Table of contents

The authors study extended symmetry properties of time-evolution partial differential equations, including Fokker-Planck-type equations. They show that, even if-in general-no non-trivial symmetry is present, in some particular interesting cases some special symmetry is allowed, and they provide the general method for finding it.

A canonical transformation is found which generates the topological term in a one-dimensional antiferromagnetic chain.

The authors investigate geometrical properties of a space curve and of its spherical images. They show that a space curve is characterised by two 'phase-like' quantities and comment on the relation of these quantities to the Berry phase.

The authors outline a probabilistic approach to modelling epitaxial growth, whereby they solve for the complete probability distribution of the surface using the Chapman-Kolmogorov equation. They describe a simple Markov growth model which is analytically solvable using this approach and exhibits damped oscillations of the step density. The model specifies that perfect layer growth occurs within subsections of an adjustable size on an atomically smooth surface. Increasing the size of a subsection increases the amplitude of the oscillations in the step density and decreases the rate of decay of the oscillations. The mechanism for the damping of the oscillations is shown to be phase incoherence between growth occurring in different subsections.

Well established integral equations already exist for both relativistic (G^(r)) and non-relativistic (G^(nr)) Green functions for independent particles moving in a given external potential V. An exact integral equation giving G^(r) in terms solely of G^(nr) and known operators is derived, by eliminating V. The connection with relativistic density functional theory is emphasised.

An attempt is made to crystallise the meaning of the maximum entropy state, when a system is subject to constraints which carry random uncertainties. Such a state is well defined, and there exists a simple statistic which allows it to be determined unambiguously, provided it is assumed that one's state of knowledge of the system fluctuates in close resemblance to the quantum uncertainty principle. Without loss of generality, the scenario is verified, for a variety of error-carrying constraints, by results from a large number of 'random die samples'.

The ideal Fermi gas in a finite volume is studied in the low-temperature regime where the thermal correlation length is comparable to the size L of the system. Physical quantities such as the density of fermions and the appropriate ordering susceptibility show oscillatory fluctuations around a smooth background as L is varied. These are investigated numerically and analytically. The amplitude of the fluctuations depends on the observable in question and on the container shape. Finite-size scaling does not hold directly for thermodynamic quantities but is recoverable by averaging the mean-squared fluctuations over an appropriate range in L.

A recombination of charged particles in disordered media in the presence of a strong external field, which causes the non-classical drift of particles, is considered. It is shown that the particles spend most of the time in randomly distributed regimes, impenetrable for diffusing particles, stretched along the field. Positively and negatively charged particles are hidden in different cavities and this causes a reduction of the recombination rate.

The authors consider Mandelbrot's fractal percolation process, characterised by a density parameter p in (0,1) and an integer subdivision index N>1. For each N, the process is known to have a percolation transition at density p_c(N) in (0,1). They prove that lim P_c(N)=P_c where p_c is the threshold value of the ordinary square lattice site percolation model.

An improved kinetic renormalisation group approach to diffusion-limited aggregation is presented. This approach is based on the growth process itself and accounts for the dispersity in the growth probabilities. It yields the multifractal spectrum D(q) with better values at smaller q. On the 2D square lattice the authors find D_f=1.694 for the fractal dimensions of the cluster and that of its interface, and D(1)=1.01 for the information dimension. The former agrees with the simulation results (D_f approximately=1.70) and the latter compares very well with the exact value D(1)=1.

The authors calculate numerically the transfer matrix spectrum of the 2D Ising model at T=T_c and in a magnetic field h not=0. In the limit h to 0, their results reproduce the mass spectrum conjectured by Zamolodchikov (1988). Scaling functions are also studied.

For neural networks in which the coupling J_ij are allowed to take on the values J_ij = 1 or J_ij = -1, the authors determine numerically the critical storage capacity for random unbiased patterns as a function of the stability. They use an exact enumeration scheme based on the Gray code and a continuous distribution for the patterns to control finite-size effects. Results are presented for N ⩽ 25; they indicate an optimal storage capacity of α_c ≈ 0.82 (N → ∞).

A neural network model in which the connection matrix is formed by summing direct products of successive patterns in cyclic sequences is studied. It is found that the noise elimination performance of this model can be better than that of an autocorrelation model due to a reduced tendency for trapping in spurious attractors.

The author shows that an explicit definition of negative-dimensional classical groups in terms of their action on Grassmann representation spaces, together with Weyl's character formula for the Grassmann tensorial representations of such groups, leads to a framework in which some surprising 'negative-dimensional' properties of group theoretic invariants arise naturally. As an application, he shows that the spectra of the Grassmann harmonic oscillator and of Grassmann angular momentum are related to their bosonic counterparts by a simple analytic continuation in the dimension.

A matrix representation of the labelling operators of Moshinsky and Nagel (1963) and of Partensky and Maguin (1978) in the non-orthogonal Draayer basis of SU(4) is derived; this allows one to solve explicitly the missing label problem for the spin-isospin multiplets in the arbitrary SU(4) supermultiplets. The simplification and degeneration of these representations are considered for two parametric irreducible representations of SU(4), as well as for some special cases of irreps typical of nuclear theory. The expansion of the linearly dependent states of the Draayer basis is discussed.

A new bosonic Holstein-Primakoff-type realisation of sp(4,R) is applied in the development of '1/N' expansions of non-relativistic spin-independent two-particle Hamiltonians. As a non-trivial example, the spectrum of helium-like ions is considered and the energies of the low-lying levels with given values of total angular momentum are computed. The calculation is explicitly performed in an approximation up to the second degree in the small expansion parameter.

For autonomous dynamical systems with two degrees of freedom the author derives an equation equivalent to the equation which follows from the condition of the vanishing Poisson bracket. The new version of the equation is very suitable for an easier and more efficient handling of terms of the same degree in the velocity components appearing in second integrals of motion. The new variables used are suggested in the light of the inverse problem of dynamics. As an application the author treats the problem of integrals of motion which are homogeneous polynomials in the velocity components x, y.

The motion of a classical particle within a billiard which rotates uniformly in the counterclockwise sense around its centre is studied. In the case of an elliptical boundary the counterclockwise motion shows a notable stability while the clockwise motion is to a large extent chaotic when the motion along the boundary in the clockwise sense is not possible. The organisation of phase space for the rotating billiard and the billiard in a constant magnetic field studied by Robnik and Berry (1985) is similar for weak fields but clearly different for strong fields where the variable curvature for the rotating billiard is manifested. For strong inertial forces the rotating stadium and the rotating ellipse behave in a similar manner.

The phase of the wavefunction on the lowest adiabatic potential energy surface of a T(X) τ₂ Jahn-Teller system is studied, and a mapping for the lowest energy states is proposed. Methods of calculating the tunnelling splitting at strong coupling are discussed, and tested by application to a soluble one-dimensional equation.

For pt.II, see ibid., vol.19, p.1863 (1986). In the perturbation theory of the general n-electron system, H(n)= Sigma ⁿf(i)+ lambda Sigma ⁿg(i,j), the subsystem method provides a finite decomposition (termed 'partition') in each order of the perturbation eigenfunctions and energies. The author shows that the third-order energy E⁽³⁾(n) is a weighted sum of third-order energies E⁽³⁾(n₁) belonging to electronic states (most of them highly excited) of the n₁=four-, three- and two-particle subsystems. The proof uses a parentage expansion of the second-order n-electron eigenfunction. The energy partitions can be regarded as the rigorous derivation of the important 'atomic energy relations' intuitively derived by Bacher and Goudsmit (1934).

The method for obtaining the superpartner potential in supersymmetric quantum mechanics (SSQM) is discussed in connection with non-linear equations and reflectionless potentials. The correspondence between a new class of soliton solutions to the U(N) non-linear Schrodinger equation, obtained via application of SSQM, and previously known soliton solutions is also discussed.

The non-linear system of equations of the sine-Gordon type, describing waves in Josephson superlattices, is investigated in the framework of the piecewise-linear model. A detailed quantitative analysis of the fluxon states and their bifurcations in the simple case of the system of two identical semi-infinite Josephson junctions, placed in an external magnetic field, is carried out.

The authors are concerned with the critical properties of the anisotropic Heisenberg chain, or XXZ model, with arbitrary integer or half-integer spin. The eigenspectra of these Hamiltonians, with periodic boundaries, are calculated for finite chains by solving numerically their associated Bethe ansatz equations. The resulting spectra are found to be in accord with the predictions of conformal invariance and the operator content is identified, for lattices with an even and odd number of sites. The results for spin 1 and spin 3/2 indicate that the conformal anomaly c for these models, in the gapless regime, has the value c=3S/(1+S), independent of the anisotropy, and the exponents vary continuously with the anisotropy as in the eight-vertex model. The operator content of these models indicate that the underlying field theory governing these critical spin-S models is described by composite fields formed by the product of Gaussian and Z(N) fields, with N=2S. Finally some of the irrelevant operators which produce the leading finite-size corrections of the eigenenergies are identified.

For pt.II, see ibid., vol.22, 205 (1989). It is shown that every n-gonal quasilattice in two dimensions has self-similarity with a complex self-similarity ratio which is equal to a non-unit PV number in the n-cyclotomic field. In this self-similarity, the original quasilattice can be inflated to a number of different quasilattices which are locally isomorphous to each other. The number is equal to the algebraic norm of the relevant PV number in the n-cyclotomic field. Several examples are presented for the cases of octagonal, decagonal and dodecagonal quasilattices. The difference between the present self-similarity and that associated with a PV unit is clarified.

An n-gonal quasilattice in two dimensions is constructed for the case of n=6m (m>or=2) as a dual to an m-tuple periodic grid G_m=G₀ union rG₀ union . . . union r^m-1G₀, where G₀ is a periodic grid with hexagonal point symmetry and r the rotation operation of the plane by angle 2 pi /n. The author investigates the case where G₀ is a honeycomb, triangular, Kagome or diced grid. The resulting quasilattice is not of 'Bravais' type unless G₀ is the honeycomb grid. It is shown also that the same quasilattice is obtained with the projection method from the 2m-dimensional lattice L=L₀*L₀*. . .*L₀, where L₀ is a triangular, honeycomb, diced or Kagome lattice being dual to G₀. In the projection method, the 2m-dimensional Euclidean space is divided into cells with a 2m-dimensional grid G which is dual to L. It is shown that G_m is identical to a two-dimensional section of G.

Starting from the Poisson summation formula in m dimensions, a class of lattice sums is evaluated analytically. The resulting formulae provide a considerable generalisation of the results reported previously and are applicable to a variety of physical problems, especially to the analysis of finite-size effects in systems undergoing phase transitions.

For an abstract thermodynamically well behaved local specification describing a lattice spin system with non-compact state space, the author gives a short proof of the independence of the limiting thermodynamics on the typical boundary conditions. This general theorem is then applied to superstable and regular spin systems studied by Lebowitz and Presutti (1980) to simplify and clarify their proof. Another application gives the uniqueness theorem for the limiting Gibbs phase for a class (in general non-superstable) of lattice systems with unbounded spins.

A standard thermodynamic theory of energy conversion is corrected by including the return fluxes from the sink to the converter and from the converter to the pump. By also allowing for the limited solid angle subtended by the pump at the converter and for general photon distributions (not only black-body), one obtains a better theory than has so far been available. It yields maximum conversion efficiencies as a function of solid angles and entropy generation rates. Comparison is made with earlier work where available.

The distinction between canonical and canonoid transformations, introduced by Currie and Saletan (1972), is emphasised through an example, which also shows that the initial condition of their theorem was quite strict. A step forward is taken by reducing the requirement that all quadratic Hamiltonians be canonoid to the requirement that a finite number of them be canonoid.

The behaviour of high-order perturbative-variational wavefunctions, with one or more free parameters, is studied. A variational criterion, and a criterion based upon variation of the norm of the perturbative approximation function with order, are used for fixing free parameters. For the quartic anharmonic oscillator, both criteria seem to be equivalent. The obtained energies, not only for the ground state but also for excited states, are in excellent agreement with previous results, although when the authors go to higher excited states they lose precision because of the worsening of the perturbative-variational wavefunctions.

An approximate value of the tunnel splitting of the methyl group torsional ground state is obtained using the traditional methods of wave mechanics.

The authors propose a generalisation of the usual bosonic loop-space formulation of gauge theory in the case of fermionic interacting matter.

A simple method for the construction of the E₈ root system with icosians is suggested. It is confronted with the root system of E₈ obtained by integral octonions.