Abstract This first chapter is an elementary introduction to the models and methods that will be used later. The ferromagnetic (Curie) transition serves as our prototype for all second-order phase transitions, and serves also to introduce the notions of broken symmetry and of order parameter. The basic model of ferromagnetism, namely the Ising model, is described in Section 1.2, and solved for an elementary case. The reason why this model is so important is that it has an exact solution in two dimensions, which exhibits spontaneous magnetization. Section 1.3 is devoted to a first approach to mean-field methods, which remain amongst the most frequently exploited approximations even today. They will lead us to define a first set of critical exponents; the listing of such exponents will then be completed in Section 1.4, once correlation functions have been defined. Finally, in Section 1.5 a qualitative description of critical phenomena will enable us, in a first approximation, to define the concept of scale invariance and its relation to the behaviour of the theory under dilatations.
Abstract This first chapter is an elementary introduction to the models and methods that will be used later. The ferromagnetic (Curie) transition serves as our prototype for all second-order phase transitions, and serves also to introduce the notions of broken symmetry and of order parameter. The basic model of ferromagnetism, namely the Ising model, is described in Section 1.2, and solved for an elementary case. The reason why this model is so important is that it has an exact solution in two dimensions, which exhibits spontaneous magnetization. Section 1.3 is devoted to a first approach to mean-field methods, which remain amongst the most frequently exploited approximations even today. They will lead us to define a first set of critical exponents; the listing of such exponents will then be completed in Section 1.4, once correlation functions have been defined. Finally, in Section 1.5 a qualitative description of critical phenomena will enable us, in a first approximation, to define the concept of scale invariance and its relation to the behaviour of the theory under dilatations.
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.
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.
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
Abstract As an introduction to the physics of phase transitions and critical phenomena, this chapter explains a number of basic and fundamental ideas such as phases, phase transitions, phase diagrams, universality, and critical phenomena. Especially important is the concept of order parameter, a quantity that measures the degree of asymmetry in the broken symmetry phase. Intuitive accounts are given to the concepts of coarse-graining, and scale and renormalization group transformations, which are powerful, systematic tools to analyze critical behaviour of macroscopic systems. Also explained are several spin and lattice gas model systems, on the basis of which phase transitions and critical phenomena will be studied.
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.
Abstract In undergraduate statistical mechanics courses the Ising model always plays an important role because it is the simplest non-trivial model used to describe magnetic systems. The one-dimensional model is easily solved analytically, while the two-dimensional one can be solved exactly by the Onsager solution. For this reason, numerical simulations are usually used to solve the two-dimensional model. Keeping in mind that the two-dimensional model is the platform for studying phase transitions, it is usually an exercise in computational undergraduate courses because its numerical solution is relatively simple to implement and its critical exponents are perfectly known. The purpose of this article is to present a detailed numerical study of the second-order phase transition in the two-dimensional Ising model at an undergraduate level, allowing readers not only to compare the mean-field solution, the exact solution and the numerical one through a complete study of the order parameter, the correlation function and finite-size scaling, but to present the techniques, along with hints and tips, for solving it themselves. We present the elementary theory of phase transitions and explain how to implement Markov chain Monte Carlo simulations and perform them for different lattice sizes with periodic boundary conditions. Energy, magnetization, specific heat, magnetic susceptibility and the correlation function are calculated and the critical exponents determined by finite-size scaling techniques. The importance of the correlation length as the relevant parameter in phase transitions is emphasized.