Table of contents

By using spin-coherent states, the author shows that correlation functions for fermions or quantum spins follow from solutions to Langevin equations associated with a functional integral representation of the partition function. His method is applicable to any number of dimensions, may also be combined with boson variables, and is suitable for computer simulations.

It is shown that the partition function of an interaction Hamiltonian for the hydrogen bonding in water molecules may be expressed as the generating function of a bond-correlated percolation model with a bond probability p depending on the hydrogen-bond strength and the temperature T. Assuming that the molecules with larger number of active bonds have a larger volume per molecule, one may show that, for temperatures near but higher than the temperature of maximum bond fluctuations, the isothermal compressibility K_T and the constant pressure specific heat C_p increase, while the thermal expansivity alpha _p becomes more negative, as T decreases. Other unusual behaviour of supercooled water can also be explained by this model.

A class of irreversible coagulation processes can be modelled by Smoluchowski's coagulation equation with rate constants K_ij=A+B(i+j)+Cij (non-negative A, B and C). For C not=0 a gelation transition occurs. The authors obtain explicit solutions for the size distribution c_k(t) with c_k(0)= delta _k1. Next, they construct and solve the equations for reversible polymerisation by incorporating break-up processes in the kinetic equation with a unimolecular fragmentation rate F_ij= lambda N_iN_jK_ij/N_i+j. The degeneracy factors N_k obey (k-1)N_k=¹/₂ Sigma K_ijN_iN_j with i+j=k and N₁=1, and the strength parameter lambda =exp(g/k_BT), where the binding energy g to - infinity for irreversible coagulation. Explicit results are only given for Flory's polymerisation models RA_f and BRA_f-1. In the vicinity of the gel point the authors verify the scaling hypothesis and calculate critical exponents.

Several criteria are discussed for a phase transition from partial to complete wetting in binary mixtures and their analogues. These criteria are based on order parameters and mechanical stability. A further geometrical description of the phase transition is given.

In the framework of the Parisi solution of an infinite-ranged Ising spin glass the author derives the equations which determine the free energy, the magnetisation and the function q(x) for the general case H not=0 and J₀ not=0. He solved these equations for two cases: (i) J₀=0, H>>1; (ii)J₀>>1, H=0. An existence of massless modes is exactly proved.

The internal state labelling problem for the d-row irreducible representations of SU(n) (where 2d<or=n), when reduced with respect to SO(n), is shown to amount to the external state labelling problem for U(d). The canonical solution of the latter due to Biedenharn et al. (1967) provides a canonical solution of the former, which reflects the operation of Littlewood's branching rule for U(n) contains/implies O(n) in a very simple way.

The authors have investigated numerically and analytically the period-doubling bifurcations and multifurcations of the periodic orbits of the conservative sine-Gordon mappings. They have derived a general equation for the appearance of multifurcations in conservative mappings. In agreement with many recent studies, they also find evidence that such mappings possess universality properties. They also discuss the role of multifurcations in conservative mappings exhibiting chaotic behaviour.

The method developed previously by the author for obtaining eigenfunctions in the form (polynomial)*exponential (polynomial) for a linear second-order differential equation in normal form is extended to embrace the general case when the interaction potential (or square of the refractive index profile) has the form (polynomial)/(polynomial). Various situations arise in the development of the theory, some of which can be concluded algebraically, but the majority require numerical calculations regarding the rank of quite a general matrix containing many unknown parameters.

By using the hodograph method the author finds almost all the solutions of the classical shallow water equations. The author also constructs an auto-Backlund transformation (superposition principle) on the set of these hodograph solutions and shows that this transformation is canonical relative to a symplectic form introduced by Manin (1978).

A general formulation of the eigenvalue condition is pointed out for a one-dimensional stationary part of a separable Schrodinger equation with finite bounds, i.e. where the wavefunction vanishes at two arbitrary values of the independent variable. The procedure is general and applies equally to other eigenvalue problems of second-order, homogeneous differential equations. Where appropriate a continuous density of eigenvalues can be defined and evaluated without first solving the eigenvalue problem. Illustrative examples are provided, including gravitational and harmonic oscillator potentials.

The temporal energy spectrum E^s,a(t) and the uncertainty product Delta p^sa(t) Delta x,^s,a(t) are derived from an analytical solution psi (x,t) of the initial-boundary-value problem for the Schrodinger equation of a particle contained between moving potential walls at x=+or-s(t), which are set in motion according to an arbitrary (non-relativistic) translation s=s(t) at time t=0. Both symmetrical (s) and antisymmetrical (a) particle states are considered as initial conditions in the region -s(0)<or=x<or=+s(0). The physical implications of the compression and expansion of the probability density by the moving walls on the wave mechanics of the particle are discussed. The results are understandable within the statistical interpretation of quantum theory.

The quantum limits to information flow are explored developing the central concept of a medium comprising several channels through which the information flows. In each channel there is an inequality between information flow I and energy flow E I²<or=E pi /(3h(cross)ln²2) a relationship which, though speculated on before or derived for only a restricted class of systems, is proven for a wide range of systems. Applications are made to the energy cost of computing and to the maximum rate of cooling, Q, which in any one channel is Q<or= pi k_B²T²/(3h(cross)).

The authors study the Aharonov-Bohm effect in a gauge for which the vector potential vanishes wherever the magnetic field does. They then show how gauge invariance implies the existence of Aharonov-Bohm scattering and excludes solutions recently discussed in the literature.

The author shows the following: (1) the proper framework for testing Rastall's theory and its generalisations is in the case of non-negligible (i.e. discernible) gravitational effects such as gravity gradients; (2) these theories have conserved integral four-momentum and angular momentum; and (3) the Nordtvedt effect then provides limits on the parameters which arise as the result of the non-zero divergence of the energy-momentum tensor.

The conformal structure of Schwarzschild's space-time is analytically extended through the hypersurfaces at null infinity. The space-times on the other side of infinity are conformally isometric to Schwarzschild's space-time with negative mass. Analogous analytic conformal extensions of any analytic asymptotically flat space-times can be made.

The field equations in the Brans-Dicke (BD) scalar-tensor theory of gravitation are solved for a spherically symmetric metric. The solutions generalise earlier conformally flat results and may all be considered as describing the field of a charged mass point surrounded by a scalar-tensor field. The conformally flat solutions are shown to be not physically meaningful for 'standard' BD theory with 2 omega +3>0.

It is shown that a system of N (N to infinity ) hard discs (of diameter sigma ) in a narrow box, of width D (D< square root 3 sigma ) and length L=Nl, is unstable for a certain range of D and l when D is variable.

The Walfisz-like formula for the number of lattice points of an arbitrary m-dimensional lattice in a hyper-ellipsoid with given semi-axes is derived from Poisson's summation formula. Applications to (i) the evaluation of certain lattice sums and (ii) the calculation of the expressions for the density of states of a single non-relativistic particle as well as of a relativistic particle enclosed in a rectangular m-dimensional box of finite size and subject to different boundary conditions are given.

It is shown that investigation of the magnetisation and the susceptibility of the finite-volume ferromagnetic Husimi-Temperley model reduces to the study of the nonlinear Burgers equation. The thermodynamic limit corresponds to vanishing diffusion coefficient and the resulting shock wave to the jump in the spontaneous magnetisation. The upper bound on the susceptibility of the finite system derived by the approximating Hamiltonian method is compared with the asymptotic form (as N to infinity ) of the susceptibility obtained from the solution of the Burgers equation.

The nature of the phase transition in the Potts model is studied when second neighbours or infinite range couplings are added. A new criterion, recently proposed, is applied.

New lower bounds on the connective constant of the square and simple cubic lattice self-avoiding walks are obtained, by enumerating a particular subset of self-avoiding walks and using a result of Kesten. The author finds mu (sq)>2.5680 and mu (SC)>4.352.

The authors complete the solution of the square lattice gas with nearest neighbour exclusion and attractive next-nearest neighbour (diagonal) interactions on the special surface corresponding to regimes III and IV of the generalised hard hexagon model. The interfacial tension, correlation length and sublattice density difference are calculated throughout these regimes by obtaining the eigenvalues of the row-to-row and corner transfer matrices. The associated critical exponents are found to be mu =v=⁵/₄, beta =³/₃₂ in regime III and mu '=v'=⁵/₂, beta '=¹/₄ in regime IV. In particular, their results confirm the recent proposal by Huse that regime III is the first-order coexistence surface (separating the disordered fluid phase from the square ordered solid phases) and that the regime III/regime IV boundary is a line of tricritical points.

The authors consider a fully frustrated Ising model on a square lattice, depending on four parameters. The partition function is shown to be equivalent to the two-parameter (ferromagnetic) Onsager partition function. This result generalises a relation established by Southern et al. (1980), and can be checked in the particular case of the (one-parameter) Villain model. In particular, the residual entropy of the Villain model is linked to the free energy of the Onsager model at criticality. The reduction from four to two parameters, occurring in this mapping, is studied in the light of the inverse relation satisfied by the partition function of the frustrated model.

Percolation problems on a Bethe lattice with Lth-nearest-neighbour bonds are treated exactly by a generalised recursive method. For the site percolation, both the cases without and with interbranch bonds are considered. Formal expressions for the critical percolation p_c, percolation probability P(p) (near p_c) and the mean cluster size S(p) (p<p_c) are obtained for any K (K+1=degree of the Bethe lattice) and L. For the bond percolation, only the case of K=2 and L=2 is considered. The method described can be extended to other more complicated branching media including decorated Bethe lattices.

A class of optimal transformations is defined through the weight function which, when implemented into simple approximations of the renormalisation group, predicts values for the thermal exponent, y_T, and the magnetic field exponent, y_H, which are in better agreement with the exact results than those obtained with the majority sign rule. Specifically, the authors find that for the ferromagnetic triangular Ising lattice y_T=0.914 and y_H=1.908 (using the same approximation, the majority sign rule predicts y_T=0.736 and y_H=1.669). This weight function is used to calculate the properties of the ferromagnetic and antiferromagnetic square Ising lattice with a two-cell cluster and a one-hypercube approximation. The weight function is the first stage in the development of a real-space renormalisation group transformation free of the 'peculiarities' observed by Griffiths (1981).

The exact solution (size distribution c_k(t) and moments M_n(t)) of Smoluchowski's coagulation equation (S-model) and of a modified equation (F-model) with a coagulation rate K_ij=ij for i- and j-clusters is obtained for arbitrary c_k(0) in the sol (t<t_c) and gel (t>t_c) phases, where t_c is the gel point. The behaviour of c_k(t) and M_n(t) is given for k to infinity , t to infinity and t to t_c. The critical exponents, critical amplitudes and scaling function that characterise the singularities near the non-equilibrium phase transition are calculated. For short-range c_k(0) (i.e. all M_n< infinity ) the F-model belongs to the universality class of classical gelation theories and of bond percolation on Cayley trees; the S-model does not.

Generating functions are defined which permit the computation of moments of the end-to-end length of random walks. These functions are applied, within the framework of the Domb-Joyce (1972) model, to the development of a two-parameter perturbation series for the contraction factor alpha _ij^-1 of the mean reciprocal distance (R_ij^-1) between the segments i and j of self-avoiding walks. The case where i or j denotes a chain end is shown to be fundamentally different from the case where neither denotes a chain end.

The Fronsdal integer spin equations (1980) for zero mass are put into first-order Duffin-Kemmer form. The minimal polynomials of the coefficient matrices are determined and compared with the massive case. It is also noted that the various antisymmetric tensor gauge field theories of recent interest are included in a general massless Duffin-Kemmer formalism of Harish-Chandra (1946).

By deriving the relativistic form of the ionisation equation for a perfect gas it is shown that the usual Saha equation is valid to 3% for temperatures below one hundred million Kelvin. Beyond 10⁹K, the regular Saha equation is seriously incorrect and a relativistic distribution function for electrons must be taken into account. Approximate forms are derived when only the electrons are relativistic (appropriate up to 10¹²K) and also for the ultrarelativistic case (temperatures greater than 10¹⁵K).