The examination of phase transitions and critical phenomena has dominated statistical physics for the latter half of this century--there is a great theoretical challenge in solving special statistical mechanical models. Additionally, beautiful experimental results have elucidated the singularities (critical behavior) that occur in phase transitions. Although a few spectacular successes in the exact solution of simple models have occurred, interesting systems have mostly proven to be intractable from the analytic perspective. Because many physically disparate systems show identical critical behavior, identifying and understanding general principles of universality and their relationship to microscopic symmetries are of fundamental importance
The examination of phase transitions and critical phenomena has dominated statistical physics for the latter half of this century--there is a great theoretical challenge in solving special statistical mechanical models. Additionally, beautiful experimental results have elucidated the singularities (critical behavior) that occur in phase transitions. Although a few spectacular successes in the exact solution of simple models have occurred, interesting systems have mostly proven to be intractable from the analytic perspective. Because many physically disparate systems show identical critical behavior, identifying and understanding general principles of universality and their relationship to microscopic symmetries are of fundamental importance
Abstract We have now seen how powerful the tools of statistical mechanics can be in the study of complex systems in thermal equilibrium and how they can be used to determine exactly the behaviour of some simplified, but nevertheless interesting, models of phase transitions. However, these exact solutions are sometimes lengthy and difficult and there is no guarantee that any particular model will yield exact results. For example, we were able to solve the two-dimensional, but not the three-dimensional, Ising model, and that only in zero magnetic field. So, it is natural to consider whether we can use a computer to simulate these systems. In this chapter we shall see that this is indeed possible; a great deal of our understanding of critical phenomena is based wholly or partly on the results of such numerical work. However, as we shall also see, such simulations are not entirely straightforward and a number of tricks are needed to obtain accurate results. This is partly a consequence of the very physics that makes the phase transitions interesting to start with. One should probably think of numerical simulation as a whole subject, almost an art, of its own; we shall have time only to scratch at the surface and discuss a few of the most important ideas.
Abstract 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.
Abstract In this chapter, the singular changes accompanying a phase transition requires the existence of macroscopic correlations. As a consequence, phase transitions mark a stark departure from the noninteracting and linear responding systems we have considered thus far. Understanding phase transi- tions requires that we study strongly interacting models, pushed far into regimes of nonlinear response. This complication renders detailed studies largely intractable analytically. However, some simplifications exist that allow for a general theory of phase transitions to be developed. These are the notions of universality that accompany phase transition phenomenology, and the ability to classify phase transi- tions simply using ideas of spontaneous symmetry breaking. The former allows us to postulate simple models bereft for molecular detail and yet still provide quantitative predictions for observable behavior. The latter allows us to predict what phases can exist, and understand why phase transitions occur.
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
Introduction In this chapter we provide a comprehensive and unified treatment of the physical concepts and quantities relevant to phase transitions and critical phenomena. Some of these concepts and quantities have already been discussed in the particular contexts of magnetism (Chapter 7), or liquid crystals (Chapter 9, We conclude the chapter with a brief reference to the application of renormalisation group ideas to critical phenomena. The resulting reconciliation of the predictions of classical theories with experiments and exact or quasi-exact solutions of model systems is a major triumph of twentieth-century physics. Phases and phase transitions Thermodynamics habitually deals with equilibrium systems that are uniform on the macroscopic scale – i.e., where the structure, density and composition are the same throughout the bulk of the system, which is many times greater than a single atom or molecule. Because the system is in equilibrium, intensive thermodynamic variables such as the temperature or chemical potential are also spatially uniform. Such a system is said to consist of a single phase – it is a one-phase system. Phases are what used to be called the states of matter. Solids, liquids, gases and plasmas are four commonly occurring phases – they correspond to the four elements of Aristotelian physics: earth, water, air and fire, respectively. There are, however, many other, less easily identified phases, some of which we have already encountered in this book, such as the ferromagnetic, antiferromagnetic, nematic or superconducting phases. On varying the temperature, pressure and chemical potential(s) of a system, properties such as its structure, density, composition, compressibility, specific heat, etc., in general change smoothly. Yet under certain conditions the changes may be abrupt – some of these properties may exhibit discontinuities, or even divergences, at a phase transition . Phase transitions are among the most spectacular phenomena of nature. Consider, for instance, the phase sequence of water on heating: from ice, an opaque, slippery, rock-like solid; to colourless, incompressible, liquid water, which nevertheless easily flows under gravity; to water vapour, an invisible and very low-density (greenhouse!) gas, hence of ill-defined volume.
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.