Table of contents

The authors have applied a systematic method for series transformation to the susceptibility of the Heisenberg model on various cubic lattices. The resulting series are sufficiently smooth for extrapolation by means of Neville tables, and the authors conclude that the exponent gamma has the value 1.40+or-0.02 for both the quantum spin-¹/₂ and classical models.

A proper treatment of the droplet model of phase transitions needs some knowledge of the degree of compactness of random clusters. It is pointed out that some numerical evidence on this is already available for both site and bond clusters on the plane square lattice. The domains occurring in a Monte Carlo solution of the percolation problem are found to be highly ramified, those for the Ising and Potts problems less so.

The authors show that the critical temperature for n-vector models with long-range interaction falling off at infinity as 1/r^{d+ sigma} vanishes when d= sigma , provided n is larger than one. As a consequence, the authors calculate the critical exponents as power series in (d- sigma ) up to second order.

For pt.I see J. Math. Phys., vol.16, p.1457 of 1975. Application of Poisson's summation formula for the analytic evaluation of a class of lattice sums in arbitrary dimensions is extended to a more generalized class of sums. The resulting formulae are applicable to a variety of problems such as electronic-structure studies of crystalline solids, the onset of Bose-Einstein condensation in finite systems, the analysis of stability of quantized vortex arrays in extreme type-II superconductors and in rotating superfluid helium, plasma oscillations in an array of filamentary conductors, etc. They also provide an alternative approach for the determination of Madelung constants and other related sums that appear in the theory of cubic lattices.

The authors have endeavoured to study, by the formulation of a relativistic kinetic theory a system in which the particles have variable rest mass. Macroscopic quantities and their evolution equations have been directly linked to the action responsible for the variation of the rest mass of the particles. The dissipative character of this 'force' gives a slight modification to the Liouville theorem and the Boltzmann equation. A detailed study of reversible processes has shown that the macroscopic quantities keep the same form as for the usual perfect fluid (no bulk viscosity appears) though they are not conserved and can give rise to shearing and vortices. This pattern could serve as a model for the study of certain dissipative processes where mass is lost.

The author studies the Potts model with a general number of states. First, he discusses the situation in which the Landau theory leads to a first order transition, but which does show fixed points of the renormalization group. Here, there are many questions which need further clarification. Then he describes, rather pedagogically, the logic behind the application of dimensional regularization to critical phenomena. He argues that this is a particularly natural approach. This technique is then applied to the Potts model, for which the critical exponents are computed to O((6-d)²), when there is a fixed point. The one state results, which correspond to the percolation problem, are compared with other calculations and with numerical simulation.

Fisher's renormalization relations (1968) between the 'ideal' critical indices and the 'true' ones are proved to be symmetric. This property is shown to be independent of the sign of the ideal specific heat index.

The second quantization method is applied to classical many-particle systems. Statistical quantities such as free energy and time correlation functions are expressed in terms of creation and annihilation operators. The method is especially useful for the system in which the number of the composite molecules changes with time, e.g. the system including chemical reaction.

A stochastic theory of a diffusion-controlled reaction is developed with the emphasis on the many-body aspects which rigorous stochastic theories inevitably encounter. The field operator method developed in the previous paper (see ibid., vol.9, no.9, p.1465 (1976)) is extensively used in the analysis. The classical Smoluchowski theory is shown to be strictly valid in the long-time scale, and its relation to the Boltzmann equation is discussed.

It is shown that supersymmetry may be applied to spin systems. A simple algebra is proposed and various examples are discussed. It is argued that certain correlation functions must vanish on account of supersymmetry.

The coherent state for charged bosons is constructed, its properties are investigated and the corresponding classical model is discussed.

Non-dynamical conserved quantities are analysed for a class of field theories and their topological character is discussed. Some examples are given.

Self-interacting, self-avoiding polymer chains and rings are generated on the f.c.c. lattice by exact enumeration. The effect of the 'good' solvent is represented by a short-range repulsive interaction between pairs of polymer units. The partition function and the moments of the end-to-end distance and mass distributions are extrapolated to obtain the limiting behaviour at various temperatures. Difficulties encountered in determining the exponents and their temperature dependence are discussed.

The authors consider the nu -dimensional one-component classical plasma model in a spherical domain. They give a heuristic derivation of the H-stability bounds which enables one to understand their physical meaning as well as the nature of the ground state. Then, accurate estimations of the binding energy of various Wigner lattices are found theoretically as well as computed numerically. For nu -2 dimensions, the lowest energy configuration of particles corresponds to the triangular lattice. For nu =3 dimensions, the results agree with those of Coldwell-Horsfall and Maradudin (1960).

An estimate is made of the fraction of the diffuse gamma -ray background originating in discrete sources rather than intergalactic space. It is shown that 'normal' galaxies probably contribute about 4% of the gamma -ray flux above 100 MeV. Radio galaxies can produce the bulk of the 1-10 MeV background if there is proportionality between their gamma -ray and radio luminosities. Seyfert galaxies and clusters can account for most of the 100 MeV observations if the X-ray and gamma -ray emissions are by the inverse Compton mechanism.