Table of contents

Dynamics of surfaces is connected with various soliton equations, in particular with a recent generalisation of the nonlinear Schrodinger equation via an elementary account of the local geometry of moving frame fields and surfaces.

The connection between diffusion-limited aggregation and the equations of dendritic growth is critically examined. A different type of Monte Carlo simulation is proposed and used to construct two-dimensional dendrite-like patterns. The wavelength occurring for short times is in good agreement with the linear stability analysis. The time dependence of the characteristic wavelengths is also determined.

The authors prove that determining the ground state of a cluster of identical atoms, interacting under two-body central forces, belongs to the class of NP-hard problems. This means that as yet no polynomial time algorithm solving this problem is known and, moreover, that it is very unlikely that such an algorithm exists. It also suggests the need for good heuristics.

Via quantum fluid dynamics the author presents a general method to obtain a solution of the nonlinear Schrodinger-Langevin equation with a multiterm potential for the description of interactions in non-conservative systems. The nonlinear complete set of equations forming the basis of the model are derived. The corresponding solutions are exhibited in terms of auxiliary ordinary differential equations which provide relations between the potential coefficients and other coefficients prescribing the fluid-particle behaviour.

The authors obtain the Lie symmetries for the equation of motion of an electric charge interacting with a magnetic monopole, fixed at the origin. They also discuss the construction of first integrals for this problem.

The authors have obtained the complete Lie-Backlund symmetry for a modified Harry-Dym system and hence deduced the bi-Hamiltonian structure associated with it. It is shown that these Lie-Backlund symmetries generate the recursion operator inherent in the theory and the conserved quantities can also be computed according to the Dorfman prescription.

Irreversible aggregation of clusters leads in a natural way to the formation of large ramified structures. The authors investigate this growth process close to the sol-gel transition, where for the first time an infinite spanning cluster appears, and find scaling behaviour. The kinetics of the model leads to properties which are distinct from standard percolation. Two critical exponents are determined numerically. The fractal dimension of the clusters is D_g=1.75+or-0.07 in two dimensions.

The authors consider a fractal network of nonlinear resistors, with the voltage V behaving as a power of the current I, mod V mod =R mod I mod ^alpha . The resistance between two points at a distance L is R(L) varies as L^{zeta ( alpha )}. They prove that zeta (0) describes the scaling of the topological-chemical distance, while zeta ( infinity ) describes that of the number of singly connected 'red' bonds. For random resistors, they also consider the width of the resistance distribution, Delta R varies as L( zeta ₂( alpha )). Values for zeta and zeta ₂ are explicitly derived for two model fractals, and Delta R/R is found to grow with L for the Sierpinski gasket and alpha >1.612. The relevance of the results to percolation clusters is discussed.

The authors study the Eden growth model on the square Manhattan lattice (SML). For this purpose it has been necessary to extend the standard Eden model in such a way that each act of growing is conceived as a directional process. By applying the position space renormalisation group technique, they find that the Eden processes on the SML and on the ordinary square lattice are in different universality classes. This finding and corresponding fractal dimensions are compared with results obtained for similar models of irreversible kinetic processes.

The analysis of exact enumeration data of self-avoiding walks attached to the surfaces of triangular and face-centred cubic lattices is consistent with gamma ₁=0.718+or-0.008+136 Delta p_c where Delta p_c=p_c-0.0995, in three dimensions and 0.9530<or= gamma ₁<or=0.9548 in two dimensions, where p_c for the triangular lattice is assumed to lie in the range 0.240 915<or=p_c<or=0.240 93.

Monte Carlo simulations are presented for one-dimensional self-avoiding Levy flights. They are in perfect agreement with the predictions of Flory theory. Some field-theoretic predictions are also verified, although much less clearly. In particular, epsilon expansions are available but their range of applicability seems to be small.

The authors present Monte Carlo and exact enumeration results for three- and four-arm stars on a variety of lattices. They estimate the exponents for the number of stars and the mean square end-to-end length of a branch, and compare these results with scaling and renormalisation group predictions.

The trail problem, a variant of the self-avoiding walk in which excluded volume is associated with lattice bonds rather than sites, is studied numerically in two and three dimensions. Monte Carlo and exact series methods are used, but the results obtained lack the consistency of similar results for self-avoiding walks. The evidence in support of the two models belonging to the same universality class is suggestive though not entirely convincing, but slowly decaying corrections to scaling could account for apparent exponent differences.

The zeros of the partition function for the zero field two-dimensional Ising model in the complex temperature plane are represented in terms of the eigenvalues of the transfer matrix. The boundaries of the limiting distribution have a particular simple representation in terms of the correlation range along the two principal axes of the quadratic lattice. Examples of the distribution in the complex plane of z=exp(-2K) are given for the anisotropic model when the horizontal and vertical interactions are in integer ratio.

The authors introduce a supersymmetric four-dimensional N=4 Yang-Mills model on the space lattice The construction involves the Dirac-Kahler formalism. They obtain the Hamiltonian on the space lattice with the correct classical continuum limit.

Let G be a simply connected Lie group, E be its Lie algebra, and phi be a continuous representation of G in a vector space V. First cohomology groups for both the group G and its Lie algebra E are shown to be isomorphic. When the group G is also semi-simple, the first cohomology group is shown to be trivial for a particular class of representations.

The object of this paper consists of finding a complementary group with respect to USp(n) within either irreducible representation ((1/2)^dn) or ((1/2)^dn-13/2) of the real symplectic group Sp(2dn,R). For such purpose, the author deals with the (full) orthogonal group O(n), n=2 nu or 2 nu +1, and the unitary symplectic group USp(n), n=2 nu , in a unified way by introducing an appropriate metric. A similar treatment applied to their complementary group determination leads to the well known Sp(2d,R) group in the O(n) case, and to the SO*(2d) group in the USp(n) case.

The authors present a covariant relativistic derivation of the equation of motion for a spinning extended charge. From it they study, in the non-relativistic limit, the rotational dynamics of a rigid spherical charge, comparing the results with previous ones. In particular some properties of this model concerning runaway and self-oscillating solutions are discussed. They show that, unlike for translational motion, there are no runaway solutions. They also analyse the rotational motion in presence of an external field.

In a recent paper, Moreno and Zepeda (1984) have applied the 1/N expansion to obtain an approximate analytic formula for the ground state energy of a particle bound by a Yukawa potential. Using the method of Mlodinow and Shatz (1982), the author calculates the energies of the ground state and the first excited state of the system to show that the results provided by the 1/N expansion are identical with those obtained by applying either the analytic perturbation theory or the hypervirial equations with the Hellman-Feynman theorem. Also an approximate analytic formula for the ground state energy is proposed.

Considers the quantum systems of N point particles moving in R³ under the influence of either their mutual Coulomb potential V or a smooth potential V_n equal to V whenever all interparticle distances are greater than 1/n and equal to zero whenever all interparticle distances are less than 1/n-1/n². The author proves that the dynamical theory based on V_n converges to that based on V as n increases indefinitely.

Constructs semi-inhomogeneous solutions of the Kac model, for which the spatial dependence is only present in the odd-velocity part of the distribution. At the macroscopic level, the local density, the local energy, and the associated current components are determined and discussed, while at the microscopic level there exist two different classes of distributions; as an illustration the author gives two solutions explicitly. For one class, the distributions relax towards a Maxwellian equilibrium solution and they correspond either to a contraction or an expansion, while for the other they go to zero when the time increases to infinity. The space variable appears linearly and, in order to maintain the positivity, stays inside a finite interval. Assuming that the distributions are zero outside such intervals, a physical interpretation can be obtained for the whole space axis, provided appropriate elastic walls, sinks and sources are introduced. These boundary conditions seem more natural for the class of distributions with Maxwellian relaxation.

For a solution of charged rods the static structure factor is derived in a weak-coupling expression. The interaction between the rods is assumed to be a superposition of screened Coulomb interactions between point charges distributed along the rod. Screening arises from the counter ions as well as from salt in the solvent. The authors treat the case of an isotropic solution as well as the nematically ordered state.

A simple criterion is given which provides disorder solutions for spin models of the Ising or Potts type. New disorder solutions are thus obtained, in particular for the Potts model on a Kagome lattice and for the general anisotropic Ising model, on a three-dimensional cubic lattice. The validity of the disorder solution, when extended outside the physical domain, is also discussed.

The authors calculate the lowest non-zero eigenvalue of the Fokker-Planck equation for an infinite square lattice of coupled overdamped anharmonic oscillators (bistable potentials) through a variational calculation. It generalises previous work for only one oscillator in a non-trivial way. They reduce that calculation to that of a suitable ratio of partition functions for the 2D Ising model, for which an exact analytical expression is obtained. Numerical results are also given and discussed.

The probability distribution function P(q) for the overlap of pairs of metastable states and the associated Parisi order function q(x) are calculated exactly at zero temperature for two simple models. The first is a chain in which each spin interacts randomly with the sum of all the spins between it and one end of the chain; the second is an infinite-range limit of a spin glass version of Dyson's hierarchical model. Both have non-trivial overlap distributions. In the first case the problem reduces to a variable-step-length random-walk problem, leading to q(x)=sin( pi x). In the second model, P(q) can be calculated by a simple recursion relation which generates devil's staircase structure in q(x). If the fraction p of antiferromagnetic bonds is less than (2)^-12/, the staircase is complete and the fractal dimensionality of the complement of the domain where q(x) is flat is ln 2/ln(1/p²). In both models the space of metastable states can be described in terms of Cayley trees, which have, however, a different physical interpretation than in the SK model.

A simple hybrid of real space renormalisation group theory and finite-lattice calculations is described. The method is based on a lattice-restructuring transformation, with the criterion that the free energies of the original and restructured lattices be equal. The authors approximate the free energy per spin of the infinite lattice by an exact evaluation of the partition function for a small cluster. As a result of this transformation, the coupling constant of the restructured lattice is found to be temperature dependent. The method is illustrated by studying the Ising model in two and three dimensions, and two-dimensional Potts models. The calculations involve little more than is required for a Migdal-Kadanoff transformation, yet, the critical properties are found to be significantly improved in comparison. Detailed comparisons are also made with the global thermodynamic properties calculated within this approximation (free energy, internal energy, specific heat, spontaneous magnetisation) and exact or best known results for the models. The simplicity of the method allows for the study of more complex lattice models, and several possible applications are discussed.

Bound states of a charged particle in the field of a dyon are studied in the non-relativistic as well as the relativistic situations. In the relativistic case all angular momentum channels except the lowest are considered. Bound state energies and wavefunctions are calculated explicitly.