Abstract In this chapter, the singular changes accompanying a phase transition requires the existence of macroscopic correlations. As a consequence, phase transitions mark a stark departure from the noninteracting and linear responding systems we have considered thus far. Understanding phase transi- tions requires that we study strongly interacting models, pushed far into regimes of nonlinear response. This complication renders detailed studies largely intractable analytically. However, some simplifications exist that allow for a general theory of phase transitions to be developed. These are the notions of universality that accompany phase transition phenomenology, and the ability to classify phase transi- tions simply using ideas of spontaneous symmetry breaking. The former allows us to postulate simple models bereft for molecular detail and yet still provide quantitative predictions for observable behavior. The latter allows us to predict what phases can exist, and understand why phase transitions occur.
Abstract In this chapter, the singular changes accompanying a phase transition requires the existence of macroscopic correlations. As a consequence, phase transitions mark a stark departure from the noninteracting and linear responding systems we have considered thus far. Understanding phase transi- tions requires that we study strongly interacting models, pushed far into regimes of nonlinear response. This complication renders detailed studies largely intractable analytically. However, some simplifications exist that allow for a general theory of phase transitions to be developed. These are the notions of universality that accompany phase transition phenomenology, and the ability to classify phase transi- tions simply using ideas of spontaneous symmetry breaking. The former allows us to postulate simple models bereft for molecular detail and yet still provide quantitative predictions for observable behavior. The latter allows us to predict what phases can exist, and understand why phase transitions occur.
The examination of phase transitions and critical phenomena has dominated statistical physics for the latter half of this century--there is a great theoretical challenge in solving special statistical mechanical models. Additionally, beautiful experimental results have elucidated the singularities (critical behavior) that occur in phase transitions. Although a few spectacular successes in the exact solution of simple models have occurred, interesting systems have mostly proven to be intractable from the analytic perspective. Because many physically disparate systems show identical critical behavior, identifying and understanding general principles of universality and their relationship to microscopic symmetries are of fundamental importance
In the previous chapters, we discussed mean-field approaches such as the Curie–Weiss and Bragg–Williams theories, which provide a microscopic justification for macroscopic order parameters and spontaneous symmetry breaking. However, these approaches were based on specific microscopic Hamiltonians. Lev Davidovich Landau introduced a far more general and elegant framework: one that does not depend on microscopic details but on symmetry principles and analyticity of the free energy near the transition.
Phase transitions are defined, and the concepts of order parameter and spontaneously broken symmetry are discussed. Simple models for magnetic phase transitions are introduced, together with some experimental examples. Critical exponents and the notion of universality are defined, and the consequences of the scaling assumptions are derived. Phase transitions and order parameters It is a fact of everyday experience that matter in thermodynamic equilibrium exists in different macroscopic phases. Indeed, it is difficult to imagine life on Earth without all three phases of water. A typical sample of matter, for example, has the temperature–pressure phase diagram presented in Fig. 1.1: by changing either of the two parameters the system may be brought into a solid, liquid, or gas phase. The change of phase may be gradual or abrupt. In the latter case, the phase transition takes place at well defined values of the parameters that determine the phase boundary. Phase transitions are defined as points in the parameter space where the thermodynamic potential becomes non-analytic. Such a non-analyticity can arise only in the thermodynamic limit, when the size of the system is assumed to be infinite. In a finite system the partition function of any system is a finite sum of analytic functions of its parameters, and is therefore always analytic. A sharp phase transition is thus a mathematical idealization, albeit one that describes the reality extremely well. Macroscopic systems typically contain ∼ 10 23 degrees of freedom, and as such are very close to being in the thermodynamic limit. The phase boundaries in Fig. 1.1, for example, for this reason represent reproducible physical quantities.
This book treats the problem of phase transitions, emphasizing the generality and universality of the methods and models used. The course is basically concentrated on the problems of vacuum degeneration in macroscopic systems and a fundamental concept of quasiaverages by Bogolubov playing a special role in the theory of phase transitions and critical phenomena. An analysis of the connection between phase transition and spontaneous symmetry breaking in a macroscopic system allows a unique description of both first- and second-order phase transitions.The unique features of this book are: (i) a unique approach of describing first — as well as second-order phase transitions, based on the Bogolubov concept of quasi-averages.(ii) a detailed presentation of the material and at the same time a review of modern problems.(iii) a general character of developed ideas that could be applied to various particular systems of condensed matter physics, nuclear physics and high-energy physics.
Among the many applications of statistical mechanics, some of the most intriguing and challenging theoretical problems arise in connection with phase transitions. These are abrupt changes of state such as occur, for example, when a liquid is transformed into a vapour, a ferromagnet loses its magnetization upon heating to its Curie temperature, or at the onset at suciently low temperatures of super uidity or superconductivity. It is within the theory of phase transitions, too, that the mathematical relationships between statistical mechanics and relativistic eld theories are most powerful. Indeed, the idea of spontaneous symmetry breaking, which lies at the heart of the theory of phase transitions, is the crucial ingredient that turns the gauge theories of Chapter 8 into a real working model of the fundamental forces of nature, to be discussed in the next chapter.
Among the many applications of statistical mechanics, some of the most intriguing and challenging theoretical problems arise in connection with phase transitions. These are abrupt changes of state such as occur, for example, when a liquid is transformed into a vapour, a ferromagnet loses its magnetization upon heating to its Curie temperature, or at the onset at sufficiently low temperatures of superfluidity or superconductivity. It is within the theory of phase transitions, too, that the mathematical relationships between statistical mechanics and relativistic field theories are most powerful. Indeed, the idea of spontaneous symmetry breaking, which lies at the heart of the theory of phase transitions, is the crucial ingredient that turns the gauge theories of chapter 8 into a real working model of the fundamental forces of nature, to be discussed in the next chapter.