Principles of Statistical Mechanics

These lectures comprise an introductory course in statistical mechanics. The Gibbs formulation of the canonical ensemble is introduced and illustrated by application to simple models of magnets and fluids, specifically the ideal gas and the magnetic Ising spin chain. In addition, the classical mean field theories of fluids and ferromagnets, namely the van der Waals theory of fluids and the Curie-Weiss theory of ferromagnets, are described and their critical behaviour elucidated. The lectures conclude with a general discussion of the principles of universality and scaling which form the cornerstones of the modern theory of phase transitions and critical phenomena. 1 Formulation of Statistical Mechanics Statistical mechanics was the topic of the first A.N.U. Physics Summer School held in 1988. On that occasion, Colin Thompson and I shared the pleasure of presenting lectures on the general principles of statistical mechanics [1, 2]. Those lectures, although unpublished, are available in the proceedings of that first A.N.U. Summer School. The principles of statistical mechanics were in fact laid down at the turn of the century and so, not surprisingly, they have not changed since 1988. I will therefore be covering much the same ground in this series of lectures. Although this introduction is brief, a more comprehensive account of the topics covered here can be found in Colin Thompson’s excellent book [3] and a review article I wrote in 1983 [4]. Other very useful general references are the books by Callen [5], Stanley [6], Huang [7], Baxter [8] and Yeomans [9]. The definitive reference for many topics in this area is of course “Phase Transitions and Critical Phenomena” edited by Domb and Green and now Domb and Lebowitz [10]. 1.1 Classical Mechanics and Phase Space Thermodynamics describes the physical properties of bulk matter (solids, liquids and gases) in terms of a few variables such as absolute temperature T and pressure P . Ultimately, of course, all of the macroscopic properties of matter (in equilibrium with its surroundings) should be derivable from a knowledge of the fundamental interactions between the constituent particles. This is the goal of statistical mechanics. Email: [email protected]