Table of contents

A solving formula of the two-dimensional Toda lattice is proved, making use of the direct method. This formula is a generalisation and unification of the Backlund transformation, the nonlinear superposition formula and the Casorati determinant solutions.

Proposes a model of population of photons in which emissions occur through a process of non-Markov evolution. The photon population statistics are shown to be antibunched in their character with an intensity correlation characteristic of resonant fluorescence and other associated phenomena.

Scaling properties of the growth probability distribution were studied for non-induced fractals in Ni-Zr alloy thin films when the irradiation dose reached a critical value. Two kinds of probability measure were used in the analysis, namely the growth-site area obtained from DLA simulation of the growth process, and the Laplacian potential gradient. The generalised dimensions D(q) with q ranging from - infinity to + infinity , as well as the f- alpha spectrum were found to be slightly different from those obtained by other authors.

Diffusion-limited aggregation clusters and the structures observed in viscous fingering experiments at high capillary numbers are tree-like fractals. The different branches may be assigned a branch order in a way that exhibits scaling, and permits a self-similar characterisation in terms of the bifurcation ratio r_N and the length ratio r_L of branches of different orders. The fractal dimension is given by D=ln(r_N)/ln(1/r_L). Good agreement between experiments and simulations is found. A crossover function characterises the branch orders.

The authors study the scaling properties of an incommensurate system which obeys simple clustering rules, the one-dimensional tight-binding Schrodinger equation with a Fibonacci sequence potential. They find that the clustering rules do not guarantee a finite number of scaling factors. They find the energy spectrum by recursion. The clustering rules and recursion relations suggest different descriptions of the same spectrum. Based on the thermodynamic formalism, they compare these descriptions, calculate their associated scaling functions, and construct the mapping between them.

Diffusion on a trifurcating hierarchical structure is studied through an exact renormalisation group procedure. The non-universal time scaling exponent of the autocorrelation function is obtained and compared with that of the simplest 1D ultradiffusion model. When a biasing term is introduced, it is found that the transition from the power law to exponential decay of the autocorrelation function is independent of the furcating number of the hierarchy. A general Z-furcating hierarchical system is also discussed briefly.

The author discusses the bond-bending model in three dimensions. He argues that for this model loops are very important in all length scales and the rigidity percolation is non-local, exhibiting similar behaviour to the central force model.

New simulations involving up to nearly 10⁹ sites for square lattice bootstrap percolation give a percolation threshold compatible with unity. In this model sites in a square lattice are initially occupied randomly, and then are removed if they have less than three occupied neighbours. The results are compared with those of similar models, such as that studied by Frobose (1989), and the earlier bootstrap and diffusion percolation models, in two, three and four dimensions, of Adler and Aharony (1986-8). In the latter models, the effective threshold approaches its asymptotic limit (one or zero) as ( lambda /ln L)^d-1, for samples of size L^d, in agreement with theoretical predictions. The coefficient lambda decreases monotonically with the parameters which facilitate the removal (addition) of sites.

The authors analyse the fractionisation for a Dirac field coupled to a ( lambda phi ⁶)₁₊₁ theory. Taking a simple Yukawa interaction they point out the no-fractionisation phenomenon in the background provided by the (anti) kink solution associated with the scalar field. If one chooses a different coupling the half-integral fermionic number is possible and the analysis is particularly transparent using a previous bosonisation procedure.

Physical applications of zeta -function regularisation require explicit knowledge of the relevant zeta function somewhere on the non-positive real axis. The author presents a simple and very general method for evaluating zeta functions Z(s)= Sigma _m lambda _m^-s at certain discrete points s=-np<or=0, n=0, 1, 2, . . ., on this axis, where p is defined by the condition that lambda _m^p be a polynomial in all summation indices. Z(-np) is shown to be a polynomial in the various parameters on which the 'eigenvalues' lambda _m depend. The polynomials Z(-np) generalise the Bernoulli polynomials and possess interesting properties. These statements apply to multidimensional (e.g. Epstein and other) zeta functions, as well as to single-sum zeta functions. Physical applications in quantum field theory are briefly indicated.

The authors present a detailed numerical study of the dynamics of high-energy kink excitations in monatomic chains, with a simple description using a microscopic model of the narrow kink core. They present velocity-amplitude and energy-momentum curves for the kink which fit well the numerical results. Kink-kink collisions are slightly inelastic depending on the velocities and point of collision within a lattice spacing.

The authors have employed algebraic methods to calculate the bound-state spectra of a non-relativistic hydrogen atom subjected to a wide class of perturbations. Their procedure exploits the linearity of the complete (perturbed) Hamiltonian in the generators of the SO(2, 2) Lie algebra which follows naturally from the separation of variables in Schrodinger's equation in parabolic coordinates. Appropriate transformations then allow the Hamiltonian to be expressed as a linear combination of the compact generators of the two underlying SO(2, 1) algebras. They give some examples for which the bound-state spectra can be obtained completely analytically.

The author recalls the problem of choosing an ordering for the terms in a Hamiltonian of the form H(p, q) of finite dimension in p, q. He demonstrates that most orderings which provide one with a Hermitian operator are in fact redundant and he gives a general form for ordering rules in terms of arbitrary parameters. He concludes that differences in the choice of an ordering contribute differences to the physics which are only of order ² and higher. He provides an example of a Hamiltonian for which the spectrum is explicitly dependent on the choice of quantisation rule (operator ordering). He compares the method of characterising operator ordering rules with that of Cohen (1966, 1970). Finally he generalises Cohen's method to the case of all linear quantisation rules and shows how his rule is a special case.

The author employs two independent methods to obtain consistent and accurate values of eigenvalues E( lambda ) of the perturbed two-dimensional oscillator Hamiltonian, H=H⁽⁰⁾+ lambda x²y²: (i) continued-fraction representations of the Rayleigh-Schrodinger perturbation series to large order, as well as a renormalised version of perturbation theory, and (ii) a method of moments based on the positivity properties of the factorised wavefunction. The latter generates converging upper and lower bounds to E( lambda ). The two methods are also used to obtain estimates for the eigenvalues F⁽⁰⁾ of the infinite-field limit Hamiltonian, H_infinity =p_x²+p_y²+x²y².

Using a comoving representation, the authors derive the conformal anomaly of string theory in an alternative path integral formulation. Accordingly, the critical dimensions for bosonic and spinning strings are obtained by the residue regularisation method. The possibility of constructing a consistent quantum string theory under the subcritical dimension is also discussed.

The formalism developed by Sarker and Krumhansl (1981) for the investigation of a phi ⁴ system with long-range interactions of the Kac-Baker type is extended to the study of the thermodynamical properties of a sine-Gordon (SG) and a double sine-Gordon (DSG) chain with long-range interactions. This extension enlarges the class of non-linear systems which can support kink solutions. In the continuum limit, the author deduces compact analytical expressions for the SG-kink solutions and the DSG-kink solutions as well as for their associated energies. Both the kink solutions and the kink energies depend on the kink width. As the interaction range increases it is found that the kink width and the kink energy increase indefinitely-the kinks disappear. Consequently one can consider the kinks as 'elementary excitations' only in the limit of not too large interaction ranges, when they will destroy long-range order in the system. For this limit he determined the phonon and kink contribution to the free-energy density at low temperatures. For the infinite interaction range limit he has found that the two systems studied undergo a second-order phase transition. The critical behaviour is-as expected-similar to that of the phi ⁴ system with long-range interactions.

The statistical properties of the quasi-energy spectrum in a simple quantum model are investigated for the case when the corresponding classical system is fully chaotic while quantum chaos is restricted by the localisation effects. It is shown that the level spacing distribution depends effectively on some parameter which is the ratio of the dimension of the eigenfunctions (mean localisation length) to the total number of the quasi-energy levels. Numerical data for a wide range of parameters of the system are given.

The diffusion-limited steady-state particle coalescence model of aggregation is solved in one dimension. The method presented can also be used to solve the time-dependent problem and is extensible to higher dimensions.

The authors calculate the asymptotic behaviour of the moments of the first-passage time and survival probability for random walks on an exactly self-similar tree, and on a quasi-self-similar comb, by applying an exact decimation approach to the master equations. For the hierarchical comb, a transition from ordinary to anomalous diffusion occurs at R=2, where R is the ratio of teeth length in successive iterations of the structure. In the anomalous regime (R>2), the positive integer moments of the first-passage time, (t^q), scale as L^{tau q}, with tau _q=1+(2q-1)ln R/ln 2, where L is the linear distance from input to output. The asymptotic behaviour of the survival probability is studied using both scaling theory and by a direct solution of the master equations. They find that the characteristic time, t*, in the asymptotic exponential decay of the survival probability, exp(-t/t*), scales as t* approximately L^{tau *=}, with tau *=ln R²/ln 2, i.e. tau * is distinct from tau ₁. However, substantial corrections to this asymptotic form for tau * exist, and these are needed to account for the recent simulation data of Havlin and Matan (1988).

On the basis of a simple model of time relaxation in disordered systems (such as glasses or spin glasses), the authors study the diffusion on clusters built on high-dimensional hypercubes. They show there exist two particular concentrations of available sites on the hypercube. The first one defines two distinct behaviours of relaxation: above p=1/2, the relaxation is purely exponential; below p=1/2, the relaxation is no longer exponential. The other case is of more interest: it is the percolation concentration. Using the unproven but argued assumption that, because of the high dimensionality of the considered space, the calculation may be mapped onto the problem of diffusion on a standard mean-field percolation cluster, they show that the relaxation should follow a stretched exponential law there with exponent 1/3. They briefly discuss the physical implications of this approach.

Calculations of the correlation functions C and C_cond for the two-dimensional Gaussian-column model of the interface in a weak gravitational field, G to 0⁺, indicate the universality of the parametrisation of the direct correlation function C, found for the SOS model by Stecki, Ciach and Dudowicz (1986). The representations of the conditional correlation function C_cond are slightly different in these two models.

For original paper see ibid., vol.19, p.3741 (1986). A comment is presented on a recent paper by Chowdhury and Naskar (1986) which describes a method that is claimed to be successful in calculating the Stokes parameters of the linear differential equation to which the Painleve IV equation defines isomonodromic deformations. However, the author shows that the simplification made by Chowdhury and Naskar is not uniform and the resulting formulae for Stokes parameters are not invariant under the deformations satisfying the Painleve IV equation.

It is shown that the periodic solution of the non-linear DABO equation can be written as an infinite sum of an equally spaced row of identical Lorentzian solitons. This may be considered as a generalisation of a similar result which has been found for the KdV equation and is related to the clean-interaction properties of colliding solitons under non-linear coupling.