H. Moraal

Kinetic theory of rotational Raman line widths

Abstract Using the generalization of the Waldmann-Snider equation to nondiagonal (in internal-energy eigenstates) distribution-function density matrices due to Tip, expressions for the line widths of the rotational Raman lines of homonuclear diatomic molecules are derived for two types of non-spherical potential, viz. , the single-P 2 and the quadrupole-quadrupole interaction. Model calculations for the H 2 modifications are presented and are shown to be in qualitative agreement with experiment. Quantitative agreement is obtained for the higher-frequency Raman lines.

Graph-theoretical characterization of invertible cellular automata

Abstract A cellular automaton is defined to be invertible if its induced global rule is surjective on the set of all finite configurations. It is shown that all one-dimensional invertible cellular automata are obtainable from invertible nearest-pair ones, i.e., cellular automata given by a rule matrix. For these, an algorithm is proposed, which runs in a time which is a polynomial function of the number of letters of the alphabet. The result is expressed in a graph-theoretical way: the cellular automaton is invertible if and only if two graphs constructed by the algorithm have no edges in common. It follows that if the number of letters in the alphabet is a prime, the only invertible cellular automata are those for which every row or column of the rule matrix is a permutation of the alphabet. For composite alphabet cardinalities, several methods (analytical and computational ones) to obtain more complicated invertible cellular automata are proposed and used to construct examples. All four-letter invertible automata are constructed.

Derivation and spectrum of a generalized Kirkwood-Salsburg operator

A generalized Kirkwood-Salsburg equation for a classical system of particles with an arbitrary stable interaction in an external potential σ(x) such that exp (- βσ) is integrable is derived. It is shown that the spectrum of the generalized Kirkwood-Salsburg operator consists of the inverses of the zeros of the grand canonical partition function.

Ising spin systems on Cayley tree-like lattices: Spontaneous magnetization and correlation functions far from the boundary

A study of an Ising model on a variety of Cayley tree-like lattices with its boundary in a magnetic field is made. It is proved rigorously that such models exhibit phase transitions only in the limit of zero field. The spontaneous magnetization far from the boundary is shown to behave as (Tinc − T)12 near Tc, and is calculated explicitly for a number of lattices with low branching ratios. Two spin-spin correlation functions (far from the boundary) are also calculated explicitly and are generally shown to be continuous with discontinuous first derivatives at Tc. A possible experiment for measuring this Cayley tree behaviour is discussed.

Spectral properties of the Kirkwood-Salsburg operator

A mathematically precise definition of the “infinite-volume” Kirkwood-Salsburg operator as a bounded linear operator in a Banach space is given. It is shown that this operator has a bounded inverse for a bounded, stable and regular pair potential. These facts are exploited to establish the connection between the Kirkwood-Salsburg and the Mayer-Montroll equations and to give a classification of the spectra and resolvents of the Kirkwood-Salsburg operator and of its inverse. The theorems proved in this article constitute a framework for the derivation of any more precise results for special potentials.

Statistical mechanics of quasi-one-dimensional systems

A definition of a quasi-one-dimensional system as a generalized Cayley or Husimi tree with a nonzero surface to bulk ratio in the thermodynamic limit is given. Sufficient conditions for the existence of the thermodynamic limit of the free energy for such a system are derived and a thorough discussion of the thermodynamic limit properties of the one-particle distribution functions is given. These results are made more precise for the case of systems with Hamiltonians which are invariant under a special type of measure-preserving group of transformations, in particular for the d-dimensional rotation group. For this latter case, the phase transitions which can occur in quasi-one-dimensional systems upon application of small external fields are studied in some detail. A number of completely solved examples is given to illustrate the general theory. These include the classical Heisenberg model on a Cayley tree and generalizations thereof.

Kinetic-theory collision cross sections for the quadrupole-quadrupole interaction

Abstract Expressions which are exact in the distorted-wave Born approximation are given for two kinetic-theory collision cross sections, viz. , σ( 0010 0010 ) and σ( 0200 0200 ), for the case of the quadrupole-quadrupole interaction. Model calculations are presented for the hydrogen isotopes and their modifications, for nitrogen and for carbon monoxide. From the calculated quadrupole-quadrupole contributions and the experimental data on HD, the total cross sections σ( 0200 0200 ) for the homonuclear H 2 and D 2 modifications are calculated, resulting in good agreement with experiment. Sound-absorption and shear-viscosity Senftleben-Beenakker effect data are shown to be consistent for the H 2 and D 2 modifications.

High-temperature expansions for classical spin models

It is shown that the transitivity of the symmetry group of a classical spin model leads to a high-temperature expansion of the free energy of such a model in terms of multiply connected graphs. For models with a completely permissible symmetry group (e.g., for Ising and Potts models), these graphs fall into equivalence classes. Such classes are classified according to their cyclomatic index. Several different methods for obtaining the high-temperature series in a practical way are also discussed. The extension of these results to models with an infinite symmetry group is indicated.

High-temperature expansions for classical spin models: III. Series and phase diagrams for the square lattice

Abstract The concepts of duality and broken symmetry are discussed briefly. It is shown how generalized colouring polynomials can be used to calculate the terms in a high-temperature expansion effectively. These results and those from two previous papers are used to obtain high-temperature series for a great variety of models. These are given explicitly for all models with two Boltzmann factors and for the Ashkin-Teller model. Many phase diagrams obtained from the series by Pade analysis are presented and their salient features discussed. A “principle” suggested by these phase diagrams is applied to the dilute Potts model.

Statistical mechanics of linear molecules II

It is shown how the Percus-Yevick and convolution-hypernetted-chain approximations for dense fluids of linear molecules can be brought into practically applicable forms by using cartesian tensor formalism and pertubation theory. The possibility of expanding the intermolecular potential and the correlation functions is discussed in detail. Also, exact reductions for the Ornstein-Zernike relation in real space and for the Percus-Yevick equation are given. Some other approaches to the statistical mechanics of linear molecules are critically discussed. It is shown in particular, that the mean spherical model is an inconsistent type of perturbation theory. The necessary cartesian tensor formulae are given in appendices as well as formally exact expressions for the second virial coefficient and for the Fourier-transformed Ornstein-Zernike relation.