In this chapter, we investigate the concept of scaling, in particular the scaling hypothesis that has proved so important in the development of the modern theory of critical phenomena. Scaling refers to the way in which properties such as the susceptibility behave in the critical region as they approach infinity (or zero). In this discussion, it is important to be specific about the thermodynamic field variables used to represent the system as well as the direction in which the critical point is approached. A useful schematic in this regard is shown in figure 9.1, which shows phase coexistence diagrams for both an Ising-type magnet and a typical fluid. The fields for the magnet are (H, T ) while for the fluid they are (μ, T ). We note immediately from this figure that the magnet’s coexistence curve is coincident with the H = 0 axis and symmetric about the M = 0 axis. However, these features do not reflect in the corresponding diagrams for the fluid, with important implications for a scaling analysis in this system, which we take up later in this chapter. We illustrate the basic critical scaling laws using the Ising (magnet) system; however, the fact that similar scaling laws also apply to fluids is an important consequence of critical-point universality, a topic we have often alluded to in this text and which is discussed in more detail in chapter 10 dealing with the renormalization-group method.
In this chapter, we investigate the concept of scaling, in particular the scaling hypothesis that has proved so important in the development of the modern theory of critical phenomena. Scaling refers to the way in which properties such as the susceptibility behave in the critical region as they approach infinity (or zero). In this discussion, it is important to be specific about the thermodynamic field variables used to represent the system as well as the direction in which the critical point is approached. A useful schematic in this regard is shown in figure 9.1, which shows phase coexistence diagrams for both an Ising-type magnet and a typical fluid. The fields for the magnet are (H, T ) while for the fluid they are (μ, T ). We note immediately from this figure that the magnet’s coexistence curve is coincident with the H = 0 axis and symmetric about the M = 0 axis. However, these features do not reflect in the corresponding diagrams for the fluid, with important implications for a scaling analysis in this system, which we take up later in this chapter. We illustrate the basic critical scaling laws using the Ising (magnet) system; however, the fact that similar scaling laws also apply to fluids is an important consequence of critical-point universality, a topic we have often alluded to in this text and which is discussed in more detail in chapter 10 dealing with the renormalization-group method.
Overview As discussed in Chapter 16 , when we approach the critical temperature of a phase coexistence line, several fascinating things start to happen which together have come to be known as “critical phenomena.” It is observed that the spatial correlation between relevant order parameters like density, magnetization, etc. becomes infinitely large. This is reflected in the divergence of inverse susceptibility of the respective order parameters. Landau’s theory explains this in terms of the flattening of the free energy surface but fails to unravel many of the other mysteries of critical phenomena that have kept scientists engaged now for about a century. Critical phenomena are characterized by certain singular properties, like the universality of the critical exponents. Several aspects of the critical phenomena are discussed in this chapter with an aim to maintain continuity with earlier chapters. A simple theory (like that of Landau) cannot explain many of the observed features and this problem remained unsolved for a long time until K. G. Wilson could develop a sophisticated theory that explained almost everything. At the core of Wilson’s method is the renormalization group technique. Physically, this technique employs the expected invariance of the Hamiltonian due to a transformation that removes local interactions and replaces them by effective longer range interactions. Because of the existence of long range correlations, the transformation is expected to converge to a fixed point. The procedure is quite mathematical and is the only robust method to obtain the critical exponents. Here we discuss the method briefly, as an introduction to the subject.
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
Condensed media show a large variety of critical phenomena, ranging from critical opalescence at the end point of the liquid–gas coexistence curve, the Curie transition of ferromagnetic materials, to the superfluid transition of helium, the behaviour of solutions of polymers, the conductivity of random media, … To this list should be added systems with local symmetries, such as those suggested by particle physics in order to understand quark confinement. Surprisingly all these phenomena can be classified into a few universality classes, characterized by specific large distance behaviour with the same critical exponents. In this chapter, we sketch the methods and ideas of the renormalization procedure. We illustrate the concepts with simple approximations in the language of classical spin models in the first section, treating in more detail the XY -model as an example in the second section. Scaling laws. Real space renormalization Homogeneity and scale invariance The discussion given in previous chapters suggests that, close to a continuous transition, critical systems exhibit universal properties. Correlations at large distance are not sensitive to the details of microscopic interactions. Their behaviour is described by a specific dimensional analysis governed by some essential characteristics of the system, such as the dimension of space, the nature of the order parameter and the underlying symmetries. The mean field approximation gave a first idea of a simple critical behaviour. Fluctuations have only a quantitative effect in dimension greater than the upper critical one.
An introductory review of the central ideas in the modern theory of dynamic critical phenomena is followed by a more detailed account of recent developments in the field. The concepts of the conventional theory, mode-coupling, scaling, universality, and the renormalization group are introduced and are illustrated in the context of a simple example---the phase separation of a symmetric binary fluid. The renormalization group is then developed in some detail, and applied to a variety of systems. The main dynamic universality classes are identified and characterized. It is found that the mode-coupling and renormalization group theories successfully explain available experimental data at the critical point of pure fluids, and binary mixtures, and at many magnetic phase transitions, but that a number of discrepancies exist with data at the superfluid transition of 4He.
The renormalization-group (RG) method discussed in this chapter has assumed a pivotal role in the modern theory of critical phenomena. It attempts to relate the partition function of a given system to that of a “similar system” with decreased degrees of freedom through a process referred to as renormalization. Exactly how these degrees of freedom are removed from the system, what we mean by a “similar” system, and how successive systems are coupled to one another are essentially the questions we take up in the introductory treatment given in this chapter. The RG method is a topic with large scope and found widely disseminated in an extensive physics literature on the topic; however, it is seldom found in engineering journals. Our purpose here is to try and make sense of some basic ideas with the RG approach so that it is more accessible to this wider community. For this we often rely upon some prior exposes of the subject in more specialized settings [1, 2, 3, 5]. In its complete sense, the RG method has only been made to work, at least analytically, for a few simple statistical-mechanical models. But aside from these numerical results, many important and quite general insights about critical phenomena can be developed from studying this approach to the problem, especially the central role played by length scale as a factor in describing the phenomenology. These ideas have significantly enhanced our understanding of ideas like scale invariance, universality classes, relevant scaling fields (as opposed to irrelevant ones), Hamiltonian renormalization, and so on; these and related concepts lie at the center of modern discourse on the subject. The essential concepts of the approach can be well illustrated using the Ising system since, with this model, lattice spins are fixed in space, which makes the analytical work quite transparent. This approach, called real space renormalization, is the RG method studied in this chapter.
Based on the foundations of thermodynamics and the equilibrium conditions for the coexistence of two phases in a magnetic Ising-like system, we show, first, that there is a critical point where the isothermal susceptibility diverges and the specific heat may remain finite, and second, that near the critical point the entropy of the system, and therefore all free energies, do obey scaling. Although we limit ourselves to such a system, we elaborate about the possibilities of finding universality, as well as the precise values of the critical exponents using thermodynamics only.