Table of contents

We conjecture the exact scaling theory for the adsorption of two-dimensional polymers by using boundary S matrices. We compute the boundary free energy (the 'g-function'), study the flow from the adsorbed to the desorbed phase and derive the crossover exponent and all the geometrical exponents at the transition. More generally, we solve the special transition in the O(n) model, the polymer case corresponding to n=0. The n=2 limit appears identical to the ordinary Kondo problem.

A covariance analysis of a scalar field in 3D and 2D turbulent flows is performed on the basis of the convective-diffusion equation. Physically distinguished scales for coherent structures in the inertial-convective range are obtained.

Starting from a Calogero-Sutherland model with the hyperbolic interaction confined by an external field with Morse potential, we construct a Heisenberg spin chain with exchange interaction varies as 1/sinh²x on a lattice given in terms of the zeros of Laguerre polynomials. Varying the strength of the Morse potential, the Haldane-Shastry and harmonic spin chains are reproduced. The spectrum of the models in this class is found to be that of a classical one-dimensional Ising chain with non-uniform nearest-neighbour coupling in a non-uniform magnetic field which allows us to study the thermodynamics in the limit of infinite chains.

We study, through numerical simulations, the spreading of damage on the bi-dimensional ferromagnetic Ising model submitted to a new kind of external field h recently introduced in the literature in the context of cellular automata. The field h is defined as the frequency at which different random numbers are used for updating the two replicas. We show that h is the conjugate field to the Hamming distance, i.e. it destroys the dynamical continuous transition observed at h=0, and the associated susceptibility diverges at the critical temperature.

We use a histogram Monte Carlo simulation method to calculate the scaling functions of the existence probability E_p and the percolation probability P of the site percolation model on square lattices with free and periodic boundary conditions. We find that different boundary conditions give quite different scaling functions near the critical region. However, they give the consistent critical point, critical exponents, and the thermodynamic order parameter from renormalization-group calculations. Similar results are found for other percolation models. The implications of our calculated results for some theoretical problems of current interest are discussed.

A statistical mechanical approach for neural networks in which couplings are the slow dynamical variables is considered. The couplings are assumed to be confined in a restricted subspace near the Hebb-rule structure corresponding to quenched patterns. We study the situation when the couplings thermalize at a temperature different from that of the spin degrees of freedom, which makes it possible to treat the system in terms of the traditional replica approach with a finite number of replicas. The structure of the model is such that the effective evolution of the couplings tends to deepen the free energy minima corresponding to the learnt patterns. The phase diagram obtained exhibits a substantial increase of the retrieval region.

An example of the Bell-Kochen-Specker argument is given for 20 rays in four dimensions.

We study the model of deposition-evaporation of trimers on a line recently introduced by Barma et al. The stochastic matrix of the model can be written in the form of the Hamiltonian of a quantum spin- 1/2 chain with three-spin couplings given by H= Sigma _i((1- sigma _i^- sigma _i+1^- sigma _i+2^-) sigma _i⁺ sigma _i+1^sigma _i+2⁺+hc). We study, by exact numerical diagonalization of H, the variation of the gap in the eigenvalue spectrum with the system size for rings of size up to 30. For the sector corresponding to the initial condition in which all sites are empty, we find that the gap vanishes as L^-z where the gap exponent z is approximately 2.55+or-0.15. This model is equivalent to an interfacial roughening model where the dynamical variables at each site are matrices. From our estimate for the gap exponent we conclude that the model belongs to a new universality class, distinct from that studied by Kardar et al.

The time evolution of the three-dimensional critical Ising model relaxing from a non-equilibrium initial state is studied by means of Monte Carlo simulation. We observe the characteristic initial increase of the (spatially) averaged magnetization predicted by Janssen et al. The exponent theta ' that governs the initial behaviour is determined, and the dependence of the long-time linear decay on the initial magnetization, m₀, is analysed. Our simulation corroborates earlier results derived from continuum models.

The entropic uncertainty relation for sets of N+1 complementary observables (A_r) existing in N-dimensional Hilbert space, Sigma _rH(A_r)>or=(N+1)ln((N+1)/2), is shown to be optimal in the case N=3 by explicit construction of the states for which equality holds. We prove that the lower bound cannot be attained when N is even, and, on the basis of numerical calculations, this is conjectured to also be the case for odd N>3.

We use a linear system of Langevin spins with disordered interactions as an exactly solvable toy model to investigate a procedure, recently proposed by Coolen and Sherrington, for closing the hierarchy of macroscopic order parameter equations in disordered spin systems. The closure procedure, based on the removal of microscopic memory effects, is shown to reproduce the correct equations for short times and in equilibrium. For intermediate time-scales the procedure does not lead to the exact equations, yet for homogeneous initial conditions succeeds in capturing the main characteristics of the flow in the order parameter plane. The procedure fails in terms of the long-term temporal dependence of the order parameters. For low-energy inhomogeneous initial conditions and near criticality (where zero modes appear) deviations in temporal behaviour are most apparent. For homogeneous initial conditions the impact of microscopic memory effects on the evolution of macroscopic order parameters in disordered spin systems appears to be mainly an overall slowing down.

We present two classes of non-equilibrium models with critical behaviour. Each model is characterized by an integer q>1, and is defined on configurations of q-valued spins on regular lattices. The definitions of the models are very similar to the updating rules in Wolff's algorithm for the Potts model, but both classes break detailed balance, except for q=2 and q= infinity . In the first case both models reduce to the Ising model, while one of them reduces to percolation (more precisely, to the general epidemic process) for q= infinity . Locations of the critical point and critical exponents are estimated in two dimensions.

The symmetry groups, generated by the inversion relations of lattice models of statistical mechanics on triangular lattices, are analysed for vertex models and for the standard scalar Potts model with two- and three-site interactions. These groups are generated by three inversion relations and are seen to be generically very large ones: hyperbolic groups. Two situations for which the representations of these groups degenerate into smaller ones, hopefully compatible with integrability, are considered. The first reduction for the vertex triangular model corresponds to the situation where the vertex of the triangular model coincides with the left- or right-hand side of a Yang-Baxter relation. In this case the representation of the group is isomorphic, up to a semi-direct product by a finite group, to Z*Z. The second reduction for q-state Ports models occurs for particular values of q, the so-called Tutte-Beraha numbers. For this model, algebraic varieties, including the known ferromagnetic critical variety, happen to be invariant under such large groups of symmetries. As a byproduct, this analysis provides nice birational representations of hyperbolic Coxeter groups.

The finite-lattice method of series expansion has been used to extend low-temperature series for the partition function, order parameter and susceptibility of the spin-1 Ising model on the square lattice. A new formalism is described which uses two distinct transfer-matrix approaches in order to significantly reduce computer memory requirements and which permits the derivation of the series to 79th order. Subsequent analysis of the series clearly confirms that the spin-1 model has the same dominant critical exponents as the spin- 1/2 Ising model. Accurate estimates for both the critical temperature and non-physical singularities are obtained. In addition, evidence for a non-analytic confluent correction with exponent Delta ₁=1.1+or-0.1 is found.

Series expansions for the longitudinal gyration radius of square-lattice directed-site animals to order N=39 are given. This doubles the length of the available series. Analysis of the series yields the estimate nu /sub ///=0.81722+or-0.00005, which excludes the conjecture nu /sub ///=9/11 based on phenomenological renormalization and shorter series, but is consistent with the later phenomenological renormalization estimate of Dhar: nu /sub ///=0.81733+or-0.00005.

The generating function for recurrent Polya walks on the four-dimensional hypercubic lattice is expressed as a Kampe-de Feriet function. Various properties of the associated walks are enumerated.

A comprehensive study of the shape of discrete random walks, considering the general case of arbitrary length and any space dimension, is presented. The probability distributions for several magnitudes such as the principal inertia moments, the asphericity and the angle between the principal axis of inertia and the end-to-end vector are evaluated numerically. In most cases, and especially in low-dimensional spaces, these probability distributions spread widely and possess a non-zero skewness. This implies that any description of the shapes of random walks which is based only on mean values of related magnitudes is incomplete. This situation does nor vary when random walks of large lengths are considered since the relative fluctuations (ratio between the standard deviation and the mean value) do not go to zero in that limit. On the other hand, when the space dimension grows, the distributions become more peaked, reaching the expected asymptotic limit for infinite dimension. The present study also indicates that the probability distributions for the principal inertia moments have approximately the form of chi-squared distributions. Analysing the probability distribution for the asphericity, it is shown that the distributions for the different moments are not completely independent.

A nonhomogeneous random walk on non-negative integers with transition probabilities p_0i= delta _0i, p_Ni= delta _Ni, P_i,i+1= lambda _i, p_i,i-1= mu _i, and p_i,i= rho _i, lambda _i+ mu _i+ rho _i=1, is studied. In particular, when the transition probabilities are independent of position, a general expression for the joint probability generating function (JPGF) of the frequency count of the stages 1,2,...,N-1 is derived. The appropriate marginal forms of this JPGF yield the PGF of the frequency count at any pair of stages, and at any particular single stage. Some moment formulae associated with the frequency count are derived. A random walk conditional on absorption at a specified boundary is also considered. The random walk model proposed is eminently suitable for the example of carcinogenesis.

A ferroelectric instability is predicted to occur in an experimentally accessible temperature range in a main-chain (semiflexible) liquid crystalline polymer with directed polar mesogenic segments. The long-range part of the dipole-dipole interaction is separated and its effect is completely accounted for in the average electric field. The remaining (local) free energy density is derived in the form of an expansion in powers of the spontaneous polarization. We derive simple expressions for the instability temperature and the polar mean-field coupling in a polar nematic polymer in terms of the molecular model parameters. Using simple estimates we show that ferroelectric ordering is more likely to occur in nematic polymers with directed polar segments than in low molecular weight nematic liquid crystals composed of polar molecules.

We examine the dynamical theory, recently proposed by Coolen and Sherrington, which describes the deterministic flow of order parameters in the memory retrieval processes of the Hopfield model. We have performed Monte Carlo simulations to investigate the limit of applicability of their theory. We have found that the qualitative behaviour of retrieval processes follows their prediction fairly well. However, the theory fails to describe the noise distribution quantitatively when the effects of non-retrieved patterns are not negligible.

We prove the almost sure convergence to zero of the fluctuations of the free energy in a class of disordered mean-field spin systems that generalize the Hopfield model in two ways: (i) Multi-spin interactions are permitted and (ii) the random variables xi _i^mu describing the 'patterns' can have arbitrary distributions with mean zero and finite (4+ epsilon )th moments. The number of patterns, M, is allowed to be an arbitrary multiple of the system size. This generalizes a previous result of Bovier, Gayrard and Picco for the standard Hopfield model, and improves a result of Feng and Tirozzi that required M to be a finite constant. Note that the convergence of the mean of the free energy is not proven.

We consider the Derrida-Higgs (DH) statistical model of species formation in the case where the population is geographically distributed in discrete locations, and mating only takes place within one location. Keeping the rate of migration between neighbouring locations at a fixed value, we change the mutation rate, changing therefore the average overlap between genotypes. When the overlap between individuals living in different locations falls below a fecundity threshold, speciation occurs. When more species coexist, the genetic structure of the population (as described by the overlap distribution P(q)) fluctuates. However, the average overlap, both within one location and among neighbouring locations, appears to vary according to the same laws as in the absence of speciation. The model provides a reasonable estimate of the parameter values necessary to observe geographic speciation, which is found to be much more likely than the sympatric speciation of the original DH model. Applications to the case of circular invasion, where the concept of biological species appears to run into difficulties, are sketched.

Irreducible finite-dimensional representations of the q-deformed algebra U_q'(SO₄), which is a generalization of Fairlie's algebra U_q'(SO₃), are studied. These representations T_rs are given by two integral or half-integral numbers r and s such that r>or= mod s mod >or=0. Spectra and eigenvectors of the operator T_rs(I₄₃) are given, where I₄₃ is one of the generators of U_q'(SO₄). By means of this result, explicit expressions for all generating operators T_rs(I_i,i-1), i=2,3,4, with respect to the basis mod x,m) corresponding to restriction onto the subalgebra U_q'(SO₂)+U_q'(SO₂) are evaluated. By analytical continuation, infinite-dimensional representations of the 'non-compact' q-deformed algebra U_q'(SO_2,2) are found. They are characterized by two complex numbers.

The classical r-matrix implied by the quantum kappa -Poincare algebra of Lukierski, Nowicki and Ruegg is used to generate a Poisson structure on the ISL(2,C) group. A quantum deformation of the ISL(2,C) group (on the Hopf algebra level) is obtained by a trivial quantization.

We propose a simple and compact method for deriving quantum groups acting on Manin's q- and h-planes. Using this method, we classify all the rigid motions of the two-dimensional quantum planes. Requiring that these inhomogenous transformations preserve differential structure of the quantum plane, we obtain two distinct inhomogenous quantum groups for the q-plane and one previously unknown inhomogenous quantum group for the h-plane, IGL_h,h'(2).

We give some general theorems, and extensions of previous results, concerning the problem of transforming an algebra of vector fields into Poincare normal form. By means of a unifying algebraic language, we show the possibility of obtaining either a 'parallel' or 'joint' normal form of the vector fields in a definite way, which simplifies the construction of normal forms, providing a precise restriction on their structure. The application to finite-dimensional dynamical systems and their Lie point symmetries is also discussed.

In a recent article, Saied and El-Wakil discussed the similarity solutions of a fragmenting system. Their article is an extension of our own work on an equation including continuous loss of mass. Saied and El-Wakil restrict their considerations to similarity solutions on a special choice of parameters. We demonstrate here that the symmetry group given in Saied and El-Wakil's article is incomplete. Furthermore, we are able to construct new similarity solutions derived from the missing subgroup.

We show that Getmanov's N-soliton-type solutions in the self-duality Yang-Mills (SDYM) equations in the (+---)-signatured M⁴-space can be derived in a unified way similar to that of deriving 't Hooft's instanton solutions in the (++++)-signatured E⁴-space. Further, we show that such N-soliton-type solutions can also be constructed for the SDYM equations in the (+-+-)-signatured real D⁴-space.

The conditions are examined under which Fourier-Bessel expansions can be used correctly in the evaluation of multiple folding integrals. A proper formulation of the method is given.

Numerical integration schemes for coupled time-dependent nonlinear Schrodinger equations are examined using exponential splitting step methods. Exponentiation of the nonlinear potential term is reduced to the exponential of a kinetic energy term which can be calculated by fast Fourier transforms. High-order iteration schemes involving a minimum number of product operators are shown to yield highly accurate amplitude and phase. These new splitting methods are shown to be highly efficient both with respect to accuracy and integration time.

We study the influence of external potentials on the solitary wave solutions of certain nonlinear partial differential equations common in physics. Our approach allows for both the translation of pulse-like envelopes through space and the 'breathing' of these pulses in the centre-of-mass frame. We find that for certain simple potentials it is possible for the familiar soliton solutions of these equations to be preserved and to execute essentially classical motion. For more general potentials, however, we find that the familiar pulse shapes cannot be preserved; shape deformations more complex than our simple breathing motion are required. When shape deformations are not too severe, our approach allows an approximate solution for the case of adiabatic following.

The dynamics of relativistic quantum vortices were recently analysed in a model based on the nonlinear wave equation for a complex scalar field. These results are presented here in the context of relativistic pure superfluids and the existence of the correct non-relativistic limit is verified. Relativistic superfluid vortices are essentially different from their Newtonian limit-their equation of motion contains an acceleration term, absent in non-relativistic vortex dynamics. Still, it is shown that under certain conditions, the relativistic Euler equation and Helmholz's theorem are obtained as limiting cases.

One can define several properties of wave equations that correspond to the absence of tails in their solutions, the most common, by far, being Huygens' principle. Not all of these definitions are equivalent, although they are sometimes assumed to be. We analyse this issue in detail for linear scalar waves, establishing some relationships between the various properties. Huygens' principle is almost always equivalent to the characteristic propagation property and in two spacetime dimensions the latter is equivalent to the zeroth-order progressing-wave propagation property. Higher-order progressing waves, in general, do have tails and do not seem to admit a simple physical characterization, but they are nevertheless useful because of their close association with exactly solvable two-dimensional equations.

There exist several definitions of the mean value and the phase uncertainty of an optical field. We show that one of these definitions is especially simple and intuitive because of its mechanical analogy. On this basis, we also derive the summation rule and the Tchebyshev inequality for random angular and phase variables and a number-phase uncertainty relation.

We discuss the Poisson-Lie structure of the integrable nonlinear O(N) sigma -model with the moving-frame method. The corresponding r- and s-matrices are given explicitly. We also perform the gauge transformation for the Lax potential and the r and s matrices. Furthermore, we discover that the field-dependent terms in our r- and s-matrices only depend on the Riemannian connection of the target manifold.

A detailed description of classical scattering in Liouville field theory (LFT) is presented. Contrary to widespread belief, LFT scattering is shown to be non-trivial, although it is finite-dimensional in some sense. In particular, for certain phase spaces, the LFT S-matrix is represented as a transformation of the Poisson group SL(2,R). The completeness and conformal invariance of the scattering are also indicated. Singular fields are treated on an equal footing with regular ones, except that only the latter are given a consistent Hamiltonian interpretation. A number of unexpected peculiarities of LFT scattering are revealed. First, for some exceptional field configurations, the asymptotic fields are not solutions of d'Alembert's equation, rather they are a sum of the d'Alembert and Liouville components. Second, the scattering occurs in 'two or three spaces'. And last, depending on the choice of the algebra of observables, the conventional splitting of the d'Alembert field into leftward and rightward components is either in general impossible or essentially non-unique.