Introduction

Thermodynamicsis a phenomenological description of properties of macroscopic systems in thermal equilibrium.

Imagine yourself as a post-Newtonian physicist intent on understanding the behavior of such a simple system as a container of gas. How would you proceed? The prototype of a successful physical theory is classical mechanics, which describes the intricate motions of particles starting from simple basic laws and employing the mathematical machinery of calculus. By analogy, you may proceed as follows:

• Idealize the system under study as much as possible (as is the case of a point particle). The concept of mechanical work on the system is certainly familiar, yet there appear to be complications due to exchange of heat. The solution is first to examine closed systems, insulated by adiabatic walls that don't allow any exchange of heat with the surroundings. Of course, it is ultimately also necessary to study open systems, which may exchange heat with the outside world through diathermic walls.

• As the state of a point particle is quantified by its coordinates (and momenta), properties of the macroscopic system can also be described by a number of thermodynamic coordinates or state functions. The most familiar coordinates are those that relate to mechanical work, such as pressure and volume (for a fluid), surface tension and area (for a film), tension and length (for a wire), electric field and polarization (for a dielectric), etc.

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Introduction

We noted in the previous chapter that the singular behavior of thermodynamic functions at a critical point (the termination of a coexistence line) can be characterized by a set of critical exponents {α, β, γ, …}. Experimental observations indicate that these exponents are quite universal, i.e. independent of the material under investigation, and to some extent, of the nature of the phase transition. For example, the vanishing of the coexistence boundary in the condensation of CO2 has the same singular behavior as that of the phase separation of protein solutions into dilute and dense components. This universality of behavior needs to be explained. We also noted that the divergence of the response functions, as well as direct observations of fluctuations via scattering studies, indicate that fluctuations have long wavelengths in the vicinity of the critical point, and are correlated over distances ξ ≫ a, where a is a typical interparticle spacing. Such correlated fluctuations involve many particles and a coarse-graining approach, in the spirit of the theory of elasticity, may be appropriate to their description. Here we shall construct such a statistical field theory.

We shall frame the discussion in the language of a magnetic system whose symmetries are more transparent, although the results are of more general applicability. Consider a material such as iron, which is experimentally observed to be ferromagnetic below a Curie temperature Tc, as in Fig. 1.4.

Many scientists and non-scientists are familiar with fractals, abstract self-similar entities which resemble the shapes of clouds or mountain landscapes. Fewer are familiar with the concepts of scale-invariance and universality which underlie the ubiquity of these shapes. Such properties may emerge from the collective behavior of simple underlying constituents, and are studied through statistical field theories constructed easily on the basis of symmetries. This book demonstrates how such theories are formulated, and studied by innovative methods such as the renormalization group.

The material covered is directly based on my lectures for the second semester of a graduate course on statistical mechanics, which I have been teaching on and off at MIT since 1988. The first semester introduces the student to the basic concepts and tools of statistical physics, and the corresponding material is presented in a companion volume. The second semester deals with more advanced applications – mostly collective phenomena, phase transitions, and the renormalization group, and familiarity with basic concepts is assumed. The primary audience is physics graduate students with a theoretical bent, but also includes postdoctoral researchers and enterprising undergraduates. Since the material is comparatively new, there are fewer textbooks available in this area, although a few have started to appear in the last few years. Starting with the problem of phase transitions, the book illustrates how appropriate statistical field theories can be constructed on the basis of symmetries.

Historically, the discipline of statistical physics originated in attempts to describe thermal properties of matter in terms of its constituent particles, but also played a fundamental role in the development of quantum mechanics. More generally, the formalism describes how new behavior emerges from interactions of many degrees of freedom, and as such has found applications in engineering, social sciences, and increasingly in biological sciences. This book introduces the central concepts and tools of this subject, and guides the reader to their applications through an integrated set of problems and solutions.

The material covered is directly based on my lectures for the first semester of an MIT graduate course on statistical mechanics, which I have been teaching on and off since 1988. (The material pertaining to the second semester is presented in a companion volume.) While the primary audience is physics graduate students in their first semester, the course has typically also attracted enterprising undergraduates. as well as students from a range of science and engineering departments. While the material is reasonably standard for books on statistical physics, students taking the course have found my exposition more useful, and have strongly encouraged me to publish this material. Aspects that make this book somewhat distinct are the chapters on probability and interacting particles. Probability is an integral part of statistical physics, which is not sufficiently emphasized in most textbooks.