Table of contents

A shift operator expansion is used to extend the range of the boson realisation method of calculation of Lie algebra generator matrix elements to those chains of algebras for which the original technique cannot be applied. The example of wsp(4,R)=w(2)(+)sp(4.R) contains/implies sp(4.R) is treated in detail and a generalisation to wsp (2d,R)=w(d) contains/implies sp(2d,R)(+)sp(2d,R) is pointed out.

The authors find the ground state energy of a particle enclosed in a regular polygon of n sides by perturbing about the equivalent circle and thereby setting up a n^-1 expansion. The first correction is O(n^-3) and hence a two-term answer is very accurate for polygons with many sides.

It is shown that an adequate Fokker-Planck equation for the transition to a convective roll pattern in a Rayleigh-Benard cell can be derived by projecting the time-reversible Liouville equation onto the centre manifold for the onset and introducing the source of irreversibility on the initial conditions. Thus the author elucidates the microscopic origin of the random source in the order-parameter evolution.

For the critical (2+1)D Ising model defined on a square lattice with antiperiodic boundary conditions, numerical studies suggest relations among the critical exponents and the finite-size scaling amplitudes of correlation lengths similar to those obtained in (1+1) dimensions from conformal invariance.

The properties of random walks on a square lattice with multifractal distributions of site-residence probabilities have been explored using computer simulations. Random walks on such substrates have well defined fractal dimensionalities D_W (obtained from the dependence of R² on N where R² is the mean square displacement from the origin and N is the number of steps in the walk) which are larger than two. Results obtained from these simulations indicate that more than two exponents (D_w and D_s where D_s is the fracton or spectral dimensionality) are needed to describe the properties of these walks. The algorithms used in this work provide a convenient way for generating walks with a fractal dimensionality greater than two and are being used to extend the scope of the diffusion-limited aggregation models.

A modified Eden model has been investigated in which the growth probabilities are determined by a fractal measure on the underlying lattice. Both the spatial distribution and probability distribution of surface sites on the growing clusters have been investigated. For the case where the multifractal substrate is constructed using a 2*2 multiplicative generator with the probabilities P₁=1, P₂=R, P₃=R³ and P₄=R³, the spatial distribution of surface sites and the sites comprising the inner and outer hulls have a fractal geometry which can be described by dimensionalities which depend on R. For the total surface, this dimensionality converges to a value of about 1.76+or-0.01 as R to 0. For the inner and outer hulls the fractal dimensionality approaches a value of about 1.48+or-0.02.

The problem of the equipartitioning of a random graph of fixed finite valence is studied by comparison with a ferromagnetic Bethe lattice with random boundary conditions. The simplicity of recursion relations for effective fields due to descendents on Bethe lattices provides simple approximations for the optimal cost, in quite good agreement with simulations.

The authors study the problem of bipartitioning a random graph of fixed finite valence using a mean-field replica-symmetric theory of an Ising ferromagnet with zero magnetisation constraint. The thermodynamics is determined by the probability distribution of an auxiliary field. The expression for the ground-state energy agrees with that proposed by Mezard and Parisi (1986) using a cavity-field method, but their expression for the fraction of crazy spins is reinterpreted.

The Smoluchowski equation with a source is analysed. Universal power-law distributions of cluster size are obtained in non-gelling and gelling systems.

The investigation of general CP transformations leads to transformations of the form U to WTUW with unitary matrices U, W. It is shown that a basis of weak eigenstates can always be chosen such that WTUW has a certain real standard form.

Some aspects of the symmetry group SL(3,R) of the simple harmonic oscillator are revisited. A realisation of the general element of the group in Newtonian spacetime is obtained, with the eight parameters included. The group is shown to be SL(3,R) in fact, without recourse to the Lie algebra. Finally, in this way, the group law of binary composition of the parameters is calculated easily in the group manifold.

For pt. I see ibid., vol.18, p.3141, 1985. A representation for unitary scattering operators acting on a symmetric Fock space and invariant under an SO(N) internal symmetry group is constructed. A group of transformations commuting with SO(N), is seen to be isomorphic to SL(2,R); the representations of SL(2,R) acting on the Fock space are shown to come from the discrete series of representations of SL(2,R). These representations are used to label the equivalent irreducible representations of SO(N) and the partial wave amplitudes of the scattering operators are shown to be matrix elements of the discrete series of representations of SL(2,R). The example of isospin internal symmetry and the pion triplet is briefly discussed.

For pt. II see ibid., vol.20, p.3565-76, 1987. Scattering operators invariant under the simplest spacetime group, namely the two-dimensional Euclidean group E(2), are investigated. Operators that commute with the action of E(2) on a Fock space generated by an infinite-dimensional representation of E(2) are constructed and shown to form a Lie algebra of SL(2,R). Unitary scattering amplitudes which include production reactions are given as matrix elements of the discrete series of representations of SL(2,R).

An exact method for constructing exact solutions of the sine-Hilbert equation is developed. It is shown that the SH equation can be transformed into a system of non-linear ordinary differential equations through a dependent variable transformation. Furthermore, this system of equations is transformed into a system of decoupled linear ordinary differential equations. Exact solutions of the SH equation are then constructed by means of a simple integration. The dynamical properties of the solutions are also studied in detail. The solution is interpreted as a superposition of N pulses which may be termed kinks, where N is a positive integer. Asymptotically for large time, it is composed of a pulse with a constant velocity and N-1 'static pulses'. This is a novel characteristic of the solution, unlike the well known N-soliton solution which behaves like N moving pulses for large time.

A classical mechanical system is analysed which exhibits complicated scattering behaviour. In the set of all incoming asymptotes there is a fractal subset on which the scattering angle is singular. Though in the complement of this Cantor set the deflection function is regular, one can choose impact parameter intervals leading to arbitrarily complicated trajectories. The authors show how the complicated scattering behaviour is caused by unstable periodic orbits having homoclinic and heteroclinic connections. Thereby a hyperbolic invariant set is created leading to horseshoe chaos in the flow. This invariant set contains infinitely many unstable localised orbits (periodic and aperiodic ones). The stable manifolds of these orbits reach out into the asymptotic region and create the singularities of the scattering function.

Hierarchies of soliton equations generated by an appropriate recursion operator are considered. N non-interacting Galilean-like point particles are connected with an N-soliton solution of an arbitrary equation from the hierarchy. The method of finding the particle representation of solitons is closely related to eigenstates of the recursion operator and master symmetries of the soliton equations.

Shear flow birefringence of simple fluids was studied by use of non-equilibrium molecular dynamics simulation, the results of which were comparable with the theoretical values calculated on the basis of the shear distortion of the radial pair distribution function for shear rates below epsilon _xz=2.0 (in Lennard-Jones reduced units).

Haldane (1982) conjectured that a simple generalisation of the Heisenberg spin chain leads to a completely integrable classical system without invoking the usual continuum approximation. Complete integrability is established for the generalised discrete chain by exhibiting a Lax pair and an infinite set of non-local conservation laws. The associated inverse scattering problem is also formulated providing explicit soliton solutions.

The modulation of a one-dimensional weakly non-linear quasimonochromatic ion acoustic plasma wave (the carrier) is considered. It is well known that the carrier is modulationally unstable for wavenumbers k larger than a critical wavenumber kc, and that when k is not near kc the modulations of the carrier are governed by a non-linear Schrodinger (NS) equation. The author shows that when k is near kc, and under certain assumptions, the modulations are governed by a modified form of the NS equation that involves higher-order non-linearities, and that the correct critical wavenumber for marginal modulational instability is slightly different from kc.

Many years ago it was pointed out by Holbourn (1936) that there is an apparent breakdown on angular momentum conservation on reflection of circularly polarised light. The author points out that the discrepancy may be resolved by considering transverse positional shifts on reflection as described by Schilling (1965), Imbert (1972) and subsequent authors, and examines the values of the shifts necessary to satisfy angular momentum conservation in various situations. It is found that shifts satisfactory in this respect may be obtained by use of a fairly general phase-shift argument-similar in formalism to that used by Julia and Neveu (1973)-at least for the case of total reflection. However, the situation is less satisfactory for partial reflection, nor is it clear that the shift values, even for the case of total reflection, agree with those measured by Imbert and others, or calculated numerically as by Ashby and Miller (1976) for instance. It is suggested that further experimental work may be useful in clarifying the situation.

A Hamiltonian formulation of magnetic field line equations is derived for arbitrary magnetic field configurations in orthogonal curvilinear coordinate systems. The canonical equivalence of the different descriptions thus obtained are explicitly demonstrated and action-angle forms are given in cases where there is a geometrical symmetry. Examples of Hamiltonians are discussed for usual plasma machines like the tokamak and the levitron. Finally, generalisation to non-orthogonal coordinate systems is worked out.

The authors discuss exactly solvable Schrodinger Hamiltonians corresponding to a surface delta interaction supported by a sphere and various generalisations thereof. First they treat the pure delta sphere model; self-adjointness of the Hamiltonian, spectral properties, stationary scattering theory, approximation by scaled short-range Hamiltonians. Next they extend the model by adding a point interaction at the centre of the sphere or, alternatively, a Coulomb interaction. Finally the whole analysis is extended to the case of a delta ' sphere interaction, first taken alone, then superimposed on a point interaction or a Coulomb potential.

A generalisation of Weyl's correspondence rule and the associated Moyal bracket for systems with non-linear phase spaces that admit symplectic transitive group actions is presented. This provides a simple account with explicit constructions of some of the general results of Bayen et al. (1978).

The authors extend their previous work on coherent paired states associated with the Lie group SU(1,1). Whereas the earlier states were defined with respect to a single type of canonical boson or (linear) quantum harmonic oscillator, the new states are defined in terms of two distinct types of bosons or oscillators. The new coherent states may again, on the one hand, be viewed as ordinary (Glauber) coherent states in the two-boson Hilbert space spanned by arbitrary numbers of two distinct Bogoliubov quasiparticles associated with the original bosons via a generalised Bogoliubov transformation. Alternatively, expressed wholly in terms of the original bosons these new coherent states are reached from the ordinary coherent states via a unitary (pairing) transformation which is shown to be associated with the entire so-called discrete series of representations of the group SU(1,1). As an important illustration of the use of these states and transformations, they study in detail a rather general class of quantum Lagrangians which includes the damped linear harmonic oscillator. They thereby illustrate their possible usefulness in applications to quantum many-body or field-theoretic processes involving fluctuation-dissipation phenomena in general.

The pure Glauber (harmonic oscillator) coherent states provide a very useful basis for many purposes. They are complete in the sense that an arbitrary state in the Hilbert space may be expanded in terms of them. Furthermore, the well known P representation provides a diagonal expansion of an arbitrary operator in the Hilbert space in terms of the projection operators onto the coherent states. The authors study the extensions of these results to the analogous mixed states which describe comparable harmonic oscillator systems in thermodynamic equilibrium at non-zero temperatures. Their results are given for the general density operator which describes the mixed squeezed coherent states of the displaced and squeezed harmonic oscillator. They show how these squeezed coherent mixed states similarly provide a very convenient complete description of a Hilbert space. In particular they show how the usual P and Q representations of operators in terms of pure states may be extended to finite temperatures with the corresponding mixed states, and various relations between them are demonstrated. The question of the existence of the generalised P representation for an arbitrary operator is further examined and some pertinent theorems are proven. They also show how their results relate to the Glauber-Lachs formalism in quantum optics for mixtures of coherent and incoherent radiation. Particular attention is focused both on the interplay between the quantum mechanical and thermodynamical uncertainties and on the entropy associated with such mixed states.

Using Feynman's polygonal paths for path integrals, the exact evaluation of the propagator for a time-dependent harmonic oscillator with a time-dependent inverse square potential becomes possible. The propagator at and beyond caustics is then evaluated by including the Maslov correction factor. Finally, the authors obtain the wavefunctions for the propagator obtained.

Moment recurrence relations are shown to be useful in perturbation calculations. General equations are developed and different cases are discussed. The anharmonic oscillator and Zeeman effect in hydrogen are used as illustrative examples.

A non-linear evolution equation for the density operator is proposed which models the irreversible dissipative time evolution of a system in contact with its surroundings. Various properties of the proposed equation are discussed. As an illustration the three-state system is studied in detail.

Using quantum mechanical generalised coherent states, mode excitation by Gauss-Hermite or Gauss-Laguerre beams with spherical wavefronts is treated. For mode coupling between two optical waveguides with offset, tilt and gap recursion relations and sum rules are found.

A correct procedure for constructing supersymmetry in three dimensions is presented. The degeneracies are found between states of the same l but different n and Z and the previous results on the Coulomb and the three-dimensional isotropic oscillator problems are reestablished. The authors also consider the hydrogen-helium problem and find supersymmetry to hold to a good approximation.

A quantal version of resonance overlap leading to global chaos is investigated. Classical chaos is reflected in irregular features of quantum beats in the phase space profile of eigenfunctions. These quantum beats are due to interference between nearby branches of bifurcated classical manifolds. Statistical properties of the quantum system qualitatively reflect the classical transition to global chaos although some discrepancies from classical ergodicity are found in the quantum mechanics even in the fully chaotic regime.

The peculiarities of the operator method in solving the Schrodinger equation with periodic potentials are discussed. Approximate analytical solutions of the Mathieu equation and Schrodinger equations for a two-level system are found on the basis of the operator method. An analogous method is used for the analytical estimation of the quasistationary state energy and width.

Spacetime is modelled as a fractal subset of Rⁿ. Analysis on homogeneous sets with non-integer Hausdorff dimensions is applied to the low-order perturbative renormalisation of quantum electrodynamics. This new regularisation method implements the Dirac matrices and tensors in R⁴ without difficulties, is gauge invariant, covariant and differs from dimensional regularisation in some aspects.

A new variational framework for the Liouville equation is presented. The Vlasov equation is obtained from the complete factorisation of the trial distribution function, while the linearised Vlasov equation arises from the additional assumption of small oscillations around an equilibrium state. It is stressed that the definition of this state does not necessarily require minimisation of the free energy. The classical energy-weighted sum rule is derived.

The author presents a statistical mechanical analysis for a one-dimensional lattice gas in which the pair interaction potential is exponential and repulsive of Kac type phi (x)= alpha exp(- gamma mod x mod ) with alpha >0. mod x mod >0 (this analysis is complementary to the one studied by Newman (1964) for a one-dimensional fluid). The main objectives of this work are the following. First, the author derives an analytical expression (in the weak long-range limit, gamma to 0) for the traces and the maximum eigenvalues of the Kac operators. Second, the author derives the equation of state for the repulsive lattice gas in the weak long-range limit. Furthermore, the author mentions the possibility of the application of this model to study classical problems in biophysics. Third, the author finds interesting properties for the non-Hermitian Kac operator which suggest that the spanning property for this operator is possible.

A method is proposed for the estimation of the optical electron response function in a polar medium without recourse to perturbation theory. It has proved possible to write the temperature Green function of the electron coupled with a short-range potential and interacting (in a dipole approximation) with long-wave polarisation fluctuations proceeding from the functional integral representation. Using the steepest descent method in the high-temperature limit, comparatively simple expressions have been obtained. These are compared with experimental data for impurity ions in polar media and temperature dependences of the edge of the medium absorption band (Urbach rule).

The authors study diffusion on a statistically self-similar fractal substrate with random transition rates possessing a hierarchical structure. The master equation is solved analytically by means of a recursion technique and different choices of random hierarchies for the transition rates are considered, leading to algebraic, stretched exponential or exponential-logarithmic behaviour for the moments and the correlation decay. Making use of the central limit theorem, analytical expressions for the fluctuation corrections to lowest order in the relative variances are derived for the first two cases where the behaviour is qualitatively the same as in the systems without fluctuations. The sign of the corrections turns out to depend on the temporal and spatial scale at which fluctuations are externally suppressed; they are negative if the diffusion is taking place at smaller scales, positive otherwise. The relevance to diffusion in a turbulent medium (intermittency corrections) is discussed.

The effects of heterogeneities on the steady state flow of a single fluid in a porous medium are examined. It is argued that incomplete knowledge of the permeability requires the use of a stochastic model of the system. It is shown that the problem may be written as a field theory which allows a perturbation series to be expressed by diagrammatic means. This allows the calculation of effective permeability, the mean pressure and the pressure variance. The method, as well as recovering familiar results, gives a formal means of improving the approximation and approaching more complex systems.

A theory based on a density-functional formalism is developed to obtain the solvent-induced potential between different sites on a polymer. The expression found for the potential surface is not pair decomposable. The theory brings to light a new approach for deriving solute-solvent correlation functions and computation of multipoint correlation functions in fluids.

The authors show how a random walk in a plane, constrained to enclose a given area, can be used to approximately represent the properties of an entangled polymer molecule. The statistical mechanical properties of the loop are calculated exactly and the distribution function for the enclosed areas is found. For the case of a random walk with free ends joined by a straight line segment, the distribution function is given by the Cauchy distribution. This implies that the area has statistical fractal properties but does not have a mean. For a genuinely closed random walk, a mean exists but the distribution of areas is not fractal. The spatial and mechanical properties of the constrained configurations have also been calculated analytically. If the unrestricted coil can be regarded as an entropic spring of zero natural length, then the area-constrained configurations behave qualitatively like springs with a finite natural length. The deformation behaviour also shows both softening and hardening dependent on the area imposed.

The energy and magnetisation density of the (2+1)D Ising model are investigated. For the thermal exponent the author finds x_epsilon =1.42+or-0.02, in agreement with hyperscaling. The Privman-Fisher universality hypothesis is confirmed by two methods. First, the author presents, at the critical point, numerical data for the finite-size scaling amplitudes of the (singular) free energy, magnetisation, susceptibility and surface tension and checks them for universality. Second, using the explicitly computed eigenvectors of the Hamiltonian, it is shown that the ratios of matrix elements of the energy and magnetisation density are universal.

A four-parameter bond-moving renormalisation is used to study the critical properties of the bond-diluted q-state Potts model on Sierpinski carpets. Fixed points, critical exponents and a phase diagram are obtained. In the authors' results, there is a borderline value q=q_t; for q<q_t, the diluted model exhibits the same critical behaviour as the pure system. However, for q>q_t there is crossover to a new diluted fixed point. This behaviour is similar to that of the diluted system on a regular lattice with translational invariance. They give some values of q_t for carpets with different (b, l ) where b and l are structure parameters of carpets.

The performance of the Hopfield model of a neural network with extensively many weighted patterns is analysed. If the system size is N, then N patterns, each provided with a suitable weight, are stored. The weights may be associated with a temporal order and, if appropriately chosen, they allow a gradual fading out of the extensively many stored patterns. Particular emphasis is put on the underlying mathematical structure.

A systematic kinetic theory is developed for calculating transport properties on 2D lattices with random site impurities in concentration c, which can be modelled by hopping models. The authors' main results are expressions in terms of lattice sums for the static and frequency-dependent conductivity and for the velocity autocorrelation function.

In a dense fluid with very large heat flow J. the thermal conductivity lambda = lambda ₀+ lambda ₂J². An estimate of lambda ₂/ lambda ₀ is made for hard spheres with mass and diameter appropriate to Ar and density just below the gas-solid transition. This estimate indicates that Fourier's law will hold in the form appropriate to low J if J<or approximately=10⁹ W m^-2 and that the J dependence of pressure and internal energy is of the same order as that of lambda . The calculation utilises an earlier derivation from the classical Liouville equation which expresses lambda at high density in terms of the thermodynamic pressure. The latter can be calculated with the aid of reciprocity and integrability conditions of extended non-equilibrium thermodynamics which treats J as a state variable.

Yoshida (1983) described a connection between scale-invariant autonomous systems of ordinary differential equations, algebraic first integrals and resonances. In his analysis it is assumed that the scale invariance determines the dominant behaviour. Here the authors discuss the case where the dominant behaviour is not determined by the scale invariance. For the constructed example they also give the Lax representation.

The author presents an asymptotic analysis of a model of polyion conformations in dilute solutions which is based on a combination of the Poisson-Boltzmann equation and the self-consistent Gaussian chain model. Logarithmic corrections to power scaling are shown to play an important role in the case of salt-free solutions.

Methyl-group rotation is a one-dimensional example of molecular rotational tunnelling. The tunnel splitting of the methyl-group torsional ground state is obtained using a path integral formulation and semiclassical approximation.

The author presents exact enumeration results for the elementary cellular automation rules 22, 30, 45 and 120 in Wolfram's notation. Obtained by a new algorithm, they improve previous results considerably. They confirm the existence of 'hidden' power-behaved long-range correlations in the patterns created by these rules.

The authors construct a simple model of ballistic motion in a random environment in an arbitrary number of dimensions. The motion is determined by a quenched set of random orthogonal matrices on a hypercubic lattice, relating incoming and outgoing directions at a particular site. They select, study and classify a set of these matrices which agree with intuitive notions of 'isotropy'. A mean field theory is constructed of the transition between localised and extended trajectories and a qualitative discussion of various features of the phase diagram is made.

For original paper see ibid., vol.20, p.3551-2. (1987). The author argues that the boundary conditions in the spin-glass phase can be chosen in analogy with the ferromagnetic case, such that the results of a previous paper, see ibid., vol.20, p.2211-5 (1987) are unchanged despite the subsequent remarks of Nishimori (1987) concerning the problem of the boundary conditions.

The authors present an explicit expression for the probability distribution for the position of a continuous-time random walker in an arbitrary number of dimensions when the interjump density has a long time tail, in contrast to earlier results which require numerical inversion of a Fourier integral. They replace this numerical procedure by one that relies on the method of steepest descents. Their results are applied to diffusion on a comb and on a percolation cluster generated on a Cayley tree at criticality and are confirmed numerically.

For original article see ibid., vol.19, p.2725 (1986). The authors have been made aware of further literature relating to their previous publication. They summarise the additional material and correct an error of attribution.