Table of contents

It is shown that a 6-j symbol for SO(2l+1) in which four primitive spinor representations (1/21/2. . .1/2) appear is directly related to an SO(3) rotation matrix possessing a rank of l+1/2 and characterised by the Euler angles (0, 1/2 pi , 0). An interpretation is given in terms of a rotation operator exp(1/2i pi T_y) acting in the combined spin and quasispin space of an atomic l shell, whose states mod T, M_T) are defined in a quasiparticle scheme in which four coupled spinors (1/21/2. . .1/2) are used.

The author introduces a new method of calculating critical amplitude ratios using series, which is both simple and powerful. This method, which gives estimates for the amplitude ratios that are neither biased by the values of the critical points nor by the critical exponents, is applied to several models. It is shown that this method produces results where no reliable estimates from series expansion exist. In particular one finds 0.025+or-0.001 for AT'/B² for the 3D Ising model and 220+or-10 for C⁺/C^- for the two-dimensional percolation model in agreement with, and with more accuracy than, values obtained by other methods.

The influence of the almost periodic excitation on the chaotic behaviour of the anharmonic oscillator is reported.

A model is presented for the interaction between an N-level atom and (N-1) modes. Under the exact resonance condition between the modes with a detuning parameter, the Hamiltonian becomes solvable. The evolution operator is calculated and the time development for the photon number operators and the occupation numbers are obtained. Multiphoton processes are discussed.

Decoupling of charge and spin density waves for a Peierls electron-phonon system with a half-filled band (polyacetylene) was achieved as a consequence of the vanishing of certain interaction terms in the fermion representation of electron field operators including spins due to the Pauli principle.

The Kauffman random networks of automata are studied on a simple cubic lattice by computer simulations. Each automaton follows random rules, depending on its six neighbours and fixed in time. A transition between the frozen and the chaotic phase is observed and the fractal dimension of the asymptotic actual damage at the phase transition is measured.

The authors study dilute spin systems with finite connectivity, i.e. Viana-Bray random bond systems, near zero temperature. The authors give a stationary free energy in terms of a global order parameter g together with the equation of motion for g and the stability matrix. For the case where replica symmetry is not broken, they diagonalise the stability matrix around the ansatz given by Mezard and Parisi (1985), and Kanter and Sompolinsky (1987), close to the point of percolation ( alpha =1) and for small admixtures of antiferromagnetic bonds. They find that the replica symmetric ansatz is unstable in the spin glass phase and in part of the ferromagnetic phase.

A renormalisation group method is presented to analyse the scaling structure of the growth probability in diffusion-limited aggregation. The recursion relation for the growth probability is derived under the renormalisation transformation. The growth probability assigned to each growth bond is represented by a random multiplicative process. The scaling of the highest growth probability is derived and the fractal dimension is found. A hierarchy of generalised dimensions D(q) is calculated to describe the growth probability. The partition of (q-1)D(q) into a density of singularities f(q) with singularity strength alpha (q) is made and the alpha -f spectrums are found.

The critical dynamics of a disordered Ising ferromagnetic chain with two coupling constants (J₁>or=J₂>0) is studied for Glauber dynamics. Using a domain wall argument the dynamical critical exponent z is found to be non-universal but independent of the disorder, namely z=1+J₁/J₂. The problem is formulated in terms of diffusion in a random medium. The diffusion is shown to be normal. Relationships with apparently very different diffusion problems, like the diffusion in hierarchically structured media, are established.

The method of thermo-field dynamics (TFD, the real time, finite-temperature quantum field theory) for quantum spin algebra has been employed to construct the coarse-grained functional action for the dynamics of the spin-¹/₂ Ising model in a transverse magnetic field. The applicability of the present approach to the unified treatment of the quantum and classical critical dynamics is pointed out.

The author shows explicitly that the representation of a finite-dimensional algebra associated with the largest eigenvalues of the four-site transfer matrix for the q-state Potts model is reducible at the first irrational Beraha q value. He shows this fits into a pattern of reducibility at the Beraha values so that the asymptotic dimensionality of representations for large lattice width n is qⁿ. For other q values the asymptotic dimensionality is 4ⁿ (q>4) or Kⁿ with K>q(0<q<4).

Two-dimensional quasiperiodic lattices with 12-fold rotational symmetry are constructed by projecting a four-dimensional hyperhexagonal lattice onto a two-dimensional subspace. The projection method in the present case is greatly simplified by representing the four-dimensional lattice as a complex two-dimensional lattice. By choosing windows with different forms one obtains several quasiperiodic tilings, which have a self-similarity. One of the tilings is quite similar to the pattern of the dodecagonal quasicrystal of a NiCr alloy.

The probability of finding a largest defect cluster of size n in a percolation network is calculated analytically using a new distribution function scaling equation. From this result and the stress (voltage) enhancement at the tip of the most critical defect in the network, the probability of failure in percolation models of breakdown is calculated. For defect fractions less than the percolation point, this distribution is found to be of the form exponential of an exponential. Numerical simulations on the two-dimensional random fuse network confirm the new distribution function and convincingly distinguish between it and the Weibull (1951) form most often used in the fitting of breakdown data.

A position space renormalisation group method is presented to analyse the scaling structure of the higher moments of the current distribution in random linear resistor networks at the percolation threshold. The recursion relation for the current distribution is derived under the renormalisation transformation. The current fraction assigned to each bond is represented by a random multiplicative process. An infinite set of exponents is calculated to describe the scaling properties of the current distribution.

By using Lie group theory, symmetries of the system of equations describing Newtonian static stars in radiative equilibrium are investigated. It turns out that the most general symmetries are those resulting from quasi-homologous transformations. These symmetries enforce a corresponding equation of state. Stromgren's (1936) homologous stars are a special case of this, more general, class of solutions.

This paper considers systems of factorisable equations for different forms of the factorisation kernel. In one dimension some of these kernels lead only to trivial factorisable equations. However, in higher dimensions they yield some novel classes of factorisable equations. A partial explicit classification of these new factorisable systems is given.

Fluctuations in dynamical systems may give rise naturally to stochastic maps which, in the absence of fluctuations, reduce to maps similar to those considered by Shilnikov (1970). Noise appears in a highly non-trivial manner. The effect of the noise on the chaotic dynamics is studied. It is shown that the different noise terms can give rise to radically different behaviour.

The authors give analytic solutions of the generalised albedo and Milne problems for the Uhlenbeck-Ornstein process, and discuss the relation between these and the original Wang-Uhlenbeck (1945) problem of determining the distribution of first-passage times.

A generating function formalism for solving the Vlasov equation for a static self-consistent potential is presented. The transition density, transition current density and linear response function are derived. As an example the response to l^pi =2⁺ and l^pi =2^- fields is studied for the case where the self-consistent potential is a harmonic oscillator.

Pseudo-symmetries were introduced by Sarlet and Cantrijn (1984) for time-dependent non-conservative systems. They are reconsidered in the context of general autonomous second-order systems, relying on the new approach to such systems which was presented by Sarlet et al. They further introduce the notion of adjoint symmetries of a second-order system, as being associated to invariant 1-forms, and show how they may be related to first integrals or to Lagrangians under appropriate circumstances. Their results enable them to clarify a rather unusual account of Noether's theorem which was recently given by Gordon (1986).

The authors extend previous results on the linear spectral problem introduced by Fordy and Kulish (1983). The odd-order isospectral flows admit both a KdV and MKdV type reduction. The non-linear terms are related to the curvature tensor of the corresponding Hermitian symmetric space. Their KdV equations are themselves reductions of known matrix KdV equations. They discuss the conserved densities and Hamiltonian structure associated with these equations.

The authors study the radiation intensity produced by a monoenergetic electron in circular motion as a function of the distance to the orbit. The analysis is carried out along two directions, namely in the radial direction, and along the tangent to the orbit. They find that, near the orbit, the intensity of radiation in the radial direction differs significantly from the value given in Schott's (1912) formula. Together with an increase of the radiation intensity as one moves closer to the orbit, the angular spread of the radiation decreases. As a result, a focusing effect of the radiation in the orbit plane is obtained. In particular, when radiation is detected almost touching the electron orbit, all the electron radiation tends to be concentrated into a line, giving rise to a very high density of energy per unit area. Except for points very close to the tangential point, they show that the intensity of radiation in the forward direction is given by Schwinger's (1949) formula, independent of the distance to the tangential point.

An application of the so-called Hill determinant method to the bound-state eigenvalue problem in the elementary quarkonium potential V(r)=-a/r+br is described, proved and illustrated for a few examples. An improvement of the method, which is based on an extended continued-fraction formulation of the eigenvalue condition, is also proposed.

A simple stable shooting method for the Schrodinger equation is described and is shown to work with various integrators, including the Numerov one. Quantum mechanical techniques are extended to apply to problems for which E is not a traditional eigenvalue and are shown to permit the extraction of useful information even from an unstable shooting method.

The exact solution of the Coulomb scattering problem in non-relativistic quantum mechanics has been known for many decades. Here its asymptotic form as h(cross) to 0, for constant energy and through all space, is discussed and shown to reproduce exactly the classical action surfaces and classical orbits for a statistical beam undergoing classical Coulomb scattering.

The authors study the semiclassical limit for the Delta ₃ statistic of integrable systems which have a homogeneous polynomial as potential. These systems possess a scale invariance which provides one with the energy dependence of the statistic using the general expressions given by Berry (1985). They also obtain the functional dependence of the universal part of the Delta ₃ statistic both for integrable and ergodic systems. They use a stationary phase approximation to evaluate the semiclassical limit of the level density when the dimension of the system is larger than one. The investigation of its validity leads to the introduction of a generalised perimeter term in the average level density. The fluctuating part of the semiclassical level density yields the semiclassical limit of the Delta ₃ statistic, which is compared numerically to results obtained for actual spectra. They find that the semiclassical approximation is excellent provided the perimeter term is taken into account exactly. They also study the dimensional dependence of Delta ₃. For energy levels around the Nth level above the ground state the position of the kink in the Delta ₃ statistic is essentially proportional to N^(d-1)d/.

A strategy of studying non-equilibrium phenomena in lattice field theory is discussed in the framework of Wigner's (1932) phase space approach. For a test model, an ensemble of anharmonic oscillators at weak coupling, an analytic non-perturbative solution is found. In this case a quasiclassical approach is shown to be valid for a wide range of parameters.

The Liouville equation for a wide class of boson Hamiltonians can be expressed in the form of partial differential equations for the quasidistributions Phi _N and Phi _A. Their coefficients are calculated by differentiation of the Hamiltonians.

The authors discuss the conformation of a linear flexible polymer chain in the vicinity of an attractive wall. They show that the structure in the adsorbing plane may be analysed in terms of a node avoiding Levy flight. This provides a complementary insight to the scaling approach.

Regular comb polymers with excluded volume interactions are studied and compared with regular star and linear polymers. First-order calculations in the excluded volume parameter, at the critical dimensionality d=4, yield the characteristic exponents of the macroscopic properties to order epsilon =4-d. The number of total configurations and the number of configurations with the backbone forming a ring are found. The evaluation of the mean end-to-end square distances of the backbone and the branches give an insight to the spatial distribution of the macromolecule.

The combinatorial factors required for performing finite lattice series expansions on the triangular lattice are discussed. Series expansions can be obtained from a linear combination of free energies for finite convex hexagonal regions. An algorithm for calculating the combinatorial weights is given. A number of results are presented, showing that many cancellations occur and that not all convex hexagons of less than a given size are required. For a given degree of computational complexity (as given by the size of the largest transfer matrix) the expansion using hexagons of less than a certain perimeter and exploiting the full triangular symmetry gives 50% more series terms than the earlier approach which adds second-neighbour bonds to the square lattice and only exploits square lattice symmetries.

The critical behaviour of an Ising model with competing first- and third-nearest-neighbour interactions ('biaxial next-nearest-neighbour Ising' or BNNNI model) on the square lattice is investigated by high- and low-temperature series.

The critical behaviour of an Ising model with competing first- and third-nearest-neighbour interactions ('biaxial next-nearest-neighbour Ising' or BNNNI model) on the square lattice is investigated by finite lattice methods. In the ferromagnetic region, the phase boundary is located with an accuracy at least equal to that of alternative methods. In the antiphase region, distinctive structure in finite lattice estimators is found over an extended temperature range. However, the nature of the transition remains unclear.

The authors consider inhomogeneous percolation models with density p_c+f(x) and examine the forms of f(x) which produce incipient structures. Taking f(x) approximately= mod x mod ^{- lambda} and assuming the existence of a correlation length exponent v for the homogeneous percolation model, they prove that in d=2, the borderline value of lambda is lambda _b=1/v. If lambda >1/v then, with probability one, there is no infinite cluster, while if lambda <1/v then, with positive probability, the origin is part of an infinite cluster. This result sheds some light on numerical and theoretical predictions of certain properties of incipient infinite clusters. Furthermore, for d>2, the models studied suggest what sort of 'incipient objects' should be examined in random surface models.

The standard percolation model (SPM) is generalised in order to take the diffusion of particles with short-range attractive interaction into account. The movement of particles depends on the temperature and the interaction energy through an external parameter tau (0<or= tau <or=1). Monte Carlo simulations in two dimensions show that the cluster size distribution for small clusters, the number of clusters per site and the percolation threshold substantially change due to the diffusion, depending continuously on tau . The fractal dimension, the correlation length and the percolation probability exponents are calculated. The study suggests that the proposed model and the SPM belong to the same universality class.

The authors investigate the dispersion of a dynamically neutral tracer flowing in a self-similar, hierarchical model of a porous medium. They consider a purely convective limit in which the time for tracer particles to traverse a given bond is strictly proportional to the inverse of the flux in the bond; the effect of molecular diffusion is neglected. Their hierarchical model contains an adjustable 'asymmetry' parameter which controls the width of the distribution of transit times for a tracer to traverse the multiplicity of paths in the network. They derive a functional recursion relation for this distribution, from which exact expressions for the moments can be obtained. They find two regimes of behaviour which are governed by the value of the asymmetry parameter. In one regime, the transit time distribution is characterised by a single timescale, so that a localised pulse of tracer spreads out at the same rate at which the pulse is convected downstream. In the second regime, the small fraction of tracer passing through the slowest bond of the network dominates in the moments of the distribution, leading to highly enhanced dispersion. The consequences of these results for the dispersion coefficient are discussed, both for self-similar and homogeneous systems.

The coupling between lattice modes and intramolecular modes in a molecular crystal is described by a mean-field model. The effect of this coupling is investigated semi-quantitatively and it is found that in certain cases a molecular crystal becomes unstable against both an ordered crystal structure and a completely disordered structure. This instability can be eliminated if the system is allowed to take on either a mesomorphic structure or a structure characterised by space-inhomogeneous order parameters.

An integrable generalisation of the XXZ Heisenberg model with arbitrary spin and with light plane type anisotropy is studied. Integral equations describing the thermodynamics of the model are found. The antiferromagnetic ground state, the excitation spectrum, the quantum numbers and scattering amplitudes of the excitations are determined.

For pt.I see ibid., vol.20, p.1565 (1987). The thermodynamics of the integrable generalisation of the XXZ Heisenberg model with arbitrary spin is studied. The low- and high-temperature heat capacities and magnetic susceptibility are computed.

Renormalisation group recursion relations are obtained to two-loop order for an n-component Landau-Ginzburg-Wilson model containing both hypercubic anisotropy and quenched random impurities. The fixed points are enumerated and their stability properties analysed, using the epsilon expansion about four spatial dimensions to second order. The existence of a new fixed point corresponding to an anisotropic impure system is reported. It is found that both isotropic and anisotropic pure systems are unstable to the addition of random impurities for n<n_R identical to 4-4 epsilon +O( epsilon ²), while both pure and random isotropic systems are unstable to anisotropy for n>n_C identical to 4-2 epsilon +O( epsilon ²). Singularities in the epsilon expansions for the exponents of the random fixed points are shown to be associated with the onset of focal behaviour (the acquisition of complex eigenvalues). These, together with the slow convergence of the expansions leads one to doubt the reliability of extrapolations to three dimensions.

The group of Lie symmetries of the Khokhlov-Zabolotskaya equation in two and three space dimensions is given.

The feasibility of recovering the fractal dimension of self-similar fractals from the back-scattering measurements is demonstrated. For this purpose, the well known Sierpinski gasket is modelled as an ensemble of lossless dielectric beads, and the scattered field computations are carried out using the coupled dipole approximation procedure.

The authors formulate the Ising model with competing first-, second- and third-neighbour interactions along the branches of a Cayley tree, in the infinite coordination limit, as a three-dimensional non-linear mapping problem. The phase diagrams display a Lifshitz point and many modulated phases. The introduction of third-neighbour interactions affects mainly the low-temperature region of the phase diagrams.

Lacunarity as a criterion of universality in phase transitions on Sierpinski carpets is studied. The approximate theoretical and numerical results suggest that systems of different lacunarity may belong to a common universality class. There seems to be a lack of solutions to the complete set of universality criteria on Sierpinski carpets.