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CRC Press eBooks

How Phase Transitions Occur in Practice

Nigel D. Goldenfeld

Published Mar 8, 2018• 1 citation
Publisher

Abstract

This chapter explores the basic notions of phase transitions, and the manner in which non-analytic thermodynamic behaviour may occur in principle. It presents some concrete calculations of phase transition behaviour in magnetic systems. The chapter is concerned with the transfer matrix method, first introduced by Hendrik Kramers, which reduces the problem of calculating the partition function to the problem of finding the eigenvalues of a certain matrix. It describes the low temperature expansion and Weiss’ mean field theory for magnets. Mean field theory is the simplest treatment of an interacting statistical mechanical system, apart from the approximation of ignoring the interactions all together. The chapter provides a simple, non-systematic derivation of mean field theory for the Ising model. This mean field theory can be easily improved by applying it to clusters of spins rather than individual spins.

DOI
10.1201/9780429493492-3

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This chapter explores the basic notions of phase transitions, and the manner in which non-analytic thermodynamic behaviour may occur in principle. It presents some concrete calculations of phase transition behaviour in magnetic systems. The chapter is concerned with the transfer matrix method, first introduced by Hendrik Kramers, which reduces the problem of calculating the partition function to the problem of finding the eigenvalues of a certain matrix. It describes the low temperature expansion and Weiss’ mean field theory for magnets. Mean field theory is the simplest treatment of an interacting statistical mechanical system, apart from the approximation of ignoring the interactions all together. The chapter provides a simple, non-systematic derivation of mean field theory for the Ising model. This mean field theory can be easily improved by applying it to clusters of spins rather than individual spins.
This chapter explains how phase transitions can occur in principle. It examines how phase transition is possible for a phase transition to occur in a simple model of a magnet-the Ising model. The chapter explores some of the wider and more complex issues associated with phase transitions, using other examples. It focuses on ergodicity breaking, using disordered systems as an example. The goal of statistical mechanics is to compute the partition function Z. In order for uniform bulk behaviour to exist, the forces in the system must satisfy certain properties, and the thermodynamic limit must be taken carefully. The casual reader of any textbook or research paper on phase transitions and statistical mechanics cannot help being struck by the frequency of the term “model”. The phase transition literature is replete with models: the Ising model, the Heisenberg model, the Potts model, the Baxter model, the F model, and even such unlikely sounding names as the non-linear sigma model!.
Abstract Methods of statistical mechanics have been enormously successful in clarifying the macroscopic properties of many-body systems. Typical examples are found in magnetic systems, which have been a test bed for a variety of techniques. This chapter introduces the Ising model of magnetic systems and explains its mean-field treatment, a very useful technique of analysis of many-body systems by statistical mechanics. Mean-field theory explained here forms the basis of the methods used repeatedly throughout this book. The arguments in the present chapter represent a general mean-field theory of phase transitions in the Ising model with uniform ferromagnetic interactions. Special features of spin glasses and related disordered systems are taken into account in subsequent chapters.
The chapter is an introduction to basic equilibrium aspects of phase transitions. It starts by reviewing thermodynamics and the thermodynamic description of phase transitions. Next, lattice models, such as the paradigmatic Ising model, are introduced as simple physical models that permit a mechano-statistical study of phase transitions from a more microscopic point of view. It is shown that the Ising model can quite faithfully describe many different systems after suitable interpretation of the lattice variables. Special emphasis is placed on the mean-field concept and the mean-field approximations. The deformable Ising model is then studied as an example that illustrates the interplay of different degrees of freedom. Subsequently, the Landau theory of phase transitions is introduced for continuous and first-order transitions, as well as critical and tricritical behaviour are analysed. Finally, scaling theories and the notion of universality within the framework of the renormalization group are briefly discussed.
This chapter describes the physics of phase transitions and critical phenomena. After introducing the thermodynamics of phase transitions, the general considerations are made explicit by presenting statistical mechanical calculations on a lattice gas model in the mean-field approximation. Phase transitions occur in macroscopic systems when one or more external parameters like temperature, pressure, electrical field, magnetic field, etc are varied. The chapter demonstrates the utility of the alternative formulations of thermodynamics and statistical mechanics in the context of phase transitions and shows how their use clarifies and unifies the theory of phase transitions. It deals with the basic thermodynamic description of a single component fluid. To understand different phases and associated transitions, one studies how the couplings flow in the parameter space under repeated Renormalization Group (RG) transformations. The flows around the fixed point are easily studied by linearizing the RG equations around the fixed points and determining the eigenvalues and eigenvectors of the linearized RG transformations.
Renormalization in pure systems In Section 2.2 we reviewed the basic facts about broken symmetry and phase transitions in pure systems, within mean field theory. This helped set the stage for the mean field description of spin glasses that we studied in Chapters 3–6. We will now want to study the phase transition and the nature of the low-temperature phase for spin glasses with fairly short-ranged interactions in low dimensionality (2 or 3). This situation is far from the region where mean field theory is a good guide, so we will have to learn new methods. Again, we start by learning about them in pure systems. Renormalization is a very general approach to problems with many strongly interacting degrees of freedom. In a statistical mechanical problem, the basic idea is to carry out the trace in the partition function over some of the variables, leaving a new problem with fewer degrees of freedom. One follows how the parameters in the Hamiltonian of the system change as this procedure is iterated many times. Generally, the partial trace corresponds to removing the short-distance degrees of freedom, so each successive effective Hamiltonian has a larger lattice constant (or equivalent microscopic length) than its predecessor: each renormalization step corresponds to a change of scale. Different thermodynamic phases are identified with flows to different fixed points in the parameter space of the Hamiltonian under this semigroup.
This chapter presents a simple approximation scheme — the mean field approximation — which allows a study of the neighbourhood of the transition temperature. It distinguishes between first and second order phase transitions. In the latter case, the correlation length diverges at the critical (transition) temperature. This has important physical consequences: in the mean field approximation, the singular behaviour of thermodynamic quantities at the critical temperature, and more generally in the neighbourhood of the critical temperature and in small magnetic field, is universal, that is, does not depend on the details of the interactions, on the dimension of space and, in the high temperature phase at least, on the symmetry of the models.