Table of contents

Vector coherent state theory is used to give an explicit construction of the irreducible representations of a group with non-commuting raising generators, the simple example of the compact unitary symplectic group USp(4) contains/implies USp(2)*USp(2).

The Bloch density matrix for an electron in a three-dimensional oscillator potential in the presence of parallel electric and magnetic fields is presented in cylindrical coordinates.

Treating as an example two-dimensional similarity invariant Hamiltonians with polynomial velocity-dependent potentials, the authors show how Painleve analysis, Yoshida's theorems and direct methods can be combined in the search for integrable Hamiltonians. New integrable systems are found, one of which has a polynomial second invariant of fifth order in the momenta. Most of the necessary calculations have been performed by applying various REDUCE packages developed for that purpose.

Adopting simple field theoretical techniques the author demonstrates how the Smoluchowski approach to reaction-diffusion models can be systematically corrected. Focusing on coagulation/fragmentation processes he identifies the upper critical dimension d_c=2 below which finite mobility corrections are important and full universality (lattice independence, etc.) can be achieved. Trends seem by direct simulation of simple lattice models for d>2 are confirmed, whilst problems of interpretation exist for d<2.

The authors show how the phase space quantisation approach can be fulfilled via the collective behaviour of quantum mechanical particles on a cubic lattice.

Kinetic gelation, the growth model for radical copolymerisation, is investigated for the dynamic properties of its clusters at the gel point. The authors find that within numerical error bars the fractal dimension d_w of a random walk of a cluster cannot be distinguished from that found for percolation clusters at p_c in three dimensions.

The acceleration-acceleration correlation function, K(t)=(a(t).a(0)) (a=d²r/dt², where r is the displacement), of a random walker on a fractal lattice is studied analytically and numerically on percolation clusters and on diffusion-limited aggregates at dimensions d=2,3. After t(>>1) discrete time steps, the authors find K(t)=A(t)/(r²(t)), with A(t) approximately (-1)^t. At a fixed distance R from the origin they find the superuniversal law K(R) approximately R^-2 on all fractals and for all d.

Fractal solid surfaces are presented as being built up from repeating fractal units of size xi equal to the surface upper self-similarity cutoff. Taking into account the difference between real diffusion pathways of particles of size lambda and their observable projections the authors distinguish three types of observable diffusion: (i) the classical law for lambda > xi , (ii) anomalous time dependence for lambda << xi and short times; (iii) normal time dependence but anomalous behaviour of the observable diffusion coefficient for lambda << xi and long times. Crossovers occur from (iii) to (i) with increasing lambda and from (ii) to (iii) with increasing time.

The type of critical points for the Potts model is found to be dependent on the structure of Sierpinski carpets. All possible types of structure for Sierpinski carpets with two interactions after bond moving are found. The author presents general Sierpinski carpets and new parameters are proposed and used to describe them. The lacunarity expression is revised. The author discusses and proposes a universal classification. Critical points, eigenvalues and flow diagrams are presented. Some fixed points display negative eigenvalues.

The authors introduce radial bias into off-lattice diffusion-limited aggregation in order to simulate radial anisotropy present in experiments on viscous fingering in liquid crystals. For large cluster sizes the model is also expected to provide information about the asymptotic shape of the ordinary off-lattice diffusion-limited aggregates. They find that for large enough radial bias the overall shape of the clusters becomes star-like with a spontaneously selected number of main arms which is either four, or less frequently five. The crossover to this structure is accompanied by a change in the radius of gyration exponent which approaches the value 1.5 previously observed for systems with axial anisotropy or very large clusters grown on the square lattice.

The mean-field renormalisation group (MFRG) has so far given poor estimates for critical indices. It is argued that this can be traced to an unsatisfactory definition of the length scaling factor. A new definition, that improves the estimates in most cases, is proposed.

The authors report molecular dynamics results for the single-particle diffusion in liquids where the interparticle interaction is a repulsive Yukawa potential U(r) varies as exp(- kappa r)/r. In supercooled liquids, atypical subdiffusive behaviour is observed in a range of times intermediate between the early-time ballistic and late-time diffusive regimes. This sub-diffusive regime is evidence for strongly correlated motion and increases in size as the glass transition is approached. Introduction of a viscous damping term to the equations of motion does not destroy this effect.

The authors present a new experimental technique for producing model two-dimensional systems subject to external fields of arbitrary symmetry and variable coupling strength. An aqueous suspension of colloidal microspheres is confined to a single layer between two quartz flats, on one of which a pattern of eroded features is etched using electron-beam lithography combined with plasma etching. By varying the depth of the etched features the strength of the electrostatic wall-particle forces can be adjusted smoothly. They demonstrate the feasibility of the method and discuss possible applications.

Because of their many physical applications, there has been considerable interest lately in the basis for irreducible representations (irreps) of symplectic groups and in the matrix elements of the generators of these groups with respect to this basis. In the present paper the authors carry out this programme for the irreps in the positive discrete series of sp(4,R) by building up this basis from powers of the raising generators applied to the lowest weight state. By using the Dyson boson realisation of the generators, their matrix representation can be obtained by direct differentiation, while the overlap of the states of the basis can be determined with the help of coherent states. The extension of the analysis to symplectic groups of arbitrary dimension is straightforward and will be implemented explicitly for sp(6,R) in a later paper.

Extended BRST supersymmetry, wherein the ghosts are treated equally, can be viewed as the Grassmann inhomogeneous rotation group in two dimensions, Sp(2) wedge-product T(2). The representations are labelled by pseudomass and pseudospin, and the physical state vectors correspond to wavepackets over the fermionic momentum. Irreducible field representations are constructed.

The authors show that any relativistic wave equations for a particle with mass m>0 and arbitrary spin s is invariant under the Lie algebra of the group GL(2s+1, C). The explicit form of basis elements of this algebra is given for any s. The complete sets of the symmetry operators of the Dirac and Maxwell equations are obtained, which belong to the classes of the first and second-order differential operators with matrix coefficients. Corresponding new conservation laws and constants of motion are found.

A Fokker-Planck description of the time evolution of a K-distributed noise process is presented, discussed and taken as the basis of an approximate analysis of the correlation properties of the noise. An equivalent set of stochastic differential equations is identified and exploited in a numerical simulation of the noise process.

A modification of the Lanczos algorithm is suggested for calculating the frequency representations of correlation functions, averaging over the canonical ensemble. The problem is reduced to a recursive construction of a set of operators so that the commutator of each operator with the Hamiltonian is equal to a linear combination of three operators: previous, given and subsequent. As an initial operator of the set the one entering the correlation function is used. The coefficients with which the operators enter the linear combinations are equal to the elements of the continued fraction representing the correlation function. On the basis of the developed algorithm formulae are obtained for the correlation function of the system of Frenkel excitons interacting with non-polar optical phonons, which describe the absorption spectrum of a dielectric crystal.

The absence of ergodicity is investigated analytically and numerically for classical field theories and for Euler equations in two dimensions. In this case the authors extend the arguments of Patrascioiu (1984) to inviscid two-dimensional fluid dynamics. They comment on the risks of truncation introduced by a numerical simulation of continuous systems reflecting on the analytical properties of the solution of the field equations. It appears that ergodicity is a property only of the discretised problem. The considerations are tested on a simple model of a radiant cavity, which shows the absence of the ultraviolet catastrophe and the possibility of wrong interpretations of numerical simulations of field theories.

A very useful concept for two-frequency dynamical systems is the rotation interval. A method is proposed and tested numerically for estimating it from a time series.

The Darboux transformation is used to generate explicit solutions of the time-dependent Schrodinger equation in one dimension. The potentials are a reflectionless potential and certain asymmetric double well potentials.

By a combination of mathematical analysis and microcomputer experiment it is shown that the hypervirial perturbation method can yield, not only the accurate quantum mechanical values, but also the zeroth-order JWKB values of E and (x²) for the case of a harmonic oscillator with a lambda x⁴ perturbation.

The conventional formulation for interaction of radiation with matter uses a zero potential Hamiltonian H₀ as an unperturbed Hamiltonian and this procedure leads to a conflict between the gauge invariance of probability amplitudes and the Born rule. It is pointed out that if one uses a dark Hamiltonian H_d as the unperturbed Hamiltonian and considers its gauge dependence, then this conflict no longer exists. From this point of view the author proves that the Lamb gauge and the Coulomb gauge in the electric dipole approximation give the same gauge-independent probability amplitudes, and thus the A.p against r.E controversy becomes unimportant.

Using the Caldirola-Kanai Hamiltonian with an external driving force for the damped driven harmonic oscillator as the quantum dissipative system, the authors have exactly evaluated the propagator, wavefunction, uncertainty relation and the transition amplitudes by the Feynman path integral method.

The authors derive a closed-form expression for the time-dependent propagator for a quantum mechanical particle which is subject to an external force which is the sum of (i) a reflecting half-plane barrier with a straight edge, and (ii) a harmonic force pointing towards a point of the edge. This new addition to the short list of exactly known quantum mechanical Green functions is a simple combination of exponential functions and Fresnel integrals, the arguments of which are combinations of trigonometric functions.

In order to create a degeneracy in a quantum mechanical system without symmetries one must vary two parameters in the Hamiltonian. When only one parameter, lambda say, is varied, there is only a finite closest approach Delta E of two eigenvalues, and never a crossing. Often the gaps Delta E in these avoided crossings are much smaller than the mean spacing between the eigenvalues, and it has been conjectured that in this case the gap results from tunnelling through classically forbidden regions of phase space and decreases exponentially as h(cross) to 0: Delta E=Ae^-S/^h(cross). The author reports the results of numerical calculations on a system with two parameters, epsilon , lambda , which is completely integrable when epsilon =0. It if found that the gaps Delta E obtained by varying lambda decrease exponentially as h(cross) to 0, consistent with the tunnelling conjecture. When epsilon =0, Delta E=0 because the system is completely integrable. As epsilon to 0, the gaps do not vanish because the prefactor A vanishes; instead it is found that S diverges logarithmically. Also, keeping h(cross) fixed, the gaps are of size Delta E=O( epsilon ^nu ), where nu is usually very close to an integer. Theoretical arguments are presented which explain this result.

It is shown that, in the Compton effect, the velocity of the recoil electron is obtained by adding relativistically the projections of the photon velocities.

A new approximation scheme has been derived for the classical statistical mechanics of one-dimensional systems with pairwise additive potentials in an external field. It enables both the thermodynamic functions and the static correlation functions to be obtained. The analysis depends on combining the recurrence relation of Baxter (1964, 1965) with an extension of the analysis of functional differential equations started by Volterra (1930).

The authors describe a method to estimate the critical supersaturation sigma _c above which a growing crystal face becomes kinetically rough. They discuss the relation of sigma _c with the step free energy kT gamma . The method is tested with computer simulations of the growth of the (001) face of a simple cubic crystal in the solid-on-solid (SOS) model. Near T_R, the roughening transition temperature, the results are in good agreement with other estimates of gamma . The method is then applied to growth of (110) faces of naphthalene for melt and solution. They find that gamma does not depend on the temperature, in contrast to gamma found for the SOS model. The (110) face of naphthalene hence does not behave according to the infinite-order roughening transition of Kosterlitz-Thouless type. The classical method to model crystal growth from solution with SOS models turns out to be fundamentally incapable to describe surface roughening in the naphthalene case.

A new type of indefinitely growing self-avoiding walk is introduced in which the walker is not allowed to perform a step pointing towards a site he has already visited. This restriction leads to a walk which is a succession of long directed parts. A Monte Carlo study on the square lattice suggests that the radius of gyration exponent nu =1 in two dimensions. The enhancement factor exponent gamma =1 in all dimensions. A self-consistent method gives nu =1 when 1<or=d<or=2, nu =2/d when 2<or=d<or=4 and nu =¹/₂ above d_c=4, the upper critical dimension.

Usual Penrose tilings are defined by a set of five phases gamma _j which add up to an integer. The authors study the general case where the phases add up to any number and show that these generalised tilings can be obtained with four tiles (two thick rhombi and two thin rhombi) with different colourings of the edges. The consideration of generalised tilings is relevant for the study of phase distortions, local structural transitions, and therefore various defects of the usual things.

The mock ANNNI model is generalized in replacing the Ising spins by q-state Potts variables. Performing exact dedecoration transformations for general q and using Monte Carlo techniques in two dimensions for q=3, a multitude of distinct spatially modulated long-range ordered phases, as well as phases with algebraic order, are found to spring from multiphase point.

Comb polymers with semi-flexible main chains and rod-like nematogenic side chains form interesting nematic phases, largely as a consequence of competition between polymer entropy and rod order. The authors model this competition taking main chains of various stiffness and side groups of various length. Depending on volume fractions, temperature, nematic coupling and stiffness they get one of three nematic phases which they call N_I, N_II or N_III. They find that at least one component, main or side chain, must be ordered toward a direction, i.e. have a positive order parameter, and the order parameter can be positive or negative. They identify the molecular trends leading to each possibility. The theory, in addition to complex phase diagrams, also predicts unusual properties such as anomalous temperature variation of optical anisotropy and molecular conformational changes. The latter, amenable to small angle neutron scattering, shows a prolate (rod-like) form for the main chain dimension in the N_II and N_III phases where the main chain order parameter is positive. The oblate (disk-like) form, the N_I phase, is examined in some detail.

A new functional approach to the stochastic quantisation of continuum Euclidean Yang-Mills field theory is presented. Non-negative integral representations for the nonequilibrium and equilibrium probability distributions are given. A new dynamical generating function is proposed and its equilibrium limit is obtained. The new dynamical Feynman rules for the generating function are derived and the superficial renormalisability of the theory is studied. Euclidean volume divergences appear. The cancellations of the latter for an Euclidean dimensions d, and, for d=4, that of all quartic ultraviolet divergences, are carried out.

The topographical invariants of monopoles introduced by Taubes (1981) are described in a Chern-Weil framework.

Calculates the Green function for a charge-monopole system and shows that a non-integrable phase factor emerges without using the classical notion of path.

Suggests modifying recent results on dynamical phase transitions on systems with hierarchically distributed barriers. If one slightly changes the transition rates, given by the barriers, one obtains a Vogel-Fulcher law for the diffusion constant.

Extensions of exact enumeration data for self-avoiding walks attached to the surfaces of face-centered cubic and triangular lattices are reported. The convergence of the estimates for the critical points p_c and exponent gamma ₁ obtained by the Baker-Hunter (1973) confluent singularity analysis is discussed and comparison made with estimates of p_c from the corresponding bulk series.