Table of contents

Critical behaviour of conduction in random networks of super-normal conductors near the percolation threshold is investigated from a geometrical point of view. A fractal dimensionality d_CA describing a contact area between typical size clusters is introduced and evaluated explicitly. Non-analytic power law behaviour (divergence) of the conductivity and dielectric constant is given a purely geometrical interpretation in terms of d_CA. Enhancement of nonlinearity in networks of super-nonlinear conductors is elucidated.

By means of quantum Monte Carlo simulation the authors show that the binding energy of small neutral (⁴He)_N clusters is a smooth monotonic function of mass number N. Magic numbers observed in a recent experiment on charged He clusters are therefore not due to enhanced stability of neutral clusters at certain sizes.

The problem of understanding the asymptotic statistical behaviour of the winding angle, theta _N, of a self-avoiding walk of N steps on a planar lattice is posed. Exact series expansion data for the square lattice up to N=21 are reported. These data and Monte Carlo estimates up to N<or approximately=170 steps are fitted well by a logarithmic growth law for ( theta _N²). The ratio ( theta _N⁴)/( theta _N²)² appears to approach a limit of 2.9 to 3.2, which is close to the Gaussian value, 3. Heuristic scaling arguments are consistent with simple logarithmic growth (and also illuminate the logarithmic behaviour known rigorously for free Brownian motion).

A Flory type theory is presented for a model of kinetic gelation in branched polymers. Fractal dimensionality of the gel cluster at the gel point is estimated to be D_g approximately (d(4x-d)+4x)/(4x-d+2) where x is the exponent for the number of monomers; this is consistent with the simulation results. It is argued that the growth in irreversible gelation is different from that in aggregation.

The author discusses the apparent failure of a Flory approximation for directed linear polymers, commenting on the details implicit in this type of picture. The relationship of a standard Flory viewpoint to a simplified application of fractal theory is explored, and a correspondence is drawn between aspects of both theories.

The well known power law for the mean square distance of a polymer chain in a good solvent is considered by using the typical behaviour of instantons. It is shown that the mean field and self consistent field exponent coincide with the instanton treatment.

Using the low-temperature series expansions, the tricritical parameters for the interacting hard squares are estimated to be t_tc=exp(- epsilon /kT_tc) approximately=5.45, y_tc approximately=1.040, values rather different from previous estimates derived from other methods.

The mean field renormalisation group approach is applied to the spin-¹/₂ anisotropic Heisenberg model. The critical temperature and the critical exponent are obtained as a function of the anisotropy for the two-dimensional model on square, triangular, and hexagonal lattices. The three-dimensional model on a simple cubic lattice is also analysed.

The author argues that the critical exponent t of random conductance networks near the percolation threshold is given by t=(d-1) nu for low dimensionalities and t=1+ beta ' for high dimensionalities, where nu is the correlation length exponent, beta ' the backbone exponent and d is dimensionality. The author argues that what separates the two regimes is a critical fractal dimensionality D_l which equals 2. The author also argues that D_l is also a critical fractal dimensionality for fractals such as lattice animals and diffusion-limited aggregates. The result for low dimensionalities has been also obtained by Aharony and Stauffer by a different argument.

The crossover exponents describing the behaviour of continuous spin models at dilutions near the percolation threshold are calculated within an epsilon expansion, where epsilon =6-d, where d is the spatial dimensionality. The result confirms a recent calculation for the random resistor network and shows that earlier calculations of the crossover exponents are incorrect.

Based on Baxter's ideas the authors present a method to solve spin models on triangular lattices. If the model has only nearest neighbour interaction and it satisfies the star-triangle relation then the exact free energy can be obtained. They apply the method to reobtain the critical triangular Potts model and to extend the self-dual Z(N) solutions of Fateev and Zamolodchikov to the triangular lattice.

The author shows that in three dimensions the consideration of single-spin distributions containing a factor of exp( lambda s⁶) can lead to statistical mechanical models of dramatically different character from those defined by the usual field theory picture of the renormalisation group theory of critical phenomena. These models plainly fall outside the usual, and previously expected, universality class.

The Clebsch-Gordan coefficients (CGC) and the Racah coefficients for the SU(2) and SU(1,1) groups are studied as functions of a discrete variable. It has been shown that CGC for SU(2) and SU(1,1) groups may be considered to be discrete analogues of wavefunctions for the one-dimensional Schrodinger equation with the Poschl-Teller potential. Expressions for CGC and 6j-symbols of the SU(1,1) group have been found through the Hahn and Racah polynomials. Consideration is given to the asymptotic properties of eigenvalues and eigenfunctions of the Hamiltonian of an asymmetric top and of the Bargmann-Moshinsky operator Omega .

A new method is presented to introduce classical mechanics elements into the problem of obtaining the spectrum of an operator H(p,q). A finite-rank functional space is created by centring complex wavepackets on a discrete number of points on an equi-energy of the classical H(p,q) and by placing real wavepackets in the classically forbidden region. The latter span the active subspace, P, and the former the inactive subspace, Q, for an application of the method of Bloch-Horowitz. Depending on the rank retained for Q, some levels can be calculated with an extreme accuracy in a simple model while others remain evasive. A semi-classical study of the Green function in the inactive subspace Q, classically allowed, gives a clear explanation of this phenomenon and sheds new light on the significance of the semi-classical approximation for the propagator. An extension to the problem of barrier penetration is proposed.

It is shown that the Kustaanheimo-Stiefel transformation which transforms the three-dimensional Kepler problem into that for a four-dimensional harmonic oscillator may be expressed in terms of quaternions In this form the transformation represents a continuous mapping of a triad of orthonormal vectors fixed in space into a rotating triad of orthogonal vectors where one of the unit vectors is mapped into the position vector of a moving particle. The reduction of the Kepler problem to that of a harmonic oscillator is straightforward and direct in the quaternion formalism. The complex stereographic transformation used recently by the author in the corresponding reduction of Schrodinger's equation for the hydrogen atom is shown to be closely related to the quaternion representation.

Elastomers and gels are characterised by a large ratio of the bulk to the shear modulus, and are consequently capable of propagating shear waves of very low velocity. The authors have elsewhere examined the normal modes excited in gels contained in rectangular cells of infinite length with rigid boundary conditions, and have shown that the fundamental mode consists of a roller structure which is periodic along the long axis of the cell. Here they extend the variational analysis to include gels of semi-infinite length and of square cross-section with one end free; it is found that the fundamental is a localised mode, in which the motion of the gel is confined to the vicinity of the free surface. Gels of finite length with a free end are also investigated. Experimental observations are made both of quasi-longitudinal and of quasi-transverse modes in polyacrylamide gels; good agreement is found between the measured frequencies of the first few modes and the calculated eigenfrequencies for this system.

The choice of a representative state in an incomplete quantal state determination is reconsidered and two main possibilities, the most probable and the expected representative state are compared. It is shown that probability densities over a set of states, underlying an expected state, are in complete agreement with the standard quantal formalism. Examples for some simple densities are given and discussed.

Lattice constants are defined in a general way such that weak lattice constants, strong lattice constants, free multiplicities and coincidable occurrence factors are cases of the generalised lattice constants. A general method to express a lattice constant of one system as a linear combination of lattice constants of another system is described. Some properties of the conversion matrices are discussed. A systematic method to express the lattice constant of a reducible graph in terms of lattice constants of irreducible graphs is also studied.

Although the two-dimensional spin-¹/₂ Ising model was solved in zero field in 1944, no exact results are yet available for the spin-1 model. This model should have the same dominant critical exponents as the spin-¹/₂ model, but appears to exhibit non-analytic corrections to scaling as expected for a phi ⁴ model and unlike the spin-¹/₂ model. In this paper new 45-term low temperature series for the magnetisation, specific heat and susceptibility of the spin-1 model on the square lattice are presented and analysed. A new 24-term staggered susceptibility series for the hard square model is presented and the extant order parameter series for this model (which is also in the phi ⁴ universality class) are also considered. For both models a non-analytic confluent correction with exponent 1.0< Delta ₁<1.3 is found. The validity of this result is enhanced by a comparison with the S=¹/₂ case.

Low-temperature series expansions have been derived for the random field Ising model with a delta -function distribution on a Bethe lattice by two independent methods: (a) the finite-cluster method which uses graph embeddings and appropriate weighting functions; (b) the use of a recursion relation specific to the Bethe lattice. Numerical values have been evaluated when the coordination number q=3, 4 and the coefficients analysed to assess critical behaviour. For small fields, and temperatures near to T_co, the critical exponent of the magnetisation seems to retain its mean-field value. But there is clear evidence of a change in critical behaviour at some point on the critical curve. It is argued that when q>3 a tricritical point is indicated as found by Aharony in his mean-field solution.

The trails problem (self-avoiding path walks) is reconsidered by the conventional series expansion approach. An exact enumeration on the square, triangular and simple cubic lattices is done by computer up to 18, 12 and 11 steps respectively. The more self-consistent and refined results, between the connective constant and the corresponding critical exponent, indicate the possibility of there being a different universality class from that of the self-avoiding walk. An average approach based on Stolz's theorem is proposed which often appears to be smoother and better than the traditional one.

It is shown that, within position-space renormalisation group (PSRG) theory, neighbour-avoiding walks (NAWs) and self-avoiding walks (SAWs) on the square lattice obey the same 'end-to-end distance' critical exponent nu . This universality is shown by applying a recent finite lattice renormalisation transformation to ordinary and oriented NAW and SAW problems and then using an exact equivalence between SAWs on an oriented lattice and NAWs on its covering lattice. The universality class includes several oriented walk problems.

The polymer size distribution in Flory's polymerisation model, A_fRB_g is in equilibrium and non-equilibrium of the general form c_k=AN_k xi ^k both below and above the gelation transition. Here explicit expressions are derived for A( alpha ) and xi ( alpha ) as a function of the extent of reaction alpha . The combinatorial factors N_k are calculated from the recursion relation 2(k-1)N_k= Sigma _i+j=kK_ijN_iN_j with K_ij=s_i(f)s_j(g)+s_i(g)s_j(f) and s_k(f)=(f-1)k+1. Using a generating function technique the authors express N_k in terms of Laguerre polynomials (g to infinity ) and Jacobi polynomials (g finite), whose large-k behaviour is of the form N_k approximately=Bk^{- tau} xi _c^-k with tau =³/₂ (no gelation occurs) if f or g equals 1, and with tau =⁵/₂ (gelation occurs) if f>1 and g>1.

Smoluchowski's coagulation equation with coagulation rate K_ij varies as ij for the process A_i+A_j to A_i+j is solved in the presence of a removal term of the general form -a(t)c_k-b(t)kc_k. The solution for the monomer initial condition has the same k-dependence as in the case without sinks, but for b>b₀>0 the solution never reaches the gelation transition.

The author considers the problem of obtaining long-time diffusivities D^L(k) from dynamic light scattering data for suspensions of spherical particles. Starting from the integro-differential memory equation for the intermediate scattering function F(k,t), the author derives an infinite-order linear differential equation for F(k,t) with coefficients mu _n(t) expressed as finite-time moments of the memory function. The author obtains an exact analytic representation of these moments for a model suspension of flow-density hard spheres; numerical results for the moments in this model suspension are presented for a range of values of wavenumber and time. The numerical study shows that at short or intermediate times t one may neglect all but the low-order moments leading to a simplified differential equation for F(k,t). At intermediate times this differential equation may be inverted to obtain the lowest moment mu ₀(t) in terms of the experimentally measured slope of ln F(k,t). The numerical study of the moments for the hard-sphere system indicates that even at quite short times mu ₀(t) gives an accurate estimate of mu ₀( infinity ) from which the diffusivities D^L(k) can be obtained.

A two-level system which is coupled to a multitude of oscillators, such that the total system formally or geometrically is governed by a mirror symmetry, can be exactly diagonalised with respect to the two-level subsystem (FG transformation). This has been thoroughly exploited for the two-site exciton localisation problem as well as for two-site quantum transport. A generalisation to the N-level system is given, provided the N levels establish a regular representation of an abelian group, and if the latter symmetry also governs the multi-oscillatory subsystem. Implications for the quantum transport problem are discussed.

Families of metrics are found in synchronous coordinates generalising Tolman's spherically symmetric solutions, both for non-coherent dust and for special cases of perfect fluid with non-zero pressure. New metrics depend on an additional arbitrary function of R, theta , and they lead specifically to solutions of the Schwarzschild and Friedmann types (but differing from them in general, especially in their Petrov types and isometries).

The lowest even- and odd-parity energy levels of Schrodinger's equation for the interaction x²+ lambda x²/(1+gx²) for arbitrary positive values of lambda and g are obtained accurately from a first-order variational perturbation treatment.

For lattice sites at integer values, the probability of a bond between sites m and n is a_{mod m-n mod} where a₀=0. The author strengthens Schulman's criterion for absence of an infinite cluster and suggest a Monte Carlo method for determining critical probabilities. The results derive from comparison of Schulman's Bethe lattice to branching processes. The author notes that a_n=pa^{mod n mod} (pa,a<1) is probably the mathematically tractable example of long-range percolation although it does not permit an infinite cluster.

The author confirms that the spin-glass transition at the AT line is a third-order transition in contradiction to the result of Elderfield. The author calculates the discontinuities of the first thermodynamic derivatives of the total susceptibility along the AT line. The local susceptibility has continuous derivatives except at T=T_c for zero external field.

The author studies the spatially homogeneous Kac model of the nonlinear Boltzmann equation in 1+1 dimensions (velocity v and time t) when an external force term is present. The author finds a closed solution. The force term decreases exponentially with time, like constant₁ exp-constant₂*t, where both constants depend explicitly on the moments of the cross section. The author finds, for the relaxation to equilibrium, that the Tjon overpopulation effect depends on both the initial condition and the microscopic model of cross section. For the existence of this effect, the author establishes a criterion which is a well defined linear combination of the moments of the author cross section.