Thermodynamics and Statistical Mechanics

Equilibrium by Entropy Maximisation

P. Attard


Thermodynamics deals with the general principles and laws that govern the behaviour of matter and with the relationships between material properties. The origins of these laws and quantitative values for the properties are provided by statistical mechanics, which analyses the interaction of molecules and provides a detailed description of their behaviour. This book presents a unified account of equilibrium thermodynamics and statistical mechanics using entropy and its maximisation.

A physical explanation of entropy based upon the laws of probability is introduced. The equivalence of entropy and probability that results represents a return to the original viewpoint of Boltzmann, and it serves to demonstrate the fundamental unity of thermodynamics and statistical mechanics, a point that has become obscured over the years. The fact that entropy and probability are objective consequences of the mechanics of molecular motion provides a physical basis and a coherent conceptual framework for the two disciplines. The free energy and the other thermodynamic potentials of thermodynamics are shown simply to be the total entropy of a subsystem and reservoir; their minimisation at equilibrium is nothing but the maximum of the entropy mandated by the second law of thermodynamics and is manifest in the peaked probability distributions of statistical mechanics. A straightforward extension to nonequilibrium states by the introduction of appropriate constraints allows the description of fluctuations and the approach to equilibrium, and clarifies the physical basis of the equilibrium state.

Although this book takes a different route to other texts, it shares with them the common destination of explaining material properties in terms of molecular motion. The final formulae and interrelationships are the same, although new interpretations and derivations are offered in places. The reasons for taking a detour on some of the less-travelled paths of thermodynamics and statistical mechanics are to view the vista from a different perspective, and to seek a fresh interpretation and a renewed appreciation of well-tried and familiar results. In some cases this reveals a shorter path to known solutions, and in others the journey leads to the frontiers of the disciplines.

The book is basic in the sense that it begins at the beginning and is entirely self-contained. It is also comprehensive and contains an account of all of the modern techniques that have proven useful in modern equilibrium, classical statistical mechanics. The aim has been to make the subject matter broadly accessible to advanced students, whilst at the same time providing a reference text for graduate scholars and research scientists active in the field. The later chapters deal with more advanced applications, and while their details may be followed step-by-step, it may require a certain experience and sophistication to appreciate their point and utility. The emphasis throughout is on fundamental principles and upon the relationship between various approaches. Despite this, a deal of space is devoted to applications, approximations, and computational algorithms; thermodynamics and statistical mechanics were in the final analysis developed to describe the real world, and while their generality and universality are intellectually satisfying, it is their practical application that is their ultimate justification. For this reason a certain pragmatism that seeks to convince by physical explanation rather than to convict by mathematical sophistry pervades the text; after all, one person's rigor is another's mortis.

The first four chapters of the book comprise statistical thermodynamics. This takes the existence of weighted states as axiomatic, and from certain physically motivated definitions, it deduces the familiar thermodynamic relationships, free energies, and probability distributions. It is in this section that the formalism that relates each of these to entropy is introduced. The remainder of the book comprises statistical mechanics, which in the first place identifies the states as molecular configurations, and shows the common case in which these have equal weight, and then goes on to derive the material thermodynamic properties in terms of the molecular ones. In successive chapters the partition function, particle distribution functions, and system averages, as well as a number of applications, approximation schemes, computational approaches, and simulation methodologies, are discussed. Appended is a discussion of the nature of probability.

The paths of thermodynamics and statistical mechanics are well-travelled and there is an extensive primary and secondary literature on various aspects of the subject. Whilst very many of the results presented in this book may be found elsewhere, the presentation and interpretation offered here represent a sufficiently distinctive exposition to warrant publication. The debt to the existing literature is only partially reflected in the list of references; these in general were selected to suggest alternative presentations, or further, more detailed, reading material, or as the original source of more specialised results. The bibliography is not intended to be a historical survey of the field, and, as mentioned above, an effort has been made to make the book self-contained.

At a more personal level, I acknowledge a deep debt to my teachers, collaborators, and students over the years. Their influence and stimulation are impossible to quantify or detail in full. Three people, however, may be fondly acknowledged: Pat Kelly, Elmo Lavis, and John Mitchell, who in childhood, school, and PhD taught me well.