Abstract Statistical mechanics is the art of predicting the behaviour of a system with a large number of degrees of freedom, given the laws governing its microscopic behaviour. It divides naturally into two parts. The first is the more fundamental. It tries to answer general questions concerning the behaviour of systems with many degrees of freedom, such as how order can arise from the apparent chaos of the microscopic behaviour, and how time-symmetric microscopic laws can give rise to the time asymmetry observed in the macroscopic world. The second part of statistical mechanics is concerned with using the results of this analysis to determine the behaviour of specific systems. This is what we will be doing in this book. Statistical mechanics will be used as a tool to study critical phenomena in particular systems.
Abstract Statistical mechanics is the art of predicting the behaviour of a system with a large number of degrees of freedom, given the laws governing its microscopic behaviour. It divides naturally into two parts. The first is the more fundamental. It tries to answer general questions concerning the behaviour of systems with many degrees of freedom, such as how order can arise from the apparent chaos of the microscopic behaviour, and how time-symmetric microscopic laws can give rise to the time asymmetry observed in the macroscopic world. The second part of statistical mechanics is concerned with using the results of this analysis to determine the behaviour of specific systems. This is what we will be doing in this book. Statistical mechanics will be used as a tool to study critical phenomena in particular systems.
The basic concepts of statistical mechanics and its application to many-body systems, phase transitions and critical phenomena will be discussed briefly in this chapter. We will be using these ideas frequently in the subsequent lectures. Statistical mechanics describes the thermodynamic behaviour of macroscopic systems from the laws which govern the behaviour of the constituent elements at the microscopic level. The microscopic elements can be atoms, molecules, dipole moments or magnetic moments, etc. A macroscopic system is generally composed of a large number of these elements (of the order of Avogadro number N A ≈ 6.022 × 1023 per mole). Each element may have a large number of internal degrees of freedom associated with different types of motion such as translation, rotation, vibration etc. The constituent elements may interact with the external field applied to the system. There can also be very complex interaction among the constituent elements. The macroscopic properties of a system in the thermodynamic limit is thus determined by the properties of the constituent molecules, their interaction with external field as well as interaction among themselves. For a system of N elements, volume V and density ρ, the thermodynamic limit is defined as lim N → ∞, lim V → ∞ keeping ρ = N/V finite. In this limit, the extensive properties of the system become directly proportional to the size of the system (N or V), while the intensive properties become independent of the size of the system.
The problems of statistical mechanics are those which involve systems with a larger number of degrees of freedom than we can conveniently follow explicitly in experiment, theory or simulation. The number of degrees of freedom which can be followed explicitly in simulations has been changing very rapidly as computers and algorithms improve. However, it is important to note that, even if computers continue to improve at their present rate, characterized by Moore's “law,” scientists will not be able to use them for a very long time to predict many properties of nature by direct simulation of the fundamental microscopic laws of physics. This point is important enough to emphasize. Suppose that, T years from the present, a calculation requiring computation time t 0 at present will require computation time t ( T ) = t 0 2 − T /2 (Moore's “law,” see Figure 1). Currently, state of the art numerical solutions of the Schrödinger equation for a few hundred atoms can be carried out fast enough so that the motion of these atoms can be followed long enough to obtain thermodynamic properties. This is adequate if one wishes to predict properties of simple homogeneous gases, liquids or solids from first principles (as we will be discussing later). However, for many problems of current interest, one is interested in entities in which many more atoms need to be studied in order to obtain predictions of properties at the macroscopic level of a centimeter or more.
Classical statistical mechanics tries to derive the macroscopic properties of matter starting from the mechanical laws that rule the behaviour of single particles. To describe equilibrium states, only observables accounting for correlations among states are considered, and only systems consisting of a large number of particles are taken into account. The observables are described by continuous functions on phase space, and the states are represented by linear assignments of a number to each observable. Within the framework of the canonical ensemble, one deals with mechanical systems in thermal equilibrium with a thermal reservoir, and the equilibrium state is reached as a result of the interaction with the external world. The external world may be really external, or equally well, the unrecognized internal degrees of freedom, like in the calculation of viscosity or thermal conductivity. If really only the external world is the cause we may expect some surface dependence while an internal unrecognized degree of freedom would have a volume effect, unless the interactions are long range. The microcanonical ensemble is instead introduced to study isolated mechanical systems, and the equilibrium is viewed as a temporal average, rather than as a limit. Attention is then focused on partition functions, the theorem of equipartition of energy and an elementary theory of specific heats. In the second part, the Planck derivation of the law of black-body radiation is analysed, presenting in chronological order the Kirchhoff theorem, the Stefan law, the Wien displacement law, the Rayleigh–Jeans formula and the Planck hypothesis. Further topics discussed are the Einstein and Debye quantum models for specific heats of solids. These topics prepare the ground for the introduction of quantum statistical mechanics. The third part is, in fact, devoted to the analysis of identical particles in quantum mechanics.
Abstract This book begins where elementary books and courses leave off and covers the advances made in statistical mechanics in the past fifty years. The book is divided into three parts. The first part is on general theory which includes a summary of the basic principles of statistical mechanics; a presentation of the physical phenomena covered and the models used to discuss them; theorems on the existence and uniqueness of partition functions; theorems on order; and critical phenomena and scaling theory. The second part is on series and numerical methods which includes derivations of the Mayer and Ree–Hoover expansions of the low density virial equation of state; Groeneveld's theorems; the application to hard spheres and discs; a summary of numerical studies of systems at high density; and the use of high temperature series expansions to estimate critical exponents for magnets. The third part covers exactly solvable models which includes a detailed presentation of the Pfaffian methods of computing the Ising partition function, magnetization, correlation functions, and susceptibility; the star-triangle (Yang–Baxter equation); functional equations and the free energy for the eight-vertex model; and the hard hexagon and chiral Potts models. All needed mathematics is developed in detail and many open questions are discussed. The goal is to guide the reader to the current forefront of research.
I. Introduction.Statistical mechanics is the science of explaining, predicting and understanding the gross, macroscopic attributes of matter (which may be taken to mean mechanical systems with essentially an infinite number of degrees of freedom) in terms of the elementary dynamical laws governing its atomic constituents.The problems that arise are sufficiently complex and intriguing, but at the same time sufficiently well posed, that the subject is nowadays as much a part of mathematics as of physics.The fields of information theory and ergodic theory had their genesis in statistical mechanical modes of thought and are now well established in the mathematics literature; there will be more to come.Ludwig Boltzmann, who died in 1906, was one of the principal founders of statistical mechanics, and his monument in Vienna contains the following eloquent testimonial to his scientific creativity:(1) S = k log W.Surely, this hypothesis of Boltzmann [1] is one of the most important and daring in statistical mechanics, for it relates S, the macroscopic entropy of a system, to W, the number of microscopic states of the system which have the same, given macroscopic properties.The number k is a universal constant, called Boltzmann's constant, and, for our purposes, we can consider it to be 1.In these lectures we shall explore some of the abstract properties of entropy, after first giving a precise formulation of it, and will include some recent results (with M. B. Ruskai) which extend formerly known facts about the strong subadditivity of entropy from the domain of classical mechanics to the quantum-mechanical domain.The presentation here will be sketchy and the reader is referred to the original papers [3], [4], [5] for more details.An expanded version of an invited address delivered before the M.I.T.
Abstract We have now seen how powerful the tools of statistical mechanics can be in the study of complex systems in thermal equilibrium and how they can be used to determine exactly the behaviour of some simplified, but nevertheless interesting, models of phase transitions. However, these exact solutions are sometimes lengthy and difficult and there is no guarantee that any particular model will yield exact results. For example, we were able to solve the two-dimensional, but not the three-dimensional, Ising model, and that only in zero magnetic field. So, it is natural to consider whether we can use a computer to simulate these systems. In this chapter we shall see that this is indeed possible; a great deal of our understanding of critical phenomena is based wholly or partly on the results of such numerical work. However, as we shall also see, such simulations are not entirely straightforward and a number of tricks are needed to obtain accurate results. This is partly a consequence of the very physics that makes the phase transitions interesting to start with. One should probably think of numerical simulation as a whole subject, almost an art, of its own; we shall have time only to scratch at the surface and discuss a few of the most important ideas.