Table of contents

It has been claimed that the solutions to the vector differential equation Del *a=ka with k constant, considered as a model for force-free magnetic fields in plasma physics and in astrophysics, exhibit several mathematical inconsistencies. This claim is repudiated.

In two recent letters Salingaros has put forward physical arguments to claim that solutions of curl B=kB cannot represent real magnetic fields. However, the first part of his assertions is only a restatement of the well known result that this equation cannot be satisfied everywhere over a non-vanishing magnetic field. However, it does not rule out the use of the equation locally over a restricted region of the field. The second part suffers from an incorrect assumption which invalidates his conclusion.

A procedure is described for constructing invariants, characteristics, and non-linear scaling fields for large-n differential renormalisation group (RG) equations. Explicit solutions are derived for the Ginsburg-Landau-Wilson and time-dependent Ginsburg-Landau models. The characteristics facilitate a natural geometric representation of the RG trajectories which, for the static case, takes the form of two-dimensional solution surfaces.

Certain aspects of the dynamics of an electrically charged mass points in the vacuum field of a magnetic dipole rotating at an angular velocity perpendicular to the dipole vector are discussed. These aspects concern the so-called acceleration boundary, i.e. the limits of the spatial region in which particles such as protons and electrons, for example, are accelerated to extremely high energies. The analytical expression found for the radius of the range of influence defined in this sense for the rotating magnetic dipole in units of the light radius is found to be proportional to the 2/3rd power of the magnetic dipole moment and to the 4/3rd power of the angular velocity.

The authors calculated the conformal anomaly number of the Blume-Capel quantum chain via finite-size corrections to the ground-state energy. In addition they compute, at the tricritical point, the lowest excitations of the spectrum to be compared with the theoretical description in terms of the irreducible representations of the corresponding Virasoro algebra.

Results based on the indirect fluctuating mean-field theory recently proposed by Suzuki (1986) are reported. The results indicate some of the important characteristics of this approximation method, which differs in significant ways from the standard mean-field approximation and recent improvements of it.

The author shows that the D_n series of a newly constructed set of models is completely integrable. In particular, the model described by D₄ is a non-critical extension of the three-state self-dual Potts model.

The author computes the local densities of the D_n series, whose integrability was shown in a previous letter (see ibid. vol.20, no.4, p.L217-20, 1987).

The possible orderings in a dilute antiferromagnetic q=3 Potts model on a triangular lattice are studied by applying to a hierarchical lattice an approximate renormalisation group scheme which incorporates and preserves all the symmetries of the Potts model. This scheme indicates that states with partial order, in which only a subset of the symmetries are broken, can exist in systems with these symmetries. They do not, however, occur in the model with nearest-neighbour pair interactions only. In this model the authors find solely the antiferromagnetically ordered states which occur for bond probabilities in excess of p₁, which is greater than p_c, the critical probability for bond percolation.

The authors present a simple argument showing that the surface energy density of a semi-infinite d-dimensional spin system has, in general, a leading thermal singularity of the same form, mod T-T_c mod ^{2- alpha} , as the bulk free energy. At bulk criticality energy-energy correlations decay as r^-2d parallel to the surface. These results hold for both free and fixed boundary spins, i.e. the 'ordinary' and 'extraordinary' transitions. The extraordinary critical behaviour of the magnetisation is the same as that of the energy density. They also confirm and generalise these results in two dimensions with an independent approach based on conformal invariance.

Chain-chain aggregation is studied on the square lattice by Monte Carlo simulations. The fractal dimension of the aggregates is found to be D=1.32+or-0.04. The results support the dynamic scaling relation n_s(t) approximately s^-2f(s/t^z) with z=0.72+or-0.02.

Strong and weak coupling series expansions are used to investigate the critical properties of the S=1 multispin coupling Ising chain in the presence of a transverse field. The order of the transition goes from second order to first order when more than three spins are coupled. According to the numerical results the two-spin and three-spin coupling models belong to the Ising and the four-state Potts universality classes, respectively. Thus the critical properties of the model do not depend on the value of the spin, even when three spins are coupled.

The discrete scale invariance of quasi-periodic systems leads to a scaling relationship Omega =L^-zf(L) between a characteristic dynamic variable Omega and length L, where f(L) is a periodic function of ln L. The dynamic exponent z can be calculated for diffusion, spin wave and phonon dynamics (where Omega is respectively Gamma , omega , omega ² where Gamma and omega are characteristic rates or frequencies respectively) by exploiting a crossover argument which results in z=d_f+t where d_f is the fractal dimension and t is the length scaling exponent for the conductance. The latter exponents and hence z, are calculated for a Fibonacci chain model ((d_f,t,z)=(1,1,2)) and, via an exact bond moving technique, for the Penrose lattice ((d_f,t,z)=(2,0,2)). The technique also provides the thermal exponent of the Heisenberg spin model on the two quasi-crystals ( nu =1, infinity , respectively).

The authors adapt the multiple hit method of noise reduction introduced earlier for diffusion-limited growth to Eden models A and C on the square lattice. The dynamic scaling is improved considerably: they obtain 0.33+or-0.015 (model A) for the dynamic exponent beta of surface roughening with relatively small computing effort. Results for model C support universality.

The state labelling problem arising in the reduction of the direct product of a p positive-row U(n) irreducible representation (h₁. . .h_p0) with a q negative-row one (0-h'_q. . .-h'₁) into a sum of mixed U(n) irreducible representations (k₁. . .k_p0-k'_q. . .-k'₁) is solved by using the complementarity between U(n) and U(p,q) within some positive discrete series irreducible representations of U(pn,qn). This complementarity allows analysis of the problem in terms of the group chain U(p,q) contains/implies U(p)*U(q) instead of U(n)*U(n) contains/implies U(n). For the most general SU(n) irreducible representations corresponding to p=q=n-1, the relevant group chain is therefore U(n-1,n-1) contains/implies U(n-1)*U(n-1). In such a case, the additional labels include those of an intermediate U(n-1) irreducible representation (h₁^s. . .h_n-1^s), as well as the additional labels solving the state labelling problems for the products (k₁. . .k_n-1)*(h₁^s. . .h_n-1^s) and (k'₁. . .k'_n-1)*(h₁^s. . .h_n-1^s) of U(n-1) irreducible representations. Hence the proposed solution reflects in a direct way the operation of King's branching rule for the chain U(n)*U(n) contains/implies U(n), supplemented, whenever necessary, with King's modification rule.

Unitary operator coherent states, as defined by Klauder (1963), Perelomov (1972) and Gilmore (1974), are considered for the harmonic series irreducible representations of U(p,q). Various properties of these states are reviewed, including their reproducing kernel and measure explicit forms, as well as the representation they lead to for the U(p,q) generators. Starting from a contraction of the su(p,q) Lie algebra, both Dyson and Holstein-Primakoff boson realisations of u(p,q) are obtained in terms of pq pairs of boson creation, annihilation operators and of the generators of a U(p)*U(q) intrinsic group. Such boson realisations are applied to determine the matrix elements of the U(p,q) generators between discrete bases classified according to the chain U(p,q) contains/implies U(p)*U(q). Finally, the isomorphism between so(4,2) and su(2,2) is employed to derive the corresponding properties of the SO(4,2) unitary operator coherent states for application in atomic physics.

In contrast to Lie algebras, Lie superalgebras contain typical and atypical irreducible representations. Character formulae for typical representations of classical Lie superalgebras have been known for a long time, but the structure of atypical representations remained unsolved. The author constructs the character formula for atypical representations of the special linear Lie superalgebras sl(m/n).

The differentiation of tensors in spherical coordinates is analysed for two distinct cases: when the result is a tensor of a rank higher, or lower, than the spherical tensor subjected to differentiation. General analytical expressions are obtained leading to equations applicable to particular problems of the theory of light scattering and absorption. The relationships obtained are used for defining spherical tensors of polarisability a_ij, hyperpolarisability B_ijk and dipole quadrupole polarisability A_i,jk for isolated molecules as well as the tensor of interaction-induced polarisability of a molecular sample taking into account intermolecular interactions in the approximation of the dipole-induced dipole (DID) model.

The eikonal solution of the radial Schrodinger equation for spherically symmetric potentials is treated by the quasilinearisation technique. The first iteration contains all the terms of the full expansion of the eikonal series obtained by Froman and Froman (1965) but with different coefficients. The strength of the centrifugal potential appropriately modified in each order of approximation is obtained stipulating the proper behaviour of the wavefunction at the origin. It leads to an iterative map which converges very fast due to quadratic convergence of quasilinearisation. The energy eigenvalues for the Coulomb potential are obtained by this technique.

A multivariable hypergeometric function which was studied recently by Niukkanen (1984) and Srivastava (1985), provides an interesting and useful unification of the generalised hypergeometric _pF_q function of one variable (with p numerator and q denominator parameters), Appell and Kampe de Feriet's hypergeometric functions of two variables, and Lauricella's hypergeometric functions of n variables, and also of many other classes of hypergeometric series which arise naturally in various physical and quantum chemical applications. Indeed, as already observed by Srivastava, this multivariable hypergeometric function is an obvious special case of the generalised Lauricella hypergeometric function of n variables, which was first introduced and studied systematically by Srivastava and Daoust (1969). By employing such useful connections of this function with much more general multiple hypergeometric functions studied in the literature rather systematically and widely, Srivastava presented several interesting and useful properties of this multivariable hypergeometric function, most of which did not appear in the work of Niukkanen. The object of this sequel to Srivastava's work is to derive a number of new Neumann expansions in series of Bessel functions for the multivariable hypergeometric function from substantially more general expansions involving, for example, multiple series with essentially arbitrary terms. Some interesting special cases of the Neumann expansions presented here are also indicated.

The author argues that it is possible to obtain the asymptotic expansion coefficients for an operator with torsion from the expansion coefficients for an operator without torsion. This is possible when both operators act on the same set of eigenfunctions in the same way, and therefore have the same spectrum. The author calculates in four dimensions (a₂(- Square Operator +B^k Del _k+X)) using the Schwinger-DeWitt ansatz. Then following the argument above he obtains (a₂(- Square Operator -B^k Del _k+X)) where tildes denote torsion. He compares results with a modest direct calculation using 'toy torsion' (totally antisymmetric and covariantly constant) and obtains agreement. He discusses the results and reviews the literature. He finds no other algorithms for (a₂(- Square Operator +B^k Del _k+X)) consistent with the limiting case (a₂(- Square Operator +B^k Del _k+X)). There are three appendices with useful coincidence limits and curvature tensor relations.

For pt.I see ibid., vol.20, no.4, p.857-74 (1987). The author obtains useful relations for calculating the a₍₁2)d/ asymptotic expansion coefficient for an associated heat kernel and elliptical operator of order greater than two in d dimensions. In the main relation he relates the a₍₁2)d/ coefficients for two commuting elliptical operators and the a₍₁2)d/ coefficient of their product. He uses this relation to obtain results (in four dimensions) for the a₂ coefficient, including total divergences, for the fourth-order operator ( Square Operator ²+E^k Del _k Square Operator +A^{(k epsilon} ) Del _k Del _epsilon +C^k Del _k+Z). He also argues that for higher-order operators, the coefficients of the loop divergences must be slightly altered. He discusses the literature.

A procedure is developed for assessing the accuracy of a truncated perturbation expansion, which leads to upper and lower bounds to some eigenvalue. Improved bounds, which require a lower bound to the overlap of the approximate and exact wavefunctions, are also considered and a lower bound to the overlap is derived. Two examples are presented, using model problems.

In view of the recently proposed entropic uncertainty relations several characterisations of 'maximal information' and 'minimal uncertainty' are compared and applied to position and momentum. It is argued that a sufficiently refined entropic uncertainty relation might require the general concept of POV observables as used in stochastic quantum mechanics.

For a simple potential, the author proposes a conversion of the standard differential radial Schrodinger equation into an infinite-dimensional matrix equation A phi =0 and describes its systematic non-numerical solution. The results may be characterised as infinite asymptotic expansions of the wavefunctions phi and/or of the secular determinant (an inverse 'Green function') det A. The inverse model-space dimension plays a role of the small perturbation parameter.

According to Jaynes (1963), statistical mechanics can be interpreted as a special type of statistical inference based on the principle of maximum entropy. The result of such an inference depends on the available information about a given physical system, but the principle itself does not decide what kind of information is essential and what is not. The well known Gibbs canonical state results from the principle when the statistical mean value of energy is supposed to be known. The maximum entropy inference employing higher-order moments of energy is examined from the point of view of the thermodynamic limit. It is shown that the contribution to the total entropy, arising from the extra information corresponding to the higher-order moments, is o(N), when N to infinity , N/V=constant (N is the number of particles and V is the volume). Consequently, from a purely thermodynamic point of view, this extra information is non-essential and can be neglected in the maximum entropy procedure. These conclusions also characterise the maximum entropy inference employing higher-order moments of (almost) arbitrary extensive quantities. One can say that the maximum entropy inference has a certain 'stability' property with respect to information corresponding to higher-order moments of extensive physical quantities.

The partition function for a quantum mechanical system can, at least formally, be expressed as a high-temperature series in terms of the zeta function of the system's Hamiltonian. This is done explicitly for a system of non-interacting harmonic oscillators. The Hamilton zeta function belongs to a class of zeta functions Sigma f(n_i)(n₁ omega ₁+n₂ omega ₂+. . .+n_N omega _N+a)^-s summed over non-negative integers n_i. Very little is known about these zeta functions, and an investigation of some of their properties is presented.

An analytic expression is obtained for the free energy of fermions bound in an anisotropic harmonic potential in the presence of an arbitrary magnetic field, at a finite temperature. The specific heat and the magnetic moment are readily calculated. The results consist of two parts, a steady part and an oscillatory one. The latter is similar to the well known de Haas-van Alphen oscillation, but persists in the absence of a magnetic field. Application of the results to the nuclear shell model and surface effects of solids are briefly discussed.

The authors present a systematic density expansion for calculating transport properties, return probabilities and spectral densities for lattice hopping models with diluted randomness. The method is applied to a disordered square lattice where the bonds carry conductances sigma ₁ and sigma ₂ in concentrations c and 1-c, respectively, and they calculate explicitly the O(c) and O(c²) contributions to the static conductivity (diffusion coefficient) and to the coefficient beta (c) of the long time tail in the velocity autocorrelation function (VACF) ( upsilon _x(0) upsilon _x(t)) approximately=- beta (c)t^-2. To O(c) the VACF is calculated for all times and shows negative correlation (cage effect) for all times.

A two-dimensional deterministic fractal model is proposed to imitate the geometric texture just below percolation threshold. The model is constructed for bond percolation via a rule of bond occupation on the square lattice. It show the typical percolation behaviour as a function of a parameter p (p is the bond concentration). The model describes the approach towards the threshold below criticality. The critical bond concentration p_c and the correlation length exponent nu are found. The scaling property of the cluster size distribution is studied below the threshold. The critical concentration and critical exponents are shown to agree with those derived from the regular model proposed for the geometric texture above the threshold. The scaling relations for the critical exponents are shown to be satisfied.

The individual principal radii of gyration are computed for unrestricted random walks in a large number of spatial dimensions. A low order expansion in one over the dimensionality yields useful information regarding the distribution and average of these measures of the extent and spatial anisotropy of high-dimensional random walks. Such an expansion may well prove useful in the study of the shapes of other random fractal objects.

The authors discuss various parallel algorithms for the enumeration of two-dimensional cluster properties in such problems as percolation and the Ising model. As an application, in a Monte Carlo simulation performed on the ICL distributed array processor (DAP), they re-examine the numerical evidence for a recent conjecture by Jug (1986) concerning percolation singularities in two dimensions.

The spectrum of the quantum Ising chain is studied in the finite-size scaling limit for periodic and antiperiodic boundary conditions. The finite-size corrections are computed for all energy gaps. The results of the model are used to test the Privman-Fisher universality hypothesis and conformal theory. At the critical point, it is shown how to build the spectrum from the irreducible representations of the Virasoro algebra. In addition, the author finds evidence for an unsuspected symmetry for systems both non-critical and of finite size.

The method of obtaining Backlund transformations proposed by Chern and Tenenblat (1986) fits completely the approach of obtaining Backlund transformations by prolongation techniques. For KdV, MKdV and sine-Gordon equations the only difference consists in the application of a non-linear representation of the prolongation algebra other than the usual one. This representation can be obtained by a coordinate transformation of the prolongation variable.

The author proposes in this comment an approximate form for the cluster size distribution function n_s. By means of this function, the author obtains not only the correct critical exponents of all moments of this distribution function, but the ratios of selected amplitudes as well, for all dimensions for which two critical exponents such as beta and gamma are known. The author calculates explicitly the ratio C₊/C_- of the second moments of n_s for dimensions d=2, 3, 4, 5 and 6- epsilon . In particular, for 6- epsilon , the result agrees to order epsilon ² with previous theoretical calculations, while for d=2 the author obtains the large value 194, in good agreement with earlier Monte Carlo results.

Hitherto the problem of random sequential packing has been treated in continuum space. In a previous paper we introduced random sequential packing into the square cellular structure, where squares with integer length are inserted at random without any overlap into the cells of a square substrate divided into square unit cells. To the packing we applied two methods A and B, which were not distinguished in the packing of continuum space. In A, contact between the packed squares is permitted, and in B such contact is forbidden. In the previous paper we independently obtained the packing fractions of A and B, and did not notice the relation between them. Here, however, we make clear the relation between the packing fractions of A and B and continuum space.