Table of contents

The author studies the critical behaviour of a family of self-dual two-dimensional spin models with Z(N) symmetry (N<or=8) in its Hamiltonian formulation. Using the relations between the mass-gap amplitudes of the Hamiltonian in a finite strip and the critical exponents he estimates the anomalous dimensions of the parafermions occurring in the underlying field theory as well as the surface exponents.

Scaling behaviour of an order parameter and its fluctuations is numerically investigated at the onset of macroscopic mutual entrainment in a population of interacting self-oscillators. In particular, evidence is presented for the power law divergence of the fluctuations with exponents near 1/8. Finite-size scaling forms are also proposed and verified.

The theory of Monte Carlo simulation of random geometric objects such as random walks and random surfaces is considered on a sequential and a parallel computer. The partition functions generated for simple Monte Carlo rules are derived and the difference between sequential and parallel computing is discussed for the case of random walks with fixed endpoints or for random loops. The implications for random surface calculations in parallel is considered in short.

Deterministic fractal models are presented to be exactly solvable for the growth probability distribution on the surface of the cluster in a hierarchical lattice. The recursion relations of the electric field on the growth bond are obtained for diffusion-limited aggregation and the dielectric breakdown models. A hierarchy of generalised dimensions D(q) is calculated to describe the growth probability, by using the recursion relations. 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 spectra are studied for different dielectric breakdown models. The scaling of the highest growth probability p_max on the growth bond is analytically derived and the dependence of the fractal dimensions is found on the parameter eta describing the different dielectric breakdown models.

A colloidal aggregate in a moving fluid may suffer an instability of the plug-flow type for a large enough shear. The resulting non-uniform structure should be allowed for in shear-melting models.

Using conformal invariance the authors calculate the leading corrections in 1/N to the magnetisation of the three-state Potts quantum chain. The results are in good agreement with the numerical finite-size results.

The effect of long-range random static fields on classical diffusion is investigated. It is shown that the long-range component of random fields leads to the appearance of a new (compared to the white noise case) fixed point in the renormalisation group equations. The critical exponents and oscillating corrections corresponding to the new fixed point are calculated.

Renormalisation group (RG) techniques are currently used to derive relevant quantities in the vicinity of the critical point. The authors present a real space RG procedure which directly yields the order parameter for all values of the external parameters (e.g. temperature). It is as simple as a mean-field calculation, although it provides non-trivial results, which can be systematically improved. The method is successfully illustrated on the square lattice Potts ferromagnet. The whole approach suggests that the order parameter on a hierarchical lattice is, on every site, proportional to its coordination number.

Polynomial identities satisfied by certain matrices a, with entries from the group algebra of a finite group G, are derived in the irreducible representations of G. In the special case of the symmetric group S_n the identities obtained parallel those previously encountered by Green (1971) for the infinitesimal generators of the general linear group. Moreover, it is demonstrated that the diagonal entries of arbitrary polynomials in the matrix a are central elements whose eigenvalues, in irreducible representations of the group G, are simply determined in terms of the dimensions of the irreducible representations.

The symmetry group for the partition function of the completely X symmetric vertex model is determined. Here X is an arbitrary finite Abelian group.

It is shown that the Lame equation, which is the simplest example of a finite-gap Hill's equation can be written in terms of the composition of two first-order matrix operators, the coefficients of which satisfy Nahm's equation. The characteristic eigenvalues of the Lame equation emerge from the representation theory of the Lie algebra so(3). A special case is that of reflectionless potentials and their associated bound states.

The authors show explicitly that the famous auto-Backlund transformation and non-auto Kruskal-Dodd-Bullough transformation for the sine-Gordon equation can be interpreted as the gauge field copies for the strength tensor associated with the Lax pair for the sine-Gordon equation. In the same way one interprets the auto and non-auto Backlund transformation for the Liouville equation.

Several versions of an algorithm for Monte Carlo simulation of solutions of partial differential equations involving a general elliptic operator are developed, with the emphasis on quadratic accuracy and computational efficiency.

Two analytic expressions for the scattering length dependent on the coupling constant lambda are proposed. These expressions possess an infinite range of convergence unlike previous attempts. As a possible application the author discusses the scattering length for collisions of electrons with rare gases, calculations of critical constants for short-range potentials and calculations of Sturmians.

The correspondence between classical and quantum theories is considered. The authors investigate the set of linear quantisations of classical Hamiltonians written in various canonical variables. The inverse problem of defining the classical limit of a quantum theory is also studied.

The one-dimensional motion of an isentropic gas has long been known to be characterised by the presence of Riemann invariants; the choice of the latter as independent variables results in an essential simplification of the system of Euler equations which becomes linear. Another case which also presents Riemann invariants was discovered by Stanyukovich (1954) and by Martin et. al. (1954) in which the LMS gas and the isentropic gas have been shown to be mathematically equivalent. These two celebrated cases of integrability are shown to be but particular cases of a more general class of entropy distributions, for which Riemann invariants are constructed.

A simple formula is derived for the calculation of the electrostatic potential of an ion near the (100) surface of a KCl crystal using a lattice sum method. The potential is found to be short ranged, tending to zero far from the surface. The potential is identical to that obtained from the solution of the Poisson equation. Although rapid convergence is achieved in the present method, it is still many times slower than the expression derived from the Poisson equation. At an adsorption site, the electrostatic potential was found to be 0.3 eV. To get the right result from a direct version of the lattice sum, 10⁴ terms must be taken.

The author has studied the mechanism of stimulated electromagnetic shock radiation (SESR) caused by the interaction between a relativistic charged particle (electron) and an externally applied electromagnetic plane wave in a dielectric under superphase conditions. He has introduced a relativistic generalisation and considered the time-dependent position of the electron in the field of the incident electromagnetic wave. He performs classical relativistically invariant calculations for electromagnetic fields of Cherenkov radiation and SESR in the linear approximation using the method of Fourier transforms. The expressions derived are more general and are consistent with earlier results in special situations. In the linear approximation, the SESR term gets modified and an additional longitudinal SESR term is obtained. There is no change in the transverse SESR term.

The octahedral Jahn-Teller system T₁⊗(ε_g⊕tau_2g) is studied using generalised Glauber states which are eigenstates of the system in the strong-coupling limit. It is shown that these states become, when limited to a total angular momentum of one, eigenstates in the weak-coupling limit. The lower-branch zero-phonon state is studied using these states and expressions as functions of coupling strength are obtained for the energy and Ham reduction factors. The results compare well with the available numerical calculations.

The Fock method is applied to the problem of a one-dimensional hydrogen atom. Integral Fock equations are obtained in discrete and continuous spectra; the case of zero energy is studied and wavefunctions and normalisation constants are calculated in the momentum representation.

The general form of the rotation, space reflection, time reversal and charge conjugation invariant separable potential for the Dirac equation is investigated. A concrete potential is constructed which reproduces the 1s_1/2 and 2p_1/2 eigenvalues of the cut-off Coulomb potential.

Terms in a few-particle wavefunction, written as an expansion of homogeneous functions, are derived by a method which resembles standard techniques for solving differential equations in one variable. The addition of solutions to the homogeneous equation, i.e. Laplace's equation, converts a particular solution to a physically acceptable form consistent with the boundary conditions. Some earlier workers had derived expansions which, although satisfying the differential equations, were not consistent with those boundary conditions. A study of these inconsistencies assisted the development of this simpler method for obtaining the terms in the wavefunction in their reduced form. The power of the method is demonstrated by its application to nsms ³S and npmp ¹S helium wavefunctions up to and including terms of fourth order in the hyper-radius.

Spin- 1/2 particles subject to external fields will often exhibit paramagnetism. This effect can be formalised in terms of an inequality involving appropriate partition functions constructed with spin Hamiltonians which are chosen to be of Pauli type. Whereas in quantum mechanics it is known that a paramagnetic inequality does not hold true in general, the paramagnetism of spin in the classical limit to 0 is investigated. The author applies previous results to derive a classical partition function which, in satisfying the relevant inequality, gives rise to general paramagnetism.

The shift in the energy of a bound electron due to an electromagnetic field is shown to be gauge invariant if eigenvalues of the energy operator, which is not in general the Hamiltonian, are used. Using eigenvalues of the energy operator, the authors show by a direct calculation that the energy shift in a zero electromagnetic field is zero, even if the potentials are not zero.

This paper presents an extension (or more exactly a prefiguration in classical treatment) of the Kramers recurrence used in quantum mechanics for mean power calculations of hydrogenic atoms. Moreover, exact direct calculations for any power of r are dealt with for the first time.

Much recent interest in the classic Casimir effect problem has urged us to treat this phenomenon carefully by giving special attention to the gauge problem, which has been almost entirely swept under the carpet in the past, and to seagull terms. By quantising in the so-called Coulomb gauge, where the dynamical degrees of freedom are apparent and the quantisation problem is unambiguous, the expression for the Casimir effect is derived by explicitly taking into account seagull terms. Although a physical quantity ought to be gauge independent a specific gauge, nevertheless, has to be chosen for its evaluation.

The authors consider the most general form of time-dependent charged harmonic oscillators in an electromagnetic field. The quantum mechanical solution is developed and the Green function is derived. Furthermore, they use gauge-invariant formulation of quantum mechanics to obtain the transition probabilities in the electric dipole approximation.

Using a fibre bundle approach as in the work of Wu and Yang (1975) one finds that the charge quantisation condition for two dyons each with Dirac magnetic charge g_i and charge q_i is q₁g₂=1/2cn₁ and q₂g₁=1/2cn₂, where n₁ and n₂ are integers. This is more restrictive than the condition q₁g₂-q₂g₁=cn found by Schwinger (1966) and Zwanziger (1968) using an infinite string of singularities. The condition does not allow a new elementary unit of electric charge for dyons contrary to their condition. The quantisation condition agrees with the result one would obtain using a minimal, semi-infinite string of singularities, just as the result of Wu and Yang agrees with the Dirac version of the quantisation condition but not the Schwinger version for the case of an uncharged magnetic monopole.

A perturbative procedure is used to calculate the transfer matrix and the tunnelling transmission coefficient through a structured potential barrier. The potential is first approximated by a steplike function constructed from a partition procedure. The rate of convergence is only l² (l is the step size). Then a Peano perturbative solution is used to achieve an l⁴ convergence rate. The accuracy of the procedure is illustrated with the following examples: the trapezoidal barrier, the exactly soluble 15 sech²(z) barrier and the biased quantum well.

The Hellmann-Feynman theorem valid for the parameter dependence of bound states is generalised to the case of Gamow states using an appropriate definition of scalar products and expectation values with such states. The one-dimensional square well potential is considered as an illustrative example.

The authors present an exact solution for the master equation for a previously studied urn model and use this example as a basis for formulating a general approach to metastability. The emphasis is on dynamics (rather than analyticity) and they also obtain a cutoff partition function that maps smoothly to the usual partition function on passing to the stable state.

The Bethe ansatz variational approach to the thermodynamics of the massive Thirring-sine-Gordon model in the attractive coupling regime and in the zero-change sector is discussed in detail.

The Gaussian white noise (GWN) model immersed in a time-dependent electric field permits an exact evaluation of the retarded advanced Green functions. The (quantum) kinetic equation can be written down with the aid of the Baym-Kadanoff formalism, and the response to a step-like electric field has been analysed recently. Exact solutions were reported which, however, are different. The discrepancy is traced to different (but plausible) ways of treating the thermal instability inherent in the GWN. It is shown how the derived kinetic equation can be used to extract other transport properties as well.

The author obtains an expression for Levinson's (1949) theorem for a charged fermion moving in the background field of a Dirac monopole.

The authors present a study of the Hopfield model of the memory characteristics of a network of interconnected two-state neuron variables. The fraction of nominated configurations which the model stores without error is calculated analytically as a function of the number, N, of neurons and the number, n, of the nominated configurations. The calculation is tested by computer simulation. The noise-free (zero-temperature) phase diagram of the model is determined within a replica-symmetric solution of the mean-field equations. The model exhibits a phase transition at alpha ( identical to n/N)= alpha _c approximately=0.069; at this point the thermodynamic states having macroscopic overlap with the nominated configurations disappear, implying a discontinuous change in the fraction of bits (of any nominated configuration) recalled correctly. Large scale Monte Carlo simulations using a distributed array processor provide some support for the existence of a phase transition close to the predicted value.

A Gaussian approximation for the synaptic noise and the n to 0 replica method are used to study spin-glass models of neural networks near saturation, i.e. when the number p of stored patterns increases with the size of the network N and p= alpha N. Qualitative features are predicted surprisingly well. For instance, at T=0 the linear Hopfield network provides effective associative memory with errors not exceeding 0.05% for alpha <or= alpha _c approximately=0.15. In a network with clipped synapses, the number of patterns which can be stored with some given error tolerance is reduced by a factor of 2/pi as compared with the linear Hopfield model. A simple learning within bounds algorithm is found to continuously interpolate between the linear Hopfield model and the network with clipped synapses.

Two recently proposed methods, the variational cluster wavefunction approximation and the mean-field-like renormalisation group transformation, are used to describe the phase transitions in the Hamiltonian version of the two-dimensional Z(q) symmetric spin models. The convergence of the two schemes to the known results is analysed. It is shown that in their usual form neither of these models can reproduce fine details of the phase diagram, although the renormalisation group method gives in general better convergence.

The renormalisation group scheme of Migdal (1976) and Kadanoff (1976) is applied to a ferromagnetic Potts model on two hierarchical lattices: in the two cases the renormalisation transformation is split into two elementary processes (parallel bonds or series bonds) and each lattice can be mapped onto the other one. It is then easy to deduce the characteristic parameters (critical exponents, repulsive fixed points, etc) of one lattice from those of the other lattice. Moreover the classical geometrical duality offers a third possibility of mapping and the relationship between the renormalisation and the duality methods is presented in the peculiar case of a two-dimensional lattice.

Transport through a random medium in an external field is modelled by particles performing biased random walks on the infinite cluster above the percolation threshold. Steps are more likely in the direction of the field-say downward-than against. A particle is allowed to move only onto an empty site (particles interact via hard core exclusion). Branches that predominantly point downwards and backbends-backbone segments on which particles must move upwards-act as traps. The authors have studied the movement of interacting random walkers in branches and backbends by Monte Carlo simulations and also analytically. In the full network, the trap-limited current flows primarily through the part of the backbond composed of paths with the smallest backbends and its magnitude in high fields is estimated. Unlike in the absence of interactions, the drift velocity does not vanish in finite fields. However, it continues to show a non-monotonic dependence on the field over a sizeable range of density and percolation probability.

An interface in the three-dimensional Blume-Capel model is studied using low-temperature series and mean-field theory. By calculating the leading-order diagrams in the low-temperature series expansion to all orders using a transfer matrix technique, the authors are able to show that the interface wets through an infinite sequence of layering transitions which become quasicontinuous as the bulk phase boundary is approached. The mean-field approximation agrees well with the low-temperature series results.

The analysis of renormalisation group equations for the q-state Potts model in two dimensions is generalised to include the presence of a surface field H_s besides the bulk one, H_B. Combined with conformal invariance results, this allows prediction of a surface susceptibility behaviour delta ^2/f delta H_s delta H_B approximately L^-1/8(1-C'lnL)^15/16 for a q=4 system of size L at the critical temperature. This prediction is checked by a Monte Carlo based finite-size scaling analysis, which also nicely reproduces the exact and conjectured magnetic surface exponents for q=2 and 3, respectively. For the Baxter-Wu model similar methods give results consistent with the q=4 behaviour without logarithmic corrections (c'=0).

Bi-partite percolation is discussed in the context of gelation. The partial generating function method is used to derive expansions for the mean number of clusters and for the weight-average molecular weight in terms of the reactant concentrations, a and b. Renormalisation group and other arguments imply that the critical behaviour in bi-partite percolation is determined by the 'ordinary' percolation fixed point at a*=b*=p_c.

The partition function of the Ising model on the Sierpinski gasket is obtained by using a decoupled Sierpinski gasket Ising model as an unperturbed system. The exact partition function can be expressed as a statistical average of a special product over an unperturbed Ising Hamiltonian. A high-temperature expansion of the partition function is produced.

The electrical conductivity of an equilibrium system of particles interacting according to a surface adhesion potential was studied by means of Monte Carlo simulations. Results for the conductivity in the region of the percolation threshold are reported as a function of particle volume fraction and temperature. The conductivity exponent mu was found to be 1.6+or-0.2, in agreement with experimental measurements on microemulsions but significantly different from the reported lattice value of about 2.0.

The orbital magnetic moment and the orbital magnetic susceptibility at finite temperature for a charged oscillator in a magnetic field of arbitrary strength are calculated exactly. Landau diamagnetism, a purely quantum mechanical non-vanishing diamagnetic susceptibility in the absence of a magnetic field and the Langevin formula are obtained as special cases.

A three-term recurrence relation is derived for the Racah coefficients or 6-j symbols based on a set of orthogonal polynomials, called Racah polynomials, that generalise these coefficients. This relation is shown to be a consequence of the well known Biedenharn-Elliot identity.

Considering the complex Burgers, Cheng chemical reaction and Liouville equations, the authors show that through Hirota's bilinearisation method solutions can be obtained straightforwardly. They also show that these systems are exactly linearisable.

The shifted 1/N expansion has been used to obtain the energy values of the Hulthen potential for non-zero values of the orbital angular momentum. The analytic expression for the energies E_n,l yield fairly accurate results for a wide range of values of n,l and the screening parameter lambda . The energy values obtained by this method have been compared and found to be in excellent agreement with Pade approximation calculations.

The author presents Monte Carlo simulations of site percolation near the percolation threshold in five to seven dimensions. In contrast to lower dimensionalities, more than one spanning cluster is found in a system of size L^d for d>6. These results confirm Coniglio's prediction (1985) that the breakdown of hyperscaling for d>6 is caused by a multiplicity of infinite clusters at the percolation threshold.

Recently Nakamura (1986) reported results for two classes of random sequential packing problems on n*n square lattices. Non-overlapping squares covering a² cells are placed on the lattice either A allowing, or B forbidding contact. Here the author demonstrates an exact isomorphism between these two classes, as well as a broader range of models (and their d-dimensional analogues). Nakamura first took the n to infinity infinite-lattice limit and then analysed convergence of the packing fraction, p (i.e. saturation or jamming coverage), to the a to infinity continuum limit (where A and B coincide). The author obtains p values for model A with n= infinity from the generalised Palasti conjecture (using exact d=1 values), and develops a corresponding a to infinity asymptotic expansion; the corresponding p values for model B are obtained from isomorphism arguments. These facilitate analysis of Nakamura's results. Isomorphisms and continuum limit behaviour are also discussed for processes on other lattices.

A new concept is proposed, termed the equiscale transformation in the same universality class. Application to the square lattice Ising model with first- and second-neighbour interactions yields critical lines which are in very satisfactory agreement with series results.