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.
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.
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.
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.
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.
In its broadest sense, “universality” is a technical term for something quite ordinary. This chapter examines what is, in the physics literature, the paradigm example of universality; namely, the so-called universality of critical phenomena—certain kinds of phase transitions that systems (e.g.,fluids and magnets) can undergo. One of the most striking examples of universal behavior concerns the pattern displayed by molecularly distinct fluids near their so-called critical points. Kadanoff states the “hypothesis of universality” as follows: All phase transition problems can be divided into a small number of different classes depending upon the dimensionality of the system and the symmetries of the order state. As Kadanoff and others have emphasized, universal behavior reflects a stability of a certain behavior under perturbation of particular details. The concept of universality also depends upon a separation of scales and the fact that no nearby characteristic length (time) scale is present.
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.
Abstract ‘The critical point’ in a phase diagram denotes the conditions under which a continuous transition between two states of matter occurs. Surprisingly, near the critical point diverse systems can have the same dependence of physical properties on parameters such as temperature. The only details that determine the critical exponents are the symmetry of the order parameter and the spatial dimensionality of the system. This independence from chemical and structural details, known as universality, presented a major challenge to theoretical physics for decades. Theory must deal with the large fluctuations in the amount of order that occurs close to the critical point. The successful theory developed by Ken Wilson exploits the fact that different states of a system and different systems can be related to one another by mathematically rescaling the length scales in the system. The mathematics involved can be made tractable by considering the abstract idea of a system with a variable number of spatial dimensions. The idea of rescaling is illustrated with the results of computer simulations of an Ising model and with images of fractals.