Table of contents

The criteria for existence of maximum entropy distributions, having prescribed the first three moments (in a semi-infinite range), are given. Some relationships between moments and Lagrange multipliers are also presented.

The authors present evidence for irregular behaviour in potential scattering and discuss a possible explanation in terms of unstable periodic orbits.

A Lorenz-like truncation of the full hydrodynamic equations is used to study the onset of the first instability of modulated Taylor-Couette flow. The linear stability analysis is carried out on the lowest order (2*2) non-autonomous dynamical system. The use of good wavefunctions for the truncation yields qualitative agreement with exact hydrodynamics even at this low order of stability analysis.

Heuristic arguments are presented supporting the conjecture that almost all quantum Hamiltonians are integrable in the sense that there exist N(N=number of freedoms) mutually commuting observables (which, in analogy with the classical action variables, can be chosen to be the number operators). This follows from perturbational considerations: the series may converge for almost all perturbations that preserve the discreteness of the spectra, because a 'quantum small denominator' almost always uniformly satisfies the condition of sufficient irrationality. The radius of convergence vanishes if h(cross)=0. The classical limit (as h(cross) to 0) of the quantum integrals of motion generically does not exist.

Introduces two wavefunctions to describe a single relativistic free electron, from which the authors can obtain the value of an arbitrary physical quantity at any time t. They find that the motion of free electrons is the same as that given by special relativity.

Non-self-dual monopole solutions are presented for a class of SU(n) Yang-Mills-Higgs models in three dimensions. The models in question are obtained by dimensional reduction of pure Yang-Mills theories in even d>4 dimensions. The Higgs field can belong to the adjoint representation or the adjoint (+) scalar representation of SU(n). In the former case, constraints on the solutions reduce the number of independent ansatz functions in the spherical symmetry ansatz.

The authors present the numerically determined distribution of the number of sites visited by thermally activated walkers on ultrametric spaces, which mimic the energetic disorder found in amorphous materials. At high temperatures the results resemble those found for Sierpinski-type fractals. At low temperatures, the distributions are highly structured, showing discontinuities which increase with the number of steps, in contrast to an intuitive picture that would lead to smoothing. They apply the results to the energy transfer in disordered systems and study the decay due to trapping.

Diffusion in the presence of a hierarchical array of barriers is defined on the Sierpinski gasket and is generalised to d dimensions. Through an exact renormalisation group procedure the anomalous long-time behaviour of the autocorrelation function is obtained and compared with that of a similar hierarchical diffusion on Euclidean d-dimensional lattices. Some evidence of a transition from anomalous to normal behaviour is presented.

The complementarity relation between the unitary groups U(d) and U(n) within the symmetrical irreducible representations of the larger unitary group U(dn) is extended to non-compact groups. It is proved that the pseudo-unitary group U(p,q) is complementary with respect to U(n) within some positive discrete series irreducible representations of the larger pseudo-unitary group U(pn,qn). The latter arise when reducing the metaplectic irreducible representations ((1/2)^dn) and ((1/2)^dn-13/2) of the real symplectic group Sp(2dn, R), where d=p+q, and they are characterised by a single label, the eigenvalue of the first order Casimir operator. Some applications of the U(p,q)-U(n) complementarity to atomic physics are outlined. For such purposes, the isomorphism between the Lie algebras of SU(2,2) and SO(4,2) is used extensively. The mathematical framework underlying the Kibler-Negadi approach (1983,4) of the hydrogen atom dynamical group is extended to the independent-electron dynamical group of intrashell many-electron states as well as to the correlated electron dynamical group of intrashell doubly excited states.

A class of periodic solutions of the two-dimensional Poisson-Boltzmann system was found. These Jacobi elliptic functions reveal a tendency to self-organisation, e.g. periodic spatial distribution, for self-consistent Coulomb ensembles.

The non-linear Fokker-Planck equation is used to study the non-equilibrium concentration of a two-step chemical reaction. The non-linearities are systematically approximated and the concentration time correlation function is calculated analytically for two reaction schemes, with and without detailed balance, respectively. The corresponding structure factors are also calculated and it is found that they exhibit the same basic Lorentzian structure. The relation of these results to other work is also mentioned.

The results of a high accuracy numerical study of the uniformity of the electric field between parallel disc electrodes are reported. Simple analytical expressions are derived describing the field behaviour at points on the electrodes near the edges of the discs and also at points between the electrodes remote from the edges. These expressions are consistent with the numerical results. A simple approximation for the fringing capacitance of parallel-plate electrode systems of arbitrary shape is also given. This expression leads to a first-order approximation for the total capacitance which, for disc electrodes, has superior accuracy to similar published approximations for radii up to five times the disc separation. A higher-order approximation for the capacitance is shown to have markedly better accuracy than any first-order approximation.

The non-linear polarisability model of dispersive optical bistability with periodic driving field is shown to have non-trivial limit cycle solutions in phase space. The model is quantised rigorously in the positive P representation. For a particular (but typical) limit cycle, the associated steady state probability distribution is shown to be compatible with a bivariate Gaussian. The inverse widths of the Gaussian are calculated as functions of position in both phase space and time.

The quantisation of the Lorenz equations is shown to take the form of a set of two complex and one real stochastic differential equations with multiplicative noise. Phase diffusion is the dominant feature for small values of the noise. Quantities such as the probability of the modulus of the variables are unchanged from those in the classical Lorenz equations. Moreover a fractal dimension can be associated with the stochastic process for sufficiently small noise. For large noises there is a radical breakdown of this picture.

The drift in the Fokker-Planck equation in the positive P representation describing a gas laser is shown to have a continuum of stable steady states. Only a small subset is physically acceptable. It is demonstrated that the diffusion does manage to average over the unphysical steady states to give correct correlation properties in the lasing region. However, in the chaotic region, the positive P representation equations have additional instabilities. As a consequence the positive P representation in the chaotic region leads to behaviour either different from known solutions or breaks down.

Fractal lattices with the 'self-duality' property are proposed to imitate the geometric texture of two-dimensional percolating networks at percolation threshold. The exponents describing the power law dependence on the scale length of the Hall and Ohmic conductivities are found. The exponent for the Hall conductivity agrees with twice that for the Ohmic conductivity. The model is extended to describe the approach towards the threshold. It is found that the model shows the typical percolation behaviour as a function of a parameter p (the bond concentration).

The properties of a simple random walk and random walks with persistence of velocity, in one dimension, are reported. The finite-fractal properties of the walks are obtained, with exact expressions for indefinitely long walks. Remarkably simple expressions are found for the length, L( epsilon ), at scale epsilon . Computer simulations of these walks are reported for finite trajectories which help to explain the properties of atomic trajectories in realistic liquids determined by molecular dynamics simulation. Some new molecular dynamics simulations are reported for a typical liquid, its coexisting vapour and the crystal, all at the same temperature. The general form of the fractal properties of the three states of matter are well represented by the present one-dimensional model. Indeed, even the quantitative nature is quite well represented, which suggests that results for random walks in three dimensions will be substantially the same.

The formula which relates the scaled interface width to the anisotropic interface tension is generalised for an arbitrary orientation of the interface in two dimensions. The relationship between the scaled interface width and the curvature of the equilibrium crystal shape is also generalised.

Nienhuis (1982,4) has shown that the critical O(n) model, on the honeycomb lattice is equivalent to a zero-temperature antiferromagnetic Potts model on the triangular lattice, i.e. to the chromatic polynomial of the triangular lattice. Here the critical O(n) model is solved by the Bethe ansatz method, thereby giving the large-lattice limit of the chromatic polynomial.

The use of the percolation model to describe the age distribution of air bubbles trapped in polar ice is discussed. An earlier Monte Carlo simulation of bubble trapping defined as a bond percolation process on the interstices of a BCC lattice is checked against series analysis. A description is given of the way in which high-density series for bond percolation on bipartite lattices can be obtained by using the definition of percolation as the q to 1 limit of the Potts model and applying the method of partial generating functions.

Fixed points of discrete vector models with icosahedral and dodecahedral symmetry are found by approximate real space renormalisation techniques. Both models have three distinct phases similar to cubic vector models. New fixed points governing the order-disorder and partial order-disorder transitions are identified. These transitions gradually become softer as the number of vector components increases.

Within the context of the l+ epsilon expansion and the near-planar interface Hamiltonian, the isotropic fixed point is found to be stable to a wide class of perturbations. In particular, the universality class includes all non-singular forms of anisotropic surface tension.

A generalisation of the so-called extended irreversible thermodynamics is proposed. It leads, in a natural manner, to evolution equations of the Ginzburg-Landau type for the fast non-conserved variables. In this generalised scheme the thermal conductivity can be written as a continued-fraction expansion. This corresponds to a macroscopic analogue of some well accepted microscopic theories.

By means of the method of coherent states and the real time Green function, the spontaneous breaking of symmetry in the (1+1)-dimensional real phi ³+ phi ⁴ field and its restoration at finite temperature are investigated. A new soliton solution which has essential differences from the kink and antikink solutions of phi ⁴ field is found. The soliton mass and the critical temperature are given.

Explicit examples are given of the false eigenvalues of the Hill determinant method, the existence of which has been suspected by workers using analytic methods. A simple criterion is discovered for distinguishing between false and physical eigenvalues.

A scaling rule has been proposed to study the energy spectrum of the Aubry model (1978) H= Sigma _n((V cos Qn)a_n^Dagger a_n+t(a_n+1^Dagger a_n+a_n+a_n-1^Dagger a_n)) for a class of irrational numbers Q=1/( mu +Q), where mu is a positive integer. The scaling rule is universal in the sense that it is independent of the ratio V/t and has been confirmed via a large scale numerical calculation. A fractal dimensionality D_f of the energy spectrum is defined and has been found to be D_f=1.