Phase Transitions

The following sections are included:

• Preface to the Second Edition

• Preface to the First Edition

• Contents

In discussing phase transitions, the first thing that we have to do is to define a phase. This is a concept from thermodynamics and statistical mechanics, where a phase is defined as a homogeneous system. As a simple example, let us consider instant coffee. This consists of coffee powder dissolved in water, and after stirring it we have a homogeneous mixture, i.e., a single phase. If we add to a cup of coffee a spoonful of sugar and stir it well, we still have a single phase — sweet coffee. However, if we add ten spoonfuls of sugar, then the contents of the cup will no longer be homogeneous, but rather a mixture of two homogeneous systems or phases, sweet liquid coffee on top and coffee-flavored wet sugar at the bottom…

The following sections are included:

• Classification of Phase Transitions

• Appearance of a Second Order Phase Transition

• Critical Phenomena

• Correlations

• Speed of Equilibration Near the Gas–Liquid Critical Point

• The Mechanism of Slowing Down Near the Critical Points

• Conclusion

We now consider a microscopic approach to phase transitions, in contrast to the phenomenological approach used in the previous chapter.In this approach, following Gibbs, we start with the interaction between particles. The first step is then to calculate the mechanical energy of the system En in each state n of all the particles, a problem which in general is far from trivial for a system of 1023 particles. In the framework of classical and quantum mechanics we must then calculate the partition function…

As we mentioned in Chapter 2, the simplest way to allow for interactions between particles is the mean field approximation, in which one assumes that the environment of each particle corresponds to the average state of the system and determines this average state self-consistently…

Scaling is a very general and well-known method for describing the response of a system to a disturbance. For physical systems, it is most readily studied in terms of dimensional analysis and the construction of dimensionless parameters, and so we consider this topic first, and start with a few simple examples…

The aim of renormalization group theory (RG) is to find the critical indices, including the indices x and y, which were still unknown after using the scaling hypothesis in the previous chapter. RG is a group of transformations (or more rigorously a semi-group, since in general there is no inverse transformation) from a site lattice with lattice constant unity and the Hamiltonian H to a block lattice with lattice constant L and the Hamiltonian H' without changing the form of the partition function,…

Our treatment of phase transitions so far has considered only phenomenological properties and model systems, without any microscopic analysis of real physical systems. In this chapter, we extend our treatment by considering many-body quantum systems, in which the two best-known phase transitions are superfluidity and superconductivity. Since these phenomena are associated with the symmetry of the wave function and quantum statistical physics, we start with a brief review of these two topics.

In the previous chapters, we have seen that different models exhibiting phase transitions, such as the mean field model and the Ising model in two and three dimensions, have different sets of critical indices associated with the transition. Phase transitions are usually classified according to their sets of critical indices, such that systems with phase transitions having the same set of critical indices belong to the same universality class. This name is used because, as we show in this chapter, there are often a range of different systems belonging to the same universality class.

The following sections are included:

• Multiple Solutions of the Law of Mass Action

• Phase Separation in Reactive Binary Mixtures

• Thermodynamic Analysis of Reactive Ternary Mixtures

• Kinetics of Phase Separation

• Change of Critical Parameters Due to a Chemical Reaction

• Modification of the Critical Indices

• Conclusion

Along the phase transitions in systems at equilibrium, there are many examples of particles undergoing phase transitions, which are carried along by the mean flow that passes through the region under study. The latter appears in different systems, such as liquids in the presence of flow [117], chemical reactions with diffusion [118], chemical waves [119], some optical systems [120] dendritic growth [121] and even population biology [122]. An additional example, which we consider in detail, is the formation of an ordered phase in the convective Ginzburg-Landau equation, which, in particular, describes the motion of vortices in superconductors [123]. This process has been observed in the superconducting films subjected to the simultaneous action of both magnetic field and bias current [124]. The vortex phase diagram in these crystals includes two distinct vortex solid phases, which are identified as ordered (B < BC) and disordered (B > BC) vortex phases, while the latter is caused by strong vortex interaction with spatial defects (pinning centers). For constant magnetic field B, the control parameter is the bias current of density J. This current will drag the vortices along the sample, helping them to destroy the disordered phase by assisting the vortices to climb the pinning barriers. The transformation of the disorder vortex phase into an ordered one in the presence of a bias current was directly observed by magneto-optical measurements of high temporal resolution, where a sharp interphase between the ordered and disordered vortex phases has been detected [124]…

Most of the systems that we have considered so far in this book have been ordered ones, with definite well-defined spatial arrangements and interactions. In this chapter, we consider the effects of disorder or randomness. We consider first a simple example of a disordered or random system, namely that of a lattice in which there is a given probability that a bond exists between a pair of adjacent sites. Although the presence or absence of a bond is a purely geometrical property, with no thermal effects involved, this system is found to have many features similar to those of a phase transition, such as a critical fraction of bonds for percolation to occur, critical indices, and the applicability of renormalization group techniques, and so we consider it first. In fact, it is formally possible (although not very instructive) to map the bond percolation problem onto the Potts model [16] mentioned in Chapter 8. After this we consider the Ising model with random interactions or sites removed at random, and spin glasses, in which there are competing interactions. Finally, we consider small world systems, in which there are a limited number of long-range interactions between randomly chosen sites…

Many-particle systems can be divided into two classes. On the one hand, one can start with a superficially simple system such as an atom, and then consider more and more complex systems as one probes further and further into its structure, electrons and a nucleus, the nucleus containing protons and neutrons, these being composed of quarks, and so on. On the other hand, one can consider large systems that are inherently complex because they contain many objects, with very many degrees of freedom and a large number of connections between them. Just as in the previous chapter, these systems can be described in terms of networks of sites and bonds. Such a network can refer not only to the atoms or molecules in a solid or a fluid but also to groups of people and their interactions, competing companies on the stock market, the sites and links of the Internet, and numerous other systems. In our analysis we consider those properties which are independent of the type of objects located at the sites and connected by the bonds, but rather depend on the topological properties of networks. Such properties are very appropriate to phase transitions, where as we saw in Chapter 8 the universality classes do not depend on the detailed properties of the objects being associated with the infinite correlation length…

The following sections are included:

• Bibliography

• Index