David A. Lavis

Boltzmann, Gibbs, and the Concept of Equilibrium

The Boltzmann and Gibbs approaches to statistical mechanics have very different definitions of equilibrium and entropy. The problems associated with this are discussed, and it is suggested that they can be resolved, to produce a version of statistical mechanics incorporating both approaches, by redefining equilibrium not as a binary property (being/not being in equilibrium) but as a continuous property (degrees of equilibrium) measured by the Boltzmann entropy and by introducing the idea of thermodynamic‐like behavior for the Boltzmann entropy. The Kac ring model is used as an example to test the proposals.

Real-Space Renormalization Group Theory

The approach to the renormalization group in this chapter is usually referred to as the use of the real-space renormalization group. This is in contrast to renormalization group methods for continuous spin distributions, using, for example, the Landau-Ginzburg local free energy density of Sect. 3.3. These latter methods, in which renormalization is performed in wave-vector space, were initiated by Wilson (1975) and owe much to the influence of quantum field theory. Real-space methods, which are conceptually rather simpler, are based, in many cases, on the block-spin method of Kadanoff (1966) described in Sect. 2.3.1

A Survey of Models

One of the main investigative interests of statistical mechanics is in the possibility of phase transitions.

Phase Transitions and Scaling Theory

Most physical systems can exist in a number of different phases, distinguished by their different types of molecular or atomic order. This order may be in the spatial configurations of one or more kinds of microsystems or it may be in the orientations or conformations of the microsystems themselves. (See Ziman (1979) for a discussion of the wide variety of possible types of order.) In the case of the vapour (gas), liquid and solid phases of, for example, water or hydrogen, the order is spatial with no order in the vapour, a short-range clustering type of order in the liquid and long-range lattice order in the solid. For water this is not the complete picture. There are at least nine different ice phases distinguished by their lattice structures and proton configurations (Volume 1, Appendix A.3, Eisenberg and Kauzmann 1969). The most well-known example of orientational order occurs in magnetic systems where ferromagnetism corresponds to the alignment of the magnetic dipoles of the microsystems. Although a simple magnetic system may possess just one type of ferromagnetic phase, more complex ferrimagnetic systems can have a large number of different magnetic phases. In the case, for example, of cerium antimonide fourteen different phases have been identified by neutron diffraction experiments (Fischer et al. 1978) and specific heat analysis (Rossat-Mignod et al. 1980).

Transfer Matrices: Incipient Phase Transitions

Let \(\mathcal {N}\) be a two-dimensional lattice of \(N\) sites with unit vectors \({\varvec{i}}\) and \({\varvec{j}}\) in the two lattice directions.

Phenomenological Theory and Landau Expansions

In Sect. 4.1, as a preliminary to the discussion of scaling theory, we gave a general heuristic treatment of the geometry of phase transitions.

Edge-Decorated Ising Models

Consider a lattice \({\mathcal {N}}\) of dimension \(d\), coordination number Open image in new window , \(N\) sites, periodic boundary conditions and an Ising variable \(\sigma ({\varvec{r}})=\pm 1\) at every \({\varvec{r}}\in {\mathcal {N}}\).

Cluster Variation Methods

Cluster variation methods are closed-form approximations which, unlike the mean-field method, make some allowance for short-range order effects.1 In first-order approximations, energies and weights for the microsystems on a cluster or group of sites are treated exactly, but the correlations due to sites shared between groups are handled by plausible assumptions. Equivalent versions were invented independently by various authors, notably Guggenheim (1935), (the quasi-chemical method), Bethe (1935), (the Bethe method) and Kasteleyn and van Kranendonk (1956a, 1956b, 1956c), (the constant-coupling method). We base our approach on a first-order method due to Guggenheim and McGlashan (1951) which allows any suitable group of sites to be used and enables quite complicated forms of ordering and intermolecular attraction to be incorporated in a natural way. For the particular problem of sublattice ordering, a similar method had been used by Yang (1945) and Li (1949). Kikuchi (1951) carried the process of approximation to a higher level by minimizing the free energy with respect to the occupation probabilities of subgroups of sites, including nearest-neighbour pairs, as well as those of the main group and this procedure was placed on a systematic basis by Hijmans and de Boer (1955, 1956) (see also Domb 1960, Burley 1972, Ziman 1979, Young 1981, Schlijper 1983 and Morita 1984). The cluster counting procedures used by Hijmans and de Boer (1955) have now been used in the finite-lattice series expansion method of de Neef and Enting (1977) (see Volume 2, Sect. 7.7).

Transfer Matrices: Exactly Solved Models

As described in Sect. 3.8.1 the arrows of this model are on a rapidity lattice \({\mathcal {N}}^\circledast \) consisting of a finite number of straight lines, which here we suppose is bounded by a simple closed curve \({\mathfrak {C}}\), as shown in Fig. 16.15. Each lattice edge is the boundary between a shaded and an unshaded face these being decorated with one site of the dual lattices \({\mathcal {N}}\) and \({\mathcal {N}}^\star \), respectively.

Graphs and Lattices

A graph \({\mathfrak {g}}\) is a set of vertices (points) some or all of which are connected by edges (lines).