Table of contents

Using Hamiltonian path integral the author obtains the spurious potential term that arise in the action when a path integral is transformed to general n-dimensional coordinates.

The authors study the critical behaviour (static and dynamic) of the Ising model, defined on a fractal Sierpinski carpet. Using Monte Carlo calculations estimates for various critical exponents are obtained from a finite-size scaling analysis. The results are compared with the available predictions for the dynamical critical exponent z. They find the dynamic exponent z near 2.2 to be lower than predicted.

An explicit formula for the spatial dependence of the pair correlation function in an Ising strip with antiperiodic boundary conditions at the bulk critical temperature is given. It follows from the conformal-invariance approach of Belevin et al. (1986), and Cardy (1986).

The dynamical behavior of the two-dimensional Ising model at the site percolation threshold on L*L lattices with L<or=64 is studied by Monte Carlo simulations. The results are consistent with the recently proposed singular dynamic scaling. The new dynamical exponents are estimated to be A_s=0.62+or-0.12 and B_s=3.92+or-0.89. Comparison with bond dilution strongly suggests that universality is violated for this model.

Studies the influence of a strong asymmetry of the synaptic strengths on the behavior of a neural network which works as an associative memory. The author finds that the asymmetry in the synaptic strengths may be crucial for the process of learning.

Discusses how one can combine a bichromatic formulation of the percolation problem with the 'pure' continuum model of water without broken H bonds. The author proposes a method for comparing the computer energy distribution with spectroscopic contours and demonstrate them to be inconsistent.

The authors present a novel transfer matrix algorithm to study convection-biased diffusion on a lattice. Average and mean-square exit time on two examples are discussed, one with uniform permeability and another with a log-uniform distribution. Different regimes (pure convection, pure diffusion, intermediate stage) are clearly identified.

The authors study diffusion on the infinite percolation clusters above the percolation threshold, p>p_c, under the influence of a constant bias field E in topological space ('topological bias'). They find that above a critical bias field E_c(p) diffusion is anomalous and non-universal: the diffusion exponent d_w^l increases with E as d_w^l=A(p) mod ln((1-E)/(1+E)) mod , while A(p) decreases monotonically with concentration p. This intrinsic anomalous behaviour is supported in a wide range of concentrations p>p_c by extensive numerical simulations using the exact enumeration method.

For a contractive matrix transformation A:R² to R², A=a_ij, a_ij in R, the set of points A=((+or-1+or-A+or-A²+or-A³+or- . . .)e for all sequences of + and -), where e=(1,0)^T, is a unique attractor of generally fractional Hausdorff dimension. The Mandelbrot set associated with this system is defined as D=(a_ij in R⁴:A is contractive, A is disconnected). The structure of D and its boundary is investigated. Computer approximations of various sections of D are presented with a discussion of the algorithm and principles involved.

The decomposition of the basic spin irreps of the special orthogonal group SO_2k into irreps of SO_D-2*K, where D is the spacetime dimension of the extended D-dimensional Poincare supersymmetry and K is the appropriate automorphism group, is determined. General results for D<or=10 capable of extending decompositions of irreps giving rise to helicities greater than +or-2 are given together with a general method for D>10. Several new branching rules for subgroups of SO_2k are developed. A number of specific results are tabulated.

The operator equation for the calculation of invariance operators (the weak symmetry operators) connecting the eigenfunctions of one degenerate eigenvalue of a linear operator is formulated. The general properties of invariance operators are described. As an example, the Schrodinger equation with some potentials of interest is considered.

The existence of the thermodynamic limit for the spectrum of the Lyapunov characteristic exponents is numerically investigated for the Fermi-Pasta-Ulam beta model (1955). The authors show that the shape of the spectrum for energy density well above the equipartition threshold epsilon _c allows the Kolmogorov Sinai entropy to be expressed simply in terms of the maximum exponent lambda _max. The presence of a power-law behaviour epsilon ^beta is investigated. The analogies with similar results obtained from the dynamics of symplectic random matrices seem to indicate the possibility of interpreting chaotic dynamics in terms of some 'universal' properties.

A new method to discretise Schrodinger equations on a mesh is described. This method is based on an accurate approximation of a variational calculation. The regularly spaced mesh and meshes based on the zeros of orthogonal polynomials are studied in detail. It is shown that with each type of mesh is associated a particular kinetic energy operator and an optimal formula for its discretized form. The applications to some simple potential problems show that the method is very accurate as well as very simple. Applications to many-body problems indicate that the accuracy of the results is improved by an order of magnitude with respect to conventional mesh calculations.

A three-level model is used for an atom in which two transitions are driven by two laser beams, having constant amplitudes and detunings that are proportional to the time. Assuming that any two of the levels are unoccupied at large negative times, the authors obtain simple formulae for the occupation probabilities at large positive times. Previous results and the early calculation of Landau and Zerner (1932) are special cases of these general formulae.

A recently developed procedure for solving the stationary Schrodinger equation in matrix representation form is proved to be useful in calculating quasibound energy levels. Results for a simple quantum mechanical model with potential energy function V=-r^-1+ lambda r( lambda <0) are shown to be very accurate.

Considers a single fermion moving in a screened Coulomb potential having the form V(r)=g(-1/r), where g is monotone increasing and concave. The method of potential envelopes is applied to this problem and it is shown that close approximations for the relativistic eigenvalues are given by the simple expression E_nj^P=min_{u in (0,1)}(D(u)-uD'(u)+V(-1/D'(u))) where D(u)=D_nj^P(u) are the known exact eigenvalues for the hydrogenic atom with potential -u/r. Specific results for the first four eigenvalues of the atoms Z=14(5)84 are compared to corresponding accurate values found numerically by the use of Dirac spinor orbits.

The authors study the existence of a stable ground state for scalar QED in four dimensions within a variational approach. The results are similar to those for lambda phi ⁴: in its regularised version there is a metastable ground state which becomes stable when the cut-off is removed. However, contrary to what happens for lambda phi ⁴ the bare coupling constant lambda _B does not now have to be negative. The theory is interacting and the renormalised coupling constant is negative, lambda <0.

Two-dimensional electrodynamics based on the spacetime S¹(X)R is formulated using Fourier series and Dirac's constraint theory. The role of the background electric field F as a dynamical variable is established. The authors show that the background field is unstable against a class of charged pair creation processes involving the global topology of S¹ provided mod F mod >1/2e, where e is the unit of electric charge. The results do not require any discussion of the Coulomb Green function or of the role of charges on spatial boundaries.

An apparent discrepancy is noted between Fujikawa's path integral analysis (1982) of anomalies and the existence of a family of distinct solutions to the Thirring model. It is proposed that this family of distinct solutions may be obtained in the path integral formalism by employing a family of distinct measures for the fermion functional integration. The new measures are constructed by means of a two-dimensional analogue of the Pauli-Gursey-Pursey transformation, and the anomalies are evaluated explicitly for those measures which are close to the usual one.

Using a model quantum clock, the author shows how the velocity of a relativistic particle can be measured. The results are used to analyse the long-standing problem of the velocity of a Dirac particle.

In addition to the fact that the spin-glass system is situated within a stationary point of the energy surface, energetic correlations have been taken into account between groups of p spins, which lead to restrictions concerning the simultaneous internal field values on different sites. The physical implications of this fact are also analysed.

An extension of the model of diffusion-limited cluster-cluster aggregation, which emphasises sticking by tips, is numerically investigated. It is argued that the model can be physically justified in the case of polarisable clusters. A quantitative agreement is found with a recent experiment of aggregation of silica microspheres in two dimensions. The particle-cluster counterpart of the model is also studied.

The authors have investigated the structure and growth kinetics of clusters formed in dilute solutions from two kinds of polyfunctional monomers. This was done by computer simulations within the framework of the cluster-cluster aggregation model with mass-independent cluster diffusion constants. The systems considered were of the type A₂+B_f and A_f+B_f, where A₂, A_f and B_f are different monomers. The monomer functionality f is equal to the coordination number of the lattice used in the simulations. The results of simulations obtained at different relative monomer concentrations are described and several novel effects are reported. Large clusters always have a fractal-like geometry but their effective fractal dimensionality, D, appears to depend on the composition of the mixture. Only in a restricted composition range does the aggregation process lead to a single large cluster. There, the cluster size distribution function N_s(t) can be described by the scaling form s^-2f(s/t^z). Outside this range, aggregation results in small clusters (oligomers) whose structure is governed by 'geometric selection rules'. Only certain non-consecutive oligomer weights are observed. The two boundary points of the interval in which kinetic scaling holds are shown to possess the properties of tricritical points.

Renormalisation group techniques are developed to treat scaling behaviour and crossover in the configurational properties of a one-dimensional N-step interacting random walk (equivalent to the two-state rotational isomeric model in polymer theory). Three approximate methods are first given for the root-mean-square value R= square root (R²) of the end-to-end length R of the walk, resulting in a crossover form R=N^nu F(N/N_c(K)) in the scaling regime in which N and N_c(K) are both large, where N_c(K) is an effective value of the number of successive steps for which the walk continues in the same direction because of the effect of the interaction K. Exact scaling equations are then derived for the distribution function P(R) for the end-to-end length and its scaling behaviour is obtained. The relationship of these results to work on block probability distributions in magnetic systems is discussed.

Like random walks on Euclidean lattices, random walks on fractal lattices are modified by a waiting time (site) disorder when the first moment of the waiting time distribution diverges. It is shown that for lattices which support recurrent walks (spectral dimension smaller than two) the inverse of the diffusion exponent in the presence of disorder is increased by the difference between the waiting time fractal dimension and the usual fractal dimension. This hyperscaling relation is derived for Sierpinski gaskets in arbitrary dimension with scale-dependent waiting times. This provides qualitative insight into this problem and the exponent relation derived should also hold for statistically self-similar structure such as percolation clusters. For lattices whose usual spectral dimension is larger than two, a mean-field result holds.

The two-dimensional modified step model with nearest-neighbour interaction C( theta )=0 for mod theta mod <or= delta pi , C( theta )=1 for delta pi < mod theta mod < pi and C( theta +2 pi )=C( theta ) is studied by Monte Carlo simulation. Using Migdal transformations Barber (1983) found evidence of a phase transition for this model for 0.05<or approximately= delta <or approximately=0.43. The author has studied the model for delta =0.1 and delta =0.25 on a number of finite lattices and also finds evidence of a phase transition, with critical temperature T_c approximately=1.25 and T_c approximately=1.5 respectively.

By using small size clusters, which carry all the basic symmetries of the Ashkin-Teller model, the authors obtain within the mean-field renormalisation group approach the complete phase diagram of the model. Estimates of the thermal critical exponent are also obtained.

The theory of light-induced drift of particles with degenerate levels is proposed. Kinetic equations which are semiclassical with respect to rotational degrees of freedom and with a phase memory accounting are obtained. The drift velocity dependence on radiation polarisation is predicted. Numerical calculation, which has been made in a strong collision model, shows that the accounting of level degeneracy leads to a velocity change of up to 10-20% under some conditions.

The nematic-isotropic phase transition of long flexible polymers using mean-field theory with the Maier-Saupe expression of the van der Waals interaction are described. The order parameter of nematic polymers is calculated as a function of reduced temperature, k_B/T(a epsilon )^1/2, where a is the quadrupolar mean-field strength per unit length of the molecules and epsilon is the bend elasticity of the polymers. The reduction factor (a epsilon )^1/2 is in contrast to the factor in the Maier-Saupe theory of conventional nematics, a, indicating via epsilon the polymer aspect to the problem. The transition temperature, critical-order parameter and latent heat are given. For very long chains the transition between the nematic and isotropic phases occurs when T=0.38775 (a epsilon )^1/2/k_B where the order parameter is equal to 0.35642. The authors predict the temperature dependence of the order parameter and transition behaviour for different lengths of the polymer chain. A detailed procedure is suggested for comparison with experiment involving order parameters, latent entropy and configurational properties as revealed by small-angle neutron and X-ray scattering.

A systematic procedure for deriving a class of potentials with the underlying symmetry group SU(1,1), starting the commutation relations for the generators of SU(1,1), is presented.

Numerical simulations of the Eden model on the Manhattan lattice show clearly that this model leads to compact objects similar to the standard Eden case. This contradicts the conclusion of a latter of Chernoutsan and Milosevic (1985).

The authors respond to the numerical simulation result, obtained by Botet (ibid., vol.19, p.2233, 1986), which implies that the Eden aggregates should be compact objects and comment on the validity of the renormalisation group approach to the same problem.