surface critical behaviour, random systems, percolation, polymer statistics, critical dynamics and conformal symmetry. The book closes with an appendix on Gaussian integration, a selected bibliography, and a detailed index. Many problems are included. The emphasis throughout is on providing an elementary and intuitive approach. In particular, the perturbative method introduced leads, among other applications, to a simple derivation of the epsilon expansion in which all the actual calculations (at least to lowest order) reduce to simple counting, avoiding the need for Feynman diagrams.
The successful calculation of critical exponents for continuous phase transitions is one of the main achievements of theoretical physics over the last quarter-century. This was achieved through the use of scaling and field-theoretic techniques which have since become standard equipment in many areas of physics, especially quantum field theory. This book provides a thorough introduction to these techniques. Continuous phase transitions are introduced, then the necessary statistical mechanics is summarized, followed by standard models, some exact solutions and techniques for numerical simulation
The sophistication of modern tools used in the study of statistical mechanics and field theory is often an obstacle to the easy understanding of new important current results reported in journals. The main purpose of this book is to introduce the reader to the methods of the fluctuation (field) theory of phase transitions and critical phenomena so as to provide a good source for research. The introductory contents are concerned with ideas of description, thermodynamic stability theory related to phase transitions, major experimental facts, basic models and their relationships. Special attention is paid to the mean field approximation and to the Landau expansion for simple and complex models of critical and multicritical phenomena. An instructive representation of the modern perturbation theory and the method of the renormalization group is developed for field models of phase transitions. The essential influence of the fluctuations on the critical behaviour is established together with the theory of correlation functions, Gaussian approximation, the Ginzburg criterion, [symbol]- and 1/n- expansions as practical realizations of the renormalization group ideas. Applications of the theory to concrete aspects of condensed matter physics are considered: quantum effects, Bose condensation, crystal anisotropy, superconductors and liquid crystals, effects of disorder of type randomly distributed quenched impurities and random fields. This volume can be used as an advanced University course book for students with a basic knowledge of statistical physics and quantum mechanics. It could be considered as a complementary text to a standard University course on statistical physics
This essay corresponds to the content of three lectures about statistical physics delivered to the audience of the 2014 section of the R. A. Salmeron School of Physics, at the UnB. Our starting point was very simple statistical models (lattice gas, spin-1/2 ferromagnet), used as illustrations of the competencies and methods in statistical physics. Thus we introduce the Gibbs ensembles, defining a connection with thermodynamics and discussing the role played by fluctuations and large numbers. We present phenomenological aspects of phase transitions and critical phenomena in simple fluids and in uniaxial ferromagnets, emphasizing the universal character of the critical exponents. We describe the phenomenological van der Waals and Curie-Weiss theories and the Landau expansion, which are present-day relevant methods, despite the fact that such theories give rise to critical exponents in disagreement with experiments. We present then the paradigmatic Ising model, which points us to a way to overcome the phenomenological results. A brief presentation of the scale phenomenological methods and the contemporaneous renormalization group are considered at the end of these lectures.