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CRC Press eBooks

Introduction

Nigel D. Goldenfeld

Published Mar 8, 2018• 1 citation
Publisher

Abstract

The Ising ferromagnet is therefore relatively simple to study, and we will devote considerable attention to it in these notes. In other words for the most part omit discussion of what are called first order phase transitions in the Ehrenfest classification . The phenomena which we will describe are often called critical phenomena and occur at second order phase transitions , although this name is inappropriate, because the Ehrenfest classification is not correct. We will see that the equation of state of a physical system near to a critical point obeys what appears to be an analogue of the law of corresponding states , encountered in the equation of state (such as that due to Van der Waals) for a fluid.

DOI
10.1201/9780429493492-1

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The Ising ferromagnet is therefore relatively simple to study, and we will devote considerable attention to it in these notes. In other words for the most part omit discussion of what are called first order phase transitions in the Ehrenfest classification . The phenomena which we will describe are often called critical phenomena and occur at second order phase transitions , although this name is inappropriate, because the Ehrenfest classification is not correct. We will see that the equation of state of a physical system near to a critical point obeys what appears to be an analogue of the law of corresponding states , encountered in the equation of state (such as that due to Van der Waals) for a fluid.
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.
To set the stage for our subsequent thorough discussion of dynamic critical phenomena, we first review the theoretical description of second-order equilibrium phase transitions. (Readers already well acquainted with this material may readily move on to Chapter 2.) To this end, we compare the critical exponents following from the van-der-Waals equation of state for weakly interacting gases with the results from the Curie–Weiss mean-field approximation for the ferromagnetic Ising model. We then provide a unifying description in terms of Landau–Ginzburg theory, i.e., a long-wavelength expansion of the effective free energy with respect to the order parameter. The Gaussian model is analyzed, and a quantitative criterion is established that defines the circumstances when non-linear fluctuations need to be taken into account properly. Thereby we identify d c = 4 as the upper critical dimension for generic continuous phase transitions in thermal equilibrium. The most characteristic feature of a critical point turns out to be the divergence of the correlation length that renders microscopic details oblivious. As a consequence, not only the correlation functions, but remarkably the thermodynamics as well of a critical system are governed by an emergent unusual symmetry: scale invariance. A simple scaling ansatz is capable of linking different critical exponents; as an application, we introduce the basic elements of finite-size scaling. Finally, a brief sketch of Wilson's momentum shell renormalization group method is presented, intended as a pedagogical preview of the fundamental RG ideas. Exploiting the scale invariance properties at the critical point, the scaling forms of the free energy and the order parameter correlation function are derived.
This and the four next chapters are devoted to the study of several aspects of the theory of phase transition. This has been for more than one century, and still remains, a subject of intense experimental and theoretical activity, with important breakthroughs having impacts on different fields in physics but also on other disciplines such as mathematics, economics, sociology, biology In the first section of this chapter, the transition between the ferromagnetic and paramagnetic phases of the Ising system will be studied in the framework of a mean field approximation. The problem will first be addressed by means of the intuitive Weiss molecular field approximation ( Section 9.1.2 ) and then by means of the more detailed Bragg-Williams approximation ( Section 9.1.3 ) which will serve as a base to develop, in Section 9.2 , a general theory of continuous phase transitions: the Landau mean field theory for which the concept of order parameter will first be introduced. Other types of phase transitions will then be studied using the mean field approximation. Phase transitions in liquid crystals will be studied in Section 9.3 . These are systems with a tensor order parameter which exhibit a rich phenomenology. Finally, in Section 9.4 , a transition of a fairly more elaborate nature than a simple first or second order phase transition will be discussed.
Phase, as the term is used in thermodynamics, refers to a spatially uniform equilibrium system. Substances undergo phase transitions, changes in phase that occur upon variations of state variables. The chemical potential of substances in coexisting phases has the same value in each of the phases in which coexistence occurs. The specific heat is discontinuous at the van der Waals critical point—a second-order phase transition in the Ehrenfest classification scheme. This chapter shows that one-dimensional systems cannot support phase transitions. Just as with liquid-vapor phase transitions, exponents characterize the critical behavior of magnetic systems. Magnets and fluids are different physical systems, yet they have the same critical behavior. Through the concept of order parameter, Landau theory subsumes the phase-transition behavior of different systems into one theory. A more physical way of demonstrating the existence of a phase transition would be to calculate the order parameter.
This essay corresponds to the content of three lectures about statistical physics delivered to the audience of the 2014 section of the R. A. Salmeron School of Physics, at the UnB. Our starting point was very simple statistical models (lattice gas, spin-1/2 ferromagnet), used as illustrations of the competencies and methods in statistical physics. Thus we introduce the Gibbs ensembles, defining a connection with thermodynamics and discussing the role played by fluctuations and large numbers. We present phenomenological aspects of phase transitions and critical phenomena in simple fluids and in uniaxial ferromagnets, emphasizing the universal character of the critical exponents. We describe the phenomenological van der Waals and Curie-Weiss theories and the Landau expansion, which are present-day relevant methods, despite the fact that such theories give rise to critical exponents in disagreement with experiments. We present then the paradigmatic Ising model, which points us to a way to overcome the phenomenological results. A brief presentation of the scale phenomenological methods and the contemporaneous renormalization group are considered at the end of these lectures.