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.

[1] In the previous chapter we dealt with the foundations and principles of knowledge of Scripture, and proved that these amount to nothing more than assembling an accurate history of it. We also showed that the ancients neglected this form of enquiry, essential though it is, or if they did write anything about it and handed it down, it has perished through the injury of time, and thus most of the foundations and principles of this knowledge have disappeared. Now we could live with this if later writers had kept within proper limits and faithfully passed on to their successors what little they had received or discovered and not contrived novelties out of their own heads. For this is how it has come about that the history of the Bible has remained not only incomplete but also rather unreliable, that is, the existing basis of our knowledge of the Scriptures is not just too sparse for us to construct an adequate history, it also teems with errors.

[2] My aim is to correct this situation and remove our prevailing theological prejudices. But my attempt, I am afraid, may be too late. For the situation has now almost reached the point that men will not allow themselves to be corrected on these questions but rather obstinately defend whatever position they have taken up, in the name of religion.

[1] Hitherto our concern has been to separate philosophy from theology and to establish the freedom to philosophize which this separation allows to everyone. The time has now come to enquire how far this freedom to think and to say what one thinks extends in the best kind of state. To consider this in an orderly fashion, we must first discuss the foundations of the state but, before we do that, we must explain, without reference to the state and religion, the natural right (jus) which everyone possesses.

[2] By the right and order of nature I merely mean the rules determining the nature of each individual thing by which we conceive it is determined naturally to exist and to behave in a certain way. For example fish are determined by nature to swim and big fish to eat little ones, and therefore it is by sovereign natural right that fish have possession of the water and that big fish eat small fish. For it is certain that nature, considered wholly in itself, has a sovereign right to do everything that it can do, i.e., the right of nature extends as far as its power extends. For the power of nature is the very power of God who has supreme right to [do] all things. However, since the universal power of the whole of nature is nothing but the power of all individual things together, it follows that each individual thing has the sovereign right to do everything that it can do, or the right of each thing extends so far as its determined power extends.

[1] We proved in chapter 2 of this treatise that the prophets possessed extraordinary powers of imagination but not of understanding, and that it was not the deeper points of philosophy that God revealed to them but only some very simple matters, adapting Himself to their preconceived beliefs. We then showed in chapter 5 that Scripture explains and teaches things in such a way that anyone may grasp them. It does not deduce and derive them from axioms and definitions, but speaks simply, and to secure belief in its pronouncements, it confirms them by experience alone, that is, by miracles and histories narrated in a language and style designed to influence the minds of the common people: on this see chapter 6 (point 3). Finally, we demonstrated in chapter 7 that the difficulty of comprehending the Bible lies solely in the language and not in the sublimity of its content. There is the further problem, though, that the prophets were not addressing the learned among the Jews but the entire people without exception, and the Apostles likewise were accustomed to proclaim the Gospel teaching in churches where there was a miscellaneous congregation of all types of people. From all this it follows that biblical teaching contains no elevated theories or philosophical doctrines but only the simplest matters comprehensible to even the very slowest.