Table of contents

A solution is found for the equations governing the space-time transformation in six-dimensional special relativity. The energy required to turn the time vector of a particle is calculated.

The frequently quoted expression for the 'mean-position' operator of L.L. Foldy and S.A. Wouthuysen (1950) is shown to be incorrect. A revised expression is given.

A formalism for discussing the notion of stochastic resonance in dynamical systems is outlined and applied to a simple two-state model.

Diffusion laws and diffusion coefficients are given for random walks on lattices whose scattering probabilities differ from site to site. Models are considered in which the direction of the particle after scattering depends on its direction before scattering.

The free energies of directed lattice animals in good and theta -solvents, and the free energy of directed percolation are found by the use of a simple Flory-type approximation, which accounts for the inherent anisotropy of these systems. From these free energies, the authors obtain the upper critical dimension below which mean-field theory breaks down. They also calculate closed-form, dimension-dependent expressions for the parallel and perpendicular correlation length exponents, which characterise the asymptotic cluster shapes. These exponents are in excellent agreement with existing numerical data.

Studies the problem of directed site animals on the square, triangular and hexagonal lattices. Closed form expressions are proposed for A(s), the number of animals of size s, on the square and triangular lattices. These expressions have been checked for s<or=33 and s<or=10 for the square and triangular lattices respectively by explicit enumeration. They imply that A(s) varies as lambda ² s^-0 for large s, where lambda =3 for the square lattice, and lambda =4 for the triangular, and theta =¹/₂ for both. For the hexagonal lattice, A(s) is found for s<or=48. The results are consistent with lambda =2.0252+or-0.0005 and theta =¹/₂.

Introduces the problem of directed lattice animals and shows that they are highly anisotropic with two different correlation lengths xi /sub /// and xi _{perpendicular to} , parallel and perpendicular to the privileged direction. A field theory is introduced and the critical exponents calculated to first order in epsilon =d_c-d where the upper critical dimension, d_c, is 7.

Within a renormalisation group strategy, different 'rescaled' mean-field approximations for the magnetisation are combined. Good qualitative estimates are obtained for the critical properties of classical and quantum spin systems. A particularly simple realisation of the method yields recursions for arbitrary dimensionality d. The critical couplings are accurate for all d, and have the correct asymptotic behaviour for d approaching infinity.

A proper lower bound approximation to the true free energy per site of an Ising model with infinite-range interaction is obtained by the Migdal-Kadanoff real space recursion formulae. The results for the critical behaviour are confirmed by mean field analysis.

Considers a semi-infinite Gaussian model with spatially inhomogeneous short-range couplings that depend on the distance z from the surface. Far from the surface the coupling constants vary as K(z)=K_B-Az^-y. For y=>2 the pair correlation function of the surface spins decays as a power law with a universal exponent eta /sub ///=2 at the bulk critical temperature. For y=2, eta /sub /// is non-universal, and for y<2 there is an anomalous exponential decay.

The authors suggest that the concept of fractal dimensionality provides a useful characterisation of the configurational properties of a single polymer. From numerical studies of long polymers traced out by self-avoiding walks on a planar lattice, they find that the fractal dimensionality is well defined and has a constant value for most scales of length of the chain. Renormalisation group theory provides a theoretical basis for the concept of fractal dimensionality in polymers.

A new method based on the concept of fractal dimensionality is used to study the problem of self-avoiding walks in a four-dimensional lattice. The authors find from Monte Carlo simulations that the confluent logarithmic exponent related to the end-to-end distance is ¹/₈+or-0.01, in excellent agreement with the prediction derived from the n to 0 vector model.

A new approach is developed to self-avoiding walks as a critical phenomenon. The approach is based on a simple assumption made on the step-step correlation function. This leads to a single scaling field related to the degree of polymerisation. As a result, all critical exponents are related to a single exponent. Moreover, the order parameter and critical fields have a clear physical meaning. The present interpretation of self-avoiding walks as critical suggests an additional meaning of the exponent v in other critical systems.

Derives the expression of the conformally covariant energy-momentum tensor for spin 2, using the author's previous analysis for spins 0, 1 and 1/2 (see ibid., vol.14, no.5, p.L125-7, 1981).

The quantum mechanical analogue of the classical integrable system, originally founded by Goryachev and Chaplygin (1900), is considered in detail. The problem is formulated in terms of the Euclid E(3) group. The Euler-Poisson equations of motion and their integrals are derived. The determination of the spectra of the integrals of motion is equivalent to construction of the special basis of representation of E(3). The eigenvalue problem admits separation in new variables, which are closely connected with two boson creation-annihilation operators. They are the same as in the Majorana representation of the Lorentz group. In the new variables the quantum mechanical equations of motion look similar to the classical ones. The constants of motion are determined by the spectral problems for the two Jacobi-type tridiagonal infinite matrices. Some numerical results are given.

It is known that the density operators rho of a quantal system are grouped into classes of equally mixed ones. The class to which rho belongs is its so-called mixing character. The set of classes is known to be a lattice. The Ruch and Mead (1976) principle of increasing mixing character in complete measurement is extended to general observables. It is shown that the principle of strictly increasing mixing character holds in general incompatible measurement, and some consequences are discussed. A sufficient condition for strictly mixing homomorphic functionals, i.e. for those which preserve the 'strictly larger' relation in the lattice, is obtained. Density-operator-dependent strengthening of the natural pre-order in the set of general observables is investigated.

An analytical formalism is developed to deal with the occurrence of jump discontinuities in the g_{mu nu} or their derivatives across a hypersurface Sigma . It is shown that the equations of relativity remain meaningful at Sigma , even when Sigma does not inherit a unique intrinsic geometry, so that the g_{mu nu} are discontinuous across Sigma in natural coordinates. The spherically symmetric surface layer at the Schwarzschild-Minkowski junction is used to illustrate these techniques, and to establish rigorously the existence of C⁰ solutions of the Einstein equations and the conservation equations. The possible validity of relativity at the microscopic level is examined, and it is concluded that, if relativity is valid at the microscopic level, then it is likely that the g_{mu nu} are not globally continuously differentiable.

Examines perfect fluid solutions in general relativity. The pressure isotropy relation possesses a discrete symmetry in the derivatives of the metric functions. A generalisation of Buchdahl's original transformation allows new physically reasonable solutions to be obtained from known solutions. An example is given. Previously noticed first by Buchdahl (1958), and later by Glass and Goldman, this symmetry was thought to lead only to unphysical solutions.

The method due to W. Israel (1976). for treating surface layers is applied to determine the gravitational field due to a rotating cylindrical shell. The interior space-time is flat while the exterior metric can be one of three types. For a given value of the stress in the cylinder, the type of the exterior metric depends on the mass per unit coordinate length of the cylinder.

The author demonstrates a very concise method to determine the force on a charged particle on the theta =0 axis of a black hole. The relevance of this result to the emission of charged particles by mini black holes is discussed.

Determines the electric and magnetic self-field of a point charge at rest on the symmetry axis of the Kerr space-time in the coordinate system in which the metric describes locally a constant, static and homogeneous gravitational field. The result differs from that of a uniformly accelerated point charge in Minkowski space-time because of the influence of global boundary conditions. It follows that one can determine the induced self-force on the point charge.

Shows that the 'non-conservative' gravitational theories of the type considered by Rastall (1972), Smalley (1974) and Malin (1975) do have conservation laws and are formally equivalent to general relativity. The supposed 'difference' between these theories and general relativity is shown to lie entirely in the question of whether the stress-energy tensor of matter fields is conserved in special relativity (flat space-time). If one chooses to interpret these theories as non-conservative, then the coefficient lambda in these theories, which measures the degree to which stress-energy is not conserved, can be constrained to values mod lambda mod <or=10^-15 by considering the propagation of sound in a fluid.

The static and dynamic force-force correlation function for the one-dimensional Toda lattice is calculated in the soliton-gas approximation. The dispersion and width of the soliton and phonon resonances are discussed in detail. Comparison with other calculations and molecular dynamics results shows that this approximation gives a reasonable description of most of the features in the correlations.

The population dynamics model known as the Malthus-Verhulst model is shown to exhibit, for particular values of the rate parameters, a new regime characterised by a relaxation time which is exponentially small in inverse of the competition rate between individuals of the species. This result is obtained by means of a representation of the time evolution of the system in terms of a stochastic process which describes the Brownian motion of a particle under the action of a unidimensional periodic potential. The long-time behaviour of the first moment of the population is calculated by a suitable extrapolation of a perturbative expansion around the Wiener process result.

Obtains upper bounds to the critical probability for percolation in a random network made of oriented diodes and resistors. It is shown that for the square lattice p_c<0.3700 and for the simple cubic lattice p_c<0.2417.

For pt.I see ibid., vol.15, no.6, p.1849-58 (1982). Studies the behaviour of the average velocity of fluid flow in a random network made of oriented diodes and resistors, as a function of the direction of the fluid flow and the concentration of resistors. It is shown that there is more than one critical value of the concentration across which there is a qualitative change in the behaviour of the wetting velocity. Also, it is shown that the directed percolation problem is related to the problem of determining the direction-dependent wetting velocity in the undirected percolation problem.

Presents several results for the directed percolation problem. The shape of the percolation cone is discussed, showing how the operating angle behaves near threshold, and explaining the orientation dependence of the correlation exponents found by Domany and Kinzel (1981). The Cayley tree model is used to discuss the high-dimensionality limit. The conductivity problem on a directed network is introduced, and the high-dimensionality values for the conductivity exponents s and tau are found.

A Potts correlated polychromatic percolation is studied. The clusters are made of sites corresponding to a given value of the q-state Potts variables, connected by bonds being active with probability p_B. To treat this problem an s-state Potts Hamiltonian diluted with q-state Potts variables (instead of lattice gas variables) is introduced to which the the Migdal-Kadanoff renormalisation group is applied. It is found for a particular choice of p_B=1-e^-K (where K is the Potts coupling constant divided by the Boltzmann factor) that these clusters, called droplets diverge at the Potts critical point with Potts exponents.

A phase diagram of the three-state Potts model on a triangular lattice with two- and three-body interactions is obtained by computer simulations. The interactions allow antiferromagnetic as well as ferromagnetic phase. The ferromagnetic transition changes from continuous to first order on varying the coupling constants. The antiferromagnetic transition and the transition along the ferro- and antiferromagnetic coexistence line are first order. The resulting phase diagram agrees with that obtained by the low temperature series by Enting and Wu (1982).

The authors have studied numerically and analytically the discrete phi ⁴ model defined by Bak and Pokrovsky (see Phys. Rev. Lett., vol.47, no.13, p.958-61, 1981). The model may describe Peierls systems, structural instabilities, metal-insulator transitions, etc., in solid state physics. The phi ⁴ theory can be formulated as an area-preserving two-dimensional mapping. This mapping exhibits infinite series of period-doubling bifurcation leading to chaos. The bifurcations are characterised by universal numbers delta =8.72109 ... and alpha =4.0180 ..., which appear to be identical to those found by Bountis (1981) for the Henon mapping, but different from the Feigenbaum numbers for dissipative systems. In addition, novel features arise because of marginally stable fixed points and the splitting of one 2-cycle orbit into two 2-cycle orbits.

The question of to what extent the conventional interpolation function for the Gaussian integral G(w) pre-determines the standard meromorphic structure within the dimensional regularisation, is examined for simplest integrals of perturbation theory. Although it is possible with some generalisation of G(w), to obtain a meromorphic structure for a simplest one-loop integral, it is not sufficient to ensure that higher-order diagrams also have meromorphic structure. An explicit example of such a case is found. All generalisations of G(w) considered lead to a violation of the gauge invariance of the theory.

For pt.V see Phys. Rev. & vol.25, no.5, p.2801-11 (1982). The Gell-Mann-Low style conformational space renormalisation method for polymers is generalised to describe the crossover between the random walk and self-avoiding walk limits, i.e. to describe the excluded volume dependence. Explicit calculations are provided to order epsilon =4-d (d the spatial dimensionality) for the full end-to-end vector distribution function, the coherent elastic scattering function, the second virial coefficients and (R²) and (S²). The crossover functions are required therefore to exhibit the correct asymptotic limits of both the random and self-avoiding walks. The theory demonstrates that the latter choice implies that the expansion factors, alpha ² and alpha _s², for the mean square end-to-end vector (R²) and radius of gyration (S²), respectively, are not universal functions of the single scaling variable describing the strength of the excluded volume interactions. Nevertheless, much of the available experimental data on long chain polymers appears to involve small renormalised dimensionless excluded volume, and therefore alpha ² and alpha _s² are approximately universal quantities. Comparisons between our theoretical predictions and experimental data on the second virial coefficient and alpha _s² show good agreement.

The authors consider j-mers A_i reacting irreversibly according to the scheme A_j+A_k to A_j+k. The kinetic equations for the concentration of A_j are examined, and particularly their behaviour near gelation. Only the case R_jk=j^alpha k^alpha (0<or= alpha <or=1) is considered; this is a variation of the usual Flory-Stockmayer model to take excluded volume and cyclisation effects roughly into account. The effect on certain critical exponents is estimated.

The diffraction of plane, cylindrical and spherical waves by a wedge is considered. Particular emphasis is placed on finding accurate and simple asymptotic field expansions which are valid in the transition regions as well as in the far field. The accuracy of previous diffraction coefficients for cylindrical and spherical waves is discussed and a more accurate derivation is presented. A new simple rational approximation of the resulting Fresnel integral is given and its special character is demonstrated.

It is shown that the power expansion of the Gibbs potential of the SK model up to second order in the exchange couplings leads to the TAP equation. This result remains valid for the general (including a ferromagnetic exchange) SK model. Theorems of power expansions and resolvent techniques are employed to solve the convergence problem. The convergence condition is presented for the whole temperature range and for general distributions of the local magnetisations.

The shape and size of a Kerr ergosphere and their reversible transformations are studied systematically. It is proved that due to the ergosphere's total angular momentum, a bulge always forms on its outer boundary either on the axis of rotational symmetry or off it. During the reversible increase of the angular momentum, because of the injection of particles into the ergosphere, the bulge can either approach the symmetry plane or recede from it, while its angular separation from the positive rotation axis increases continuously, approaching the value of 30 degrees for the extreme Kerr ergosphere. The reversible changes of the ergosphere's linear dimensions parallel to the rotation axis and perpendicular to it are studied and the corresponding ranges of their permitted values are specified.

Enumeration of lattice animals embeddable in a square lattice has been extended to 24 cell animals by Redelmeier (1981). It is shown that the number of animals per site a_n approximately 0.317(4.0626)ⁿn^-1 exp(-0.465 n^-0.87) to a high degree of accuracy.