Here, in brief outline, is the run of the chapter. Section 16.1 presents two experiments to introduce the topic of “critical phenomena,” as that phrase is used in physics. The next section illustrates the mathematical behavior of certain physical quantities near a critical point. Section 16.3 constructs a theoretical model: the Ising model. The following two sections develop methods for extracting predictions from the model. The last three sections draw distinctions, outline some general results, and collect the chapter's essentials. Achieving a sound theoretical understanding of critical phenomena was the major triumph of thermal physics in the second half of the twentieth century. Thus the topic serves admirably as the culmination of the entire book. Experiments Liquid–vapor We begin with an experiment that you may see as a lecture demonstration, performed either with carbon dioxide or with Freon, a liquid once used in refrigerators. A sealed vertical cylinder contains a carefully measured amount of CO 2 (say) under high pressure. To begin with, the system is in thermal equilibrium at room temperature, as sketched in part (a) of figure 16.1. A distinct meniscus separates the clear liquid phase from the vapor phase above it. Now heat the system with a hair dryer. The meniscus rises (because the liquid expands), and gentle boiling commences. Next the meniscus becomes diffuse, and then it disappears. The CO 2 has become spatially homogeneous in density, as illustrated in figure 16.1 (b).
Here, in brief outline, is the run of the chapter. Section 16.1 presents two experiments to introduce the topic of “critical phenomena,” as that phrase is used in physics. The next section illustrates the mathematical behavior of certain physical quantities near a critical point. Section 16.3 constructs a theoretical model: the Ising model. The following two sections develop methods for extracting predictions from the model. The last three sections draw distinctions, outline some general results, and collect the chapter's essentials. Achieving a sound theoretical understanding of critical phenomena was the major triumph of thermal physics in the second half of the twentieth century. Thus the topic serves admirably as the culmination of the entire book. Experiments Liquid–vapor We begin with an experiment that you may see as a lecture demonstration, performed either with carbon dioxide or with Freon, a liquid once used in refrigerators. A sealed vertical cylinder contains a carefully measured amount of CO 2 (say) under high pressure. To begin with, the system is in thermal equilibrium at room temperature, as sketched in part (a) of figure 16.1. A distinct meniscus separates the clear liquid phase from the vapor phase above it. Now heat the system with a hair dryer. The meniscus rises (because the liquid expands), and gentle boiling commences. Next the meniscus becomes diffuse, and then it disappears. The CO 2 has become spatially homogeneous in density, as illustrated in figure 16.1 (b).
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.
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.
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.
Introduction We all observe phase transitions in our daily lives, with hardly a second thought. When we boil water for a cup of tea, we observe that the water is quiescent until it reaches a certain temperature (100 ℃), and then bubbles appear vigorously until all the water has turned to steam. Or after an overnight snowfall, we have watched the snow melt away when temperatures rise during the day. The more adventurous among us may have heated an iron magnet above about 760 ℃ and noted the disappearance of its magnetism. Familiar and ubiquitous as these and many related phenomena are, a little reflection shows that they are quite mysterious and not easy to understand: indeed, the outlines of a theory did not emerge until the middle of the twentieth century, and, although much has been understood since then, active research continues. Ice and water both consist of molecules of H 2 O, and we can look up all the physical parameters of a single molecule, and of the interaction between a pair of molecules, in standard reference texts. However, no detailed study of this information prepares us for the dramatic change that occurs at 0 ℃. Below 0 ℃, the H 2 O molecules of ice are arranged in a regular crystalline lattice, and each H 2 O molecule hardly strays from its own lattice site. Above 0 ℃, we obtain liquid water, in which all the molecules are moving freely throughout the liquid container at high speeds. Why do 10 23 H 2 O molecules cooperatively “decide” to become mobile at a certain temperature, leading to the phase transition from ice to water?
Abstract Chapter 16 explores the statistical mechanics of discrete lattice models and their use in describing critical phenomena. The chapter begins with a review of phase diagrams, coexistence curves, crititical points, and the behavior of certain thermodynamic quantities in the neighborhood of a critical point. The notion of universality classes is introduced. To set up the class of lattice models, magnetic systems are described; simplifications are made to these models to yield the so-called Ising model. Solution of this model is demonstrated within the mean-field theory approximation, which is shown to yield a qualitative description of critical behavior. Exact solutions are derived in one and two dimensions, and predictions of critical exponents from them are obtained. Following this, the renormalization group is introduced, and approaches for understanding critical behavior and universality from renormalization group transformations are discussed. The chapter concludes with a discussion of the use of discrete models in areas ranging from molecular crystals to tissue morphogenesis
In the preceding chapter, we have introduced several levels for the mathematical description of stochastic dynamics. We now use the kinetic Ising models introduced in Section 2.3.3 to formulate the dynamic scaling hypothesis which appropriately generalizes the homogeneity property of the static correlation function in the vicinity of a critical point, as established in Chapter 1. The dynamic critical exponent z is defined to characterize both the critical dispersion and the basic phenomenon of critical slowing-down. As a next step, and building on the results of Section 2.4, a continuum effective theory for the mesoscopic order parameter density, basically the dynamical analog to the Ginzburg–Landau approach, is constructed in terms of a non-linear Langevin equation. The distinction between dissipative and diffusive dynamics for the purely relaxational kinetics of either a non-conserved or conserved order parameter field, respectively, defines the universality classes A and B. Following the analysis of these models in the Gaussian approximation, they also serve to outline the construction of a dynamical perturbation theory for non-linear stochastic differential equations through direct iteration. In general, however, the order parameter alone does not suffice to fully capture the critical dynamics near a second-order phase transition. Additional hydrodynamic modes originating from conservation laws need to be accounted for as well. The simplest such situation is entailed in the relaxational models C and D, which encompass the static coupling of the order parameter to the energy density. Further scenarios emerge through reversible non-linear mode couplings in the Langevin equations of motion.