Table of contents

It is shown that the Henon-Heiles and two coupled quartic anharmonic oscillator systems possess non-trivial generalised Lie symmetries for specific sets of parametric values, for which second integrals of motion can also be constructed directly using Noether's theorem, thereby establishing their complete integrability.

The authors give two fundamental methods for quantum or classical free energies of integrable models. Periodic boundary conditions induce an integral equation for classically allowed momenta. Generalisations of the Berthe ansatz and a method of functional integration on the classical action in action-angle variables follow, giving identical (Bose-Fermi equivalent) results. For sinh-Gordon the Bose classical limits agree with the transfer integral method.

The authors give two extensions of Pomeau's additive invariant (1984) for reversible cellular automata and networks.

The total energy (and its spectral density) emitted during the collision of two solitons described by the conservatively perturbed non-linear Schrodinger equation (NSE) is calculated by means of perturbation theory based on the inverse scattering transform. It is shown that the total emitted energy (at large relative velocity) does not depend on the relative internal phase of the colliding solitons. The same results are obtained for solitons in a system of two weakly coupled non-linear Schrodinger equations.

The authors describe an up till now unrecognised phenomenon in kinetic growth models which leads to observable oscillations in such quantities as the density and velocity of growth. These oscillations, which can occur on length scales of many lattice spacings, arise because of an induced incommensuration in the growth mechanism. To illustrate the phenomenon, they present results for a particularly simple model, but the phenomenon is expected to be quite general and appear in a wide range of growth processes. The essential ingredients for the existence of the oscillations are that the growth take place at a reasonably well defined interface and that the growth process be discrete (e.g. that the cluster grows by the addition of discrete particles of finite size). The growth process is related to a functional stochastic iterative map so that the growth oscillations play the role of limit cycles. They suggest that the fixed point of this map is related to critical fractal kinetic growth.

The influence of a static disorder substrate and a static disorder external field on thin-film layering transition is investigated. The author finds that when the disorder is confined to the substrate alone, the influence appears only in the lowest transition line and acts to encourage film growth at finite temperatures. Random external fields influence thick, as well as thin, films and act to discourage the film development, resulting in a re-entrant-like effect.

A Hilbert curve is constructed on the Sierpinski gasket. It is shown that despite the fact that the Sierpinski gasket is rigorously self-similar, the Hilbert curve is merely self-affine.

Temporal intermittency in a chaotic dynamical system can be regarded as a manifestation of the multifractal properties of the set of histories. The Renyi entropies K_q which generalize the Kolmogorov entropy K₁ play a role analogous to the exponents tau (q) characterising the scaling properties of the moments of the mass distribution. The authors introduce the topological entropy h( lambda ) of the subset of histories with the same local expansion parameter. They show that K_q and h( lambda ) are related to each other by means of a Legendre transformation.

The authors show that the Renyi entropies characterise the temporal intermittency in chaotic systems and are linked to a set of generalized Lyapounov exponents related to the time fluctuations of the responses to slight perturbations on the trajectory. It is also briefly indicated how a simple extension of a numerical algorithm proposed by Grassberger and Procaccia (1983) allows one to compute these entropies from a signal.

Kauffman's model describes a random network of automata. The authors calculations indicate that the multivalley structure of the basins of attraction in Kauffman's model is very similar to that of infinite-range spin glasses. The similarity with spin glasses is tested quantitatively by computing the probability that two initial configurations fall into the same valley.

Using conformal invariance and Coulomb gas results, the author gives the exact value in two dimensions of the eta exponent of L dense polymers, attached by their extremities: eta _L=(L²-4)/8. The value in two dimensions of the gamma exponent of a dense branched polymer of fixed topology with n_L L-leg vertices, L>or=1 is then deduced to be gamma = Sigma _L>or=1 n_L(2-L)(L+18)/32. These values correspond to a conformally invariant theory with central charge C=-2.

A critical condition is obtained for the ferromagnetic model on the hypercubic lattice in d dimensions with different Potts interactions along the different lattice axes. It is done by extending a simple procedure which is shown to be exact for the square lattice in d=2 dimensions.

The authors present a simple analytical approach to studying diffusion on hierarchical and fractal structures. They show that hopping on one-dimensional hierarchical structures can be mapped onto diffusion on a family of loopless fractals.

The authors consider two-dimensional clustering for a high density of clusters. They find that three regimes can be distinguished and give the criteria for crossover between them. The numerical value of the gelation threshold strongly depends on the system size.

An experimental study of uniaxial compression of a mixture of 'hard' and 'soft' cylinders is made. The macroscopic strain-stress law strongly depends on the geometrical and compositional heterogeneities. Two changes in the behaviour are observed: one is weak, at the percolation threshold, and the other is strong, at the 'rigidity' threshold.

The finite-size limit of the lower part of the spectrum of the Ashkin-Teller chain with free boundary conditions is studied numerically and interpreted from the point of view of conformal invariance. Several irreducible representations of the Virasoro algebra with the central charge c=1 are identified. For two special values of the coupling constant, higher degeneracies occur and the whole spectrum can be understood in terms of a few irreducible representations of the N=2 superconformal algebra.

An upper bound for the number of metastable states in the Hopfield model is calculated as a function of the Hamming fraction from an input pattern. For all finite values of alpha , the ratio of number of patterns to nodes, the hamming fraction from the input pattern to the nearest metastable state is infinite. When alpha <0.113, the bound also implies that there is a gap between a set of states close to the input pattern and another set centred around the Hamming fraction 0.5 from it.

The author has used Lie's extended group method to obtain the maximal symmetry groups of the Dirac equation for finite mass spin 1/2 particles and the Weyl equation for zero mass spin 1/2 particles. In both cases the maximal symmetry group is an infinite parameter Lie group having an invariant subgroup also of an infinite number of generators. The corresponding factor group for the Dirac equation is an eleven-parameter Lie group isomorphic to the Weyl group. In the case of the Weyl equation the corresponding factor group is a sixteen-parameter Lie group containing a proper subgroup isomorphic to the conformal group.

In an axisymmetric approximation the author demonstrates a parity breaking transition to a twisted configuration in tangentially anchored nematic liquid crystal droplets. The twisted phase occurs when K₁₁>or=K₂₂+0.431K₃₃. In a magnetic field there is another transition corresponding to the droplet axis changing from parallel to normal to the field.

Necessary and sufficient conditions for the existence of localised solutions of the form psi (x,t)=exp(-i omega t) phi (x), with phi real, of the classical equations of motion for (1+1)-dimensional nonlinear spinor field are presented. Furthermore, the authors give existence conditions for non-linear second-order equations obtained as a Klein-Gordon limit of the considered spinor field equations.

The exact density matrix elements and the correlation functions of the one-dimensional non-stationary singular oscillator (the potential U(x,t)= omega ²(t)x²/2+g/x²) excited from the thermodynamic equilibrium states are obtained. Generating functions for rho _nm and also for transition probabilities are constructed.

The authors show how complex paths can be consistently introduced into sums over Feynman histories by using the notion of functional contour integration. For a k-dimensional system specified by a potential with suitable analyticity properties, each coordinate axis is replaced by a copy of the complex plane, and at each instant of time a contour is chosen in each plane. This map from the time axis into the set of complex contours defines a functional contour. The family of contours labelled by time generates a (k+1)-dimensional submanifold of the (2k+1)-dimensional space defined by the cartesian product of the time axis and the coordinate planes. The complex Feynman paths lie on this submanifold. The convergence problems encountered in previous proposals for complex path integrals are avoided by the requirement that each contour is asymptotically pinched to the real coordinate axis. An application of this idea to systems described by absorptive potentials yields a simple derivation of the correct WKB result in terms of a complex path that extremalises the action. The method can also be applied to spherically symmetric potentials by using a partial wave expansion and restricting the contours appropriately.

A quantum mechanical version of a kinetic equation is derived which accounts for three-particle collisions. It is shown that the total energy is conserved in the binary collision approximation.

The authors study the problem of directed compact site animals in two dimensions using exact enumeration methods. The numerical analysis of the animal number series strongly indicates that the critical exponent theta equals zero. They also estimate the non-universal growth parameter lambda =2.66185+or-0.00050.

The author explains with examples, the relationship between zeros of the partition function, analyticity and degeneracy of absolute magnitude of eigenvalues of the transfer matrix for statistical mechanical models. It is shown how to write down a polynomial representation of the partition function for any model with a two-sheet largest eigenvalue. The staggered ice model representation of the q-state Potts model, on a sequence of semi-infinite strips, is solved and it is shown that for 0<q<4 the Potts model result is only projected out by special boundary conditions. The non-Potts part of the result is obtained for a 4* infinity strip and gives Baxter's antiferromagnetic critical curve (1982).

Numerical simulations show that the time lag between the quench and the establishment of a steady-state regime of nucleation depends on the quench rate. For the fast quenches the time lag is found to be exponentially, rather than linearly, dependent on the radius of the critical nucleus. It is shown qualitatively that the time lag for slow quenches is smaller than that for the fast quenches. All these results are obtained in the framework of Glauber dynamics. For Kawasaki dynamics it is much harder to detect the beginning of the steady-state regime, and the results are less convincing.

The author reports series expansion analyses of self-avoiding walks with nearest neighbour bond interactions. The estimates 2 nu _t=1.07+or-0.05 and phi =0.64+or-0.05 for the correlation and crossover exponents at the theta point were obtained by examining the number of walks and the end-to-end distance data up to 16 steps on the triangular lattice.

The author studies the diffraction spectrum and the structure factor of quasicrystalline, aperiodic, linear arrays with two arbitrary characteristic measure lengths, produced by a fairly general family of generating rules depending on two parameters. The calculation is performed by actually constructing the two-dimensional periodic structure whose projection will result in the desired linear aperiodic array. Distributions of some sequences of irrational numbers modulo 1 are derived as a byproduct of the main subject.

A generic model, which applies rigorously to finite chains in the absence of inter-segmental interactions, and to subchains of infinite chains with practically arbitrary interactions, furnishes a simple recurrence relation. Self-avoiding walks on the diamond lattice (SAW-D) form the paradigmatic example for polymer science. The generic model as a whole is solved (in terms of parameters) for configurational statistics, asymptotically (large number n of segments) and on stated plausible conjectures, by classical methods, i.e. without postulating a singularity of power law form. The appropriate generalisation of the power law form of scaling theory emerges very simply, e.g. in terms of Kummer's hypergeometric function. Extensive computations on the SAW-D model lead to two correction terms to the crude scaling form. In this way, some arguments in the literature on asymptotic theories can be settled. Several examples illustrate the danger of mistaking non-asymptotic experimental results for those desired in the asymptotic range. Thus, contrary to his own conclusions and those of Fleming (1979) Brun's data (1977) data for freely hinged hard-sphere chains are here reconciled with the 'universal' exponent gamma of Le Guillou and Zinn-Justin (1977). For the SAW-D model subject always to further refinements (which will never produce a final answer), the present experimental estimate for gamma is also shown to be about 1.2, in line with conjectural estimates from many forms of modern theory.

The method of de Vega and Woynarovitch (1985) is used to calculate finite-size corrections to the ground-state energy in different sectors for the XXZ Heisenberg chain. Finite-size scaling amplitudes and correction-to-scaling exponents in the critical region are derived. Using conformal invariance, a scaling dimension x=( pi - gamma )/2 pi is extracted corresponding to the electric field operator in the 8-vertex model: this confirms a conjecture of Baxter and Kelland (1974). Finite-size scaling properties near the Kosterlitz-Thouless critical point Delta =-1 are discussed.

The authors employ the results of conformal invariance at critical points to calculate the structure friction S(k), bulk susceptibility x and correlation lengths (defined via moments of the correlation function) for infinite strips with periodic boundaries. Analytic formulae for operators with anomalous dimension 0<or=x<or=1 are obtained. The correction to the leading asymptotic form of S for large mod k mod is determined. The convergence to the leading 1/k^2-2x behaviour is shown to depend on x and be slow for x=1/8 the Ising spin-spin correlation function value. For x=1, S is found to display a logarithmic lineshape.

A recent analytic theory of two-dimensional isotropic percolation indicates that the singular behaviour near p_c is identical to that appropriate for the associated dilute Ising critical point. A specific prediction, supported by recent series expansion studies, is that the mean number of clusters, K(p), presents a singularity of the form K(p) approximately mod p-p_c mod ²ln mod ln mod p-p_c//, rather than the accepted form K(p) approximately mod p-p_c mod ^{2- alpha} with alpha =-2/3. A novel numerical and renormalisation group finite-size scaling analysis of the nature of the singularity in K(p) is presented in support of the new theory, which implies the absence of a separate universality class for two-dimensional percolation processes. This study is consistent with the effective values of the exponents alpha =0 and nu =1.

The phase diagram and the critical behaviour of an (intersecting) random surface gas model in three dimensions is studied by means of Monte Carlo simulations. The critical exponents alpha , beta , gamma and delta are evaluated in the 'critical window' between the finite-size rounding and the correction-to-scaling regime. Within error bars, Ising exponents for the self-avoiding case (and along critical lines) and mean-field behaviour at tricritical points are obtained. For the self-avoiding planar surface model the Hausdorff dimension is calculated (d_H=2.30+or-0.05).

The author studies studies the q-state antiferromagnetic Potts model on fractals, namely the Sierpinski carpets (1<D_f<2) and pastry shells (2<D_f<3). For both families a critical value q_c was found, above which the system is always in the paramagnetic phase whereas for q at or less than q_c the system exhibits a low temperature critical phase. The phase diagrams are analyzed from the geometrical point of view.

For pt.IV see ibid., vol.19, p.2431 (1986). The method of partial generating functions is used to derive, for the simple cubic lattice, the number of connected strong embeddings through 13 sites, the number of connected weak embeddings through 14 bonds and three new bond perimeter polynomials D₁₀, D₁₁, D₁₂ for the bond percolation problem. For the body-centred cubic and simple cubic lattices an expression is derived for the mean number of clusters for the site percolation problem in powers of the probabilities of occupation of A and B sites.

The derivation of series expansions for a study of percolation processes for both site and bond mixtures is reviewed. For the bond problem, low density expansions for the mean size of clusters on the simple cubic and body-centred cubic lattices are given through p¹⁴. High density expansions for the mean number and size of finite clusters and the percolation probability are given for the simple cubic lattice through q⁴¹ and for the body-centred cubic lattice through q⁶¹.

A general formalism for the practical application of Sykes' partial generating function method (1986) is given. The formalism is suitable for the study of problems where the generating functions depend on detailed 'local' properties and is constructed in a manner which allows easy programming of the method.

The partial generating function method is applied to the enumeration of lattice animals classified according to their valence distribution. This detailed information allows one to study the effects of the number and nature of branch points on the properties of branched polymers. Several series analysis techniques are applied to the data. Particular attention is given to the effects of correction-to-scaling terms.

The author demonstrates in a simple way, using a unitary transformation, the general gauge invariance of linear response theory.

Migdal-Kadanoff bond-moving renormalisation is used to study the q-state Potts model on Sierpinski carpets. A general approximate recursion relation including q as a parameter is given. The property of 'bond-interchanging invariance' is found and used in deriving the recursion relation. Fixed points and critical exponents for some carpets are presented. Marginal fixed points instead of unstable ones are found. Several typical flow diagrams are also shown.

Some results (Mathieu J. Phys. A:Math. Gen. 18 L1061) concerning the existence of static soliton solutions are corrected and illustrated by a general example.