General definitions

The laws of thermodynamics are based on observations of macroscopic bodies, and encapsulate their thermal properties. On the other hand, matter is composed of atoms and molecules whose motions are governed by more fundamental laws (classical or quantum mechanics). It should be possible, in principle, to derive the behavior of a macroscopic body from the knowledge of its components. This is the problem addressed by kinetic theory in the following chapter. Actually, describing the full dynamics of the enormous number of particles involved is quite a daunting task. As we shall demonstrate, for discussing equilibrium properties of a macroscopic system, full knowledge of the behavior of its constituent particles is not necessary. All that is required is the likelihood that the particles are in a particular microscopic state. Statistical mechanics is thus an inherently probabilistic description of the system, and familiarity with manipulations of probabilities is an important prerequisite. The purpose of this chapter is to review some important results in the theory of probability, and to introduce the notations that will be used in the following chapters.

General definitions

Kinetic theorystudies the macroscopic properties of large numbers of particles, starting from their (classical) equations of motion.

Thermodynamics describes the equilibrium behavior of macroscopic objects in terms of concepts such as work, heat, and entropy. The phenomenological laws of thermodynamics tell us how these quantities are constrained as a system approaches its equilibrium. At the microscopic level, we know that these systems are composed of particles (atoms, molecules), whose interactions and dynamics are reasonably well understood in terms of more fundamental theories. If these microscopic descriptions are complete, we should be able to account for the macroscopic behavior, that is, derive the laws governing the macroscopic state functions in equilibrium. Kinetic theory attempts to achieve this objective. In particular, we shall try to answer the following questions:

1. How can we define “equilibrium” for a system of moving particles?

2. Do all systems naturally evolve towards an equilibrium state?

3. What is the time evolution of a system that is not quite in equilibrium?

The simplest system to study, the veritable workhorse of thermodynamics, is the dilute (nearly ideal) gas. A typical volume of gas contains of the order of 1023 particles, and in kinetic theory we try to deduce the macroscopic properties of the gas from the time evolution of the set of atomic coordinates.

General definitions

Statistical mechanicsis a probabilistic approach to equilibrium macroscopic properties of large numbers of degrees of freedom.

As discussed in chapter 1, equilibrium properties of macroscopic bodies are phenomenologically described by the laws of thermodynamics. The macrostate M depends on a relatively small number of thermodynamic coordinates. To provide a more fundamental derivation of these properties, we can examine the dynamics of the many degrees of freedom comprising a macroscopic body. Description of each microstate µ requires an enormous amount of information, and the corresponding time evolution, governed by the Hamiltonian equations discussed in chapter 3, is usually quite complicated. Rather than following the evolution of an individual (pure) microstate, statistical mechanics examines an ensemble of microstates corresponding to a given (mixed) macrostate. It aims to provide the probabilities PM(µ) for the equilibrium ensemble. Liouville's theorem justifies the assumption that all accessible microstates are equally likely in an equilibrium ensemble. As explained in chapter 2, such assignment of probabilities is subjective. In this chapter we shall provide unbiased estimates of PM(µ) for a number of different equilibrium ensembles. A central conclusion is that in the thermodynamic limit, with large numbers of degrees of freedom, all these ensembles are in fact equivalent. In contrast to kinetic theory, equilibrium statistical mechanics leaves out the question of how various systems evolve to a state of equilibrium.

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.

• […]

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.

We showed in the previous chapter that the divine law which makes men truly happy and teaches the true life, is universal to all men. We also deduced that law from human nature in such a way that it must itself be deemed innate to the human mind and, so to speak, inscribed upon it. As for ceremonies, or those at least which are narrated in the Old Testament, these were instituted for the Hebrews alone and were so closely accommodated to their state that in the main they could be practised not by individuals but only by the community as a whole. It is certain, therefore, that they do not belong to the divine law and hence contribute nothing to happiness and virtue. They are relevant only to the election of the Hebrews, that is (as we showed in chapter 3), to the temporal and material prosperity and peace of their state, and therefore could have relevance only so long as that state survived. If in the Old Testament they are ascribed to the law of God, that is only because they were instituted as the result of a revelation or on revealed foundations. But since reasoning, no matter how sound, carries little weight with ordinary theologians, I propose now to adduce the authority of the Bible to confirm what I have just proved. Then, for yet greater clarity, I will show why and how these ceremonies served to establish and preserve the Jewish state.