Table of contents

The authors have studied numerically the parallel dynamics of nonsymmetric Sherrington-Kirkpatrick spin glasses, varying the degree of symmetry eta :=(J_i,kJ_{k, i})/(J_{i, k}²) of the coupling coefficients between 0 and 1. For systems of finite size N, in the limit t to infinity and at 'zero temperature', T=0, they find subtle behaviour of the function (C₂(t)):=(s_i(t-1)s_i (t+1)), which characterizes the appearance of 2-cycles or fixed-point attractors quantitatively. One has to distinguish the two cases eta >0.5, where (C₂( infinity )) to 1, i.e. the system is eventually trapped with probability 1 in a fixed-point or a 2-cycle, if after t to infinity the limit N to infinity is taken, and eta <0.5, where, in contrast, (C₂( infinity )) is <1, since longer cycles appear. However, the 'trapping' for eta >0.5 happens only for T=0, and at time scales tau _N which increase exponentially with N, whereas for T>0, or if for T=0 the limit N to infinity would be taken before t to infinity , the quantity (C₂(t to infinity )) would decrease smoothly and monotonically with decreasing eta right from eta =1, in quantitative agreement with the mean-field simulation of Eissfeller and Opper. For T=0, the transient behaviour of (C₂(t)) between the mean-field value, which is reached already after typically 100 time steps, and the trapping event, is found to be governed by log-normal statistics with size-depending parameters.

Using large lattices and different methods of simulation and analysis, the kinetic critical exponent z for damage spreading is shown to be consistent with z=2 in two to five dimensions, without the logarithmic problems seen in some earlier work.

Diagonal solutions for the reflection matrices associated to the elliptic R matrix of the eight-vertex free fermion model are presented. They lead through the second derivative of the open chain transfer matrix to an XY Hamiltonian in a magnetic field which is invariant under a quantum deformed Clifford-Hopf algebra.

The authors examine the relation between the representation theory of a two-parameter deformation of the oscillator algebra and certain bibasic Laguerre functions and polynomials.

The paper studies the relationship between NMR measurements of the magnetization decay in porous media and random-walk problems. The behaviour of this decay can be related to the statistical properties of random walks which interact with the pore-solid interface via perfectly reflecting boundary conditions. An efficient numerical technique measures these statistics using a variable step size random-walk simulation. This technique is applied to three simple models for porous materials. The results indicate that there is a regime of stretched-exponential relaxation of the magnetization for a percolation model.

The authors study nonequilibrium critical relaxation properties of model C (purely dissipative relaxation of an order parameter coupled to a conserved density) starting from a macroscopically prepared initial state with short-range correlations. Using a field-theoretic renormalization group approach they show that all the stages of growth of the correlation length display universal behaviour governed by a new critical exponent theta . This exponent is calculated to second order in in =4-d where d is the spatial dimension of the system.

A recently developed method is used to calculate the main effects of excluded volume on the distribution of ions around a charged central sphere in thermal equilibrium. The authors find significant corrections to the results of the conventional Gouy-Chapman theory when the electrostatic energy due to the charge of the sphere is large compared with the thermal energy. The concentration shows a distinct saturation effect, while at the surface of the sphere the known saturation of the potential is lifted. Furthermore, the effect of excluded volume is found to be strongly dominated by the excluded volume of ions with a charge opposite to the charge of the sphere.

The authors apply the Monte Carlo renormalization group (MCRG) analysis of self-avoiding walks (SAWs) on fractals to calculate the critical exponent gamma , associated with the total number of distinct SAWs. In the case of the Sierpinski gasket family of fractals (whose members are labelled by an integer b, 2<or=b< infinity ) they have calculated gamma for 2<or=b<or=80. Their MCRG results deviate at most 0.2% from the available exact results (2<or=b<or=8). The entire set of their results demonstrates that gamma , being always larger than the Euclidean value ⁴³/_32, monotonically increases with b.

The authors introduce a general class of random spin systems which are symmetric under local gauge transformations. Their model is a generalization of the usual Ising spin glass and includes the Z_q, XY, and SU (2) gauge glasses. For this general class of systems, the internal energy and an upper bound on the specific heat are calculated explicitly in any dimensions on a special line in the phase diagram. Although the line intersects a phase boundary at a multicritical point, the internal energy and the bound on the specific heat are found to be written in terms of a simple function. They also show that the boundary between the ferromagnetic and nonferromagnetic phases is parallel to the temperature axis in the low-temperature region of the phase diagram. This means the absence of re-entrant transitions. All these properties are derived by simple applications of gauge transformations of spin and randomness degrees of freedom.

The authors examine the nature of the tunable family of patterns obtained from the discrete eta -DLA model in the deterministic zero-noise limit. The eta -DLA model is a variant of the standard diffusion-limited aggregation (DLA) model in which the DLA growth probabilities are raised to the power eta . The observed morphologies, which range from compact Eden clusters for eta =0 through to sharp needle-like clusters with increasing eta , can be characterized by a sequence of step lengths in a stable staircase structure proceeding back from the tip. Side-branch whiskers, which are found on the triangular lattice but not on the square lattice, occur closer to the tip as eta is increased. Beyond a value eta _c, whiskers are found immediately behind the tip. They derive the length of the exposed tip as a function of eta using a stationary contour approximation and conformal mapping methods. A theoretical estimate for eta _c is derived by refining this approach to incorporate the possible shielding of surface sites by aggregate sites. Their theoretical results are in excellent agreement with the numerical results on both lattices.

The authors study the long-time behaviour of the transient before the collapse on the periodic attractors of a discrete deterministic asymmetric neural network model. The system has a finite number of possible states so it is not possible to use the term chaos in the usual sense of sensitive dependence on the initial condition. Nevertheless, on varying the asymmetry parameter, k, one observes a transition from ordered motion (i.e. short transients and short periods on the attractors) to 'complex' temporal behaviour. This transition takes place for the same value k_c at which one has a change in behaviour for the mean transient length from a power law in the size of the system (N) to an exponential law in N. The 'complex' behaviour during the transient shows strong analogies with chaotic behaviour: decay of temporal correlations, positive Shannon entropy, nonconstant Renyi entropies of different orders. Moreover the transition is very similar to the intermittent transition in chaotic systems: a scaling law for the Shannon entropy and strong fluctuations of the 'effective Shannon entropy' along the transient, for k>k_c.

Recently Hansen et al. (1993) derived a Fokker-Planck equation (FPE) for the learning dynamics of neural networks, which differs from a previously given version by Radons et al. (1993). It is shown that the discrepancies are due to different implicit assumptions for the distribution of time intervals between the discrete learning events. Both approximations are therefore equally justified from a general point of view. The long-time properties, however, are independent of this distribution and are in general more accurately described in the original FPE of Radons et al. Especially, mean and variance of the synaptic parameter distributions are exact only in the latter approach.

The authors compute the maximum Lyapunov exponent lambda of an earthquake model which exhibits deterministic chaos and they discuss its relation with the predictability time of the system. A method is proposed to estimate lambda by the calculation of the entropy of Markov processes which mimic (i) a Poincare map of the model and (ii) a random map related to the seismic signal. The latter map can be obtained using experimental records generated by low-dimensional chaotic system where Poincare maps are not feasible.

The authors investigate the predictions of random-matrix theory for eigenvector statistics and compare them with numerical results obtained for the dissipative periodically kicked top. Different types of statistics are found. In contrast to conservative dynamics they are connected with the types of eigenvalue of the propagator rather than with the symmetries of Hamiltonian evolution.

From the extension of the covariant vertex operator construction for the affine Kac-Moody algebras to Lorentzian algebras, it is shown that the parafermionics realization of arbitrary level affine algebras can be interpreted in terms of bosonic realization. This connection is explicitly illustrated for level k=2 SU(2) algebra.

A path-integral formulation in the representation of coherent states for the unitary U₂ group and U_2/1 supergroup is introduced. U₂ and U_2/1 path integrals are shown to be defined on the coset spaces U₂/U₁(X)U₁ and U_2/1/U_1/1(X)U₁ respectively. These cosets appear as curved classical phase spaces. Partition functions are expressed as path integrals over these spaces. In the case when U₂ and U_2/1 are the dynamical groups, the corresponding path integrals are evaluated with the help of linear fractional transformations that appear as the group (supergroup) action in the coset space (superspace). Possible applications for quantum models are discussed.

A novel genetic algorithm has been developed and applied to the solution of ordinary differential equations. The algorithm solves the equations by a process of breeding better candidate solutions from a family of estimates, and learns to retain the best features as it progresses. This self-learning system is intrinsically parallel, and capable of handling linear and nonlinear equations, both stiff and non-stiff. Genetic algorithms are a key element in artificial intelligence and machine learning, playing a significant role in optimization and robotics. In this document, the application of genetic algorithms to the solution of ordinary differential equations is presented as a radically different way of approaching numerical simulations, and is of importance to many disciplines.

Calculates in closed form a family of definite integrals I_n(b,a;c), n=0,1,2,3,. . ., which arise in the calculation of regularized functional determinants associated with a compact space form X of a rank 1 Riemannian symmetric space. In the special case when n=0, b=¹/₂, a= pi , c=1, and X is a Riemann surface, the integral I₀(¹/₂, pi ;1) is known, for example in the context of Polyakov string theory.

The dynamics and symmetries of the two-dimensional vortex gas on compact Riemann surfaces are analysed using Lagrangian dynamics. As the vortex Lagrangian is linear in the canonical momenta, Dirac's theory of constraints is then used to form the Hamiltonian dynamics for the system.

The authors consider the spectral statistics of independent electrons moving at zero temperature in a weakly disordered metallic ring threaded by a magnetic flux. The analysis is based on the supersymmetry method involving both commuting and anticommuting variables. Besides, they consider an ensemble of Gaussian distributed symmetric random matrices (Gaussian orthogonal ensemble) which are perturbed by a small time reversal symmetry breaking contribution. For energies smaller than the inverse diffusion time around the ring E_c, the spectral correlation functions of both models can be represented in terms of supermatrix integrals of identical structure. In conformity with recent numerical results, this implies that the spectral properties of the two models coincide. These matrix integrals are to a large extent universal, i.e. they depend only on two physical parameters: the mean level spacing and a symmetry breaking parameter which is identified as the typical sensitivity of levels to the time reversal symmetry breaking perturbation. The authors parametrize the relevant matrix coset space of the nonlinear sigma -model in a novel way which is particularly convenient for treating models in the crossover between the two symmetry classes. As an example, they present a detailed calculation of the level-level correlation function. The basic formalism, however, applies quite generally and can be used for the investigation of different types of correlation functions and system geometries as well.

The Grassmannian model for the Korteweg-de Vries hierarchy is used to describe the translational, Galilean and scaling self-similar solutions of this hierarchy. These solutions are characterized by the string equations appearing in two-dimensional quantum gravity. In particular, the subsets of the Sato Grassmannian corresponding to solutions of the string equations are found. The well known Adler-Moser rational solutions are obtained as well as a three-parameter family of solutions associated with the Painleve II equation.

The authors have found similarity reductions for the deformed Maxwell-Bloch system to the fifth and second Painleve equations. Asymptotics of the solutions of these equations, which are relevant to two-level atomic systems with pumping, have also been derived.

Studies a model based on N scalar complex fields coupled to a scalar real field, where all fields are treated classically as c-numbers. The model describes a composite particle made up of n constituents with bare mass m₀ interacting both with each other and with themselves via the exchange of a particle of mass mu ₀. The stationary states of the composite particle are described by relativistic Hartree equations. Since the self-interactions is included, the case of an elementary particle is a non-trivial special case of this model. Using an integral transform method, the author derives the exact ground-state solution and prove its local stability. The mass of the composite particle is calculated as the total energy in the rest frame. For the case of a massless exchange particle the mass formula is given in closed form. The mass, as a function of the coupling constant, possesses a well pronounced minimum for each value of mu ₀/m₀, while the absolute minimum occurs at mu ₀=0.

Combinatorial relations can be used to convert the non-relativistic time-sliced Feynman path integral into perturbation expansions. These methods reveal that when the time interval is sliced into N increments, each order of perturbation theory sustains an error O(1/ square root N). In this way the author provides exact path integral results for the following potentials: delta-function comb, finite well, tunnelling barrier, and a generalized exponential cusp. For the tunnelling barrier it is seen how the celebrated (-1) reflection factor arises in the limit of infinite barrier height. The one-dimensional Coulomb problem is solved as a limiting case of the exponential cusp. In addition, for power potentials the author indicates how this path integral approach yields sometimes divergent, nevertheless asymptotic perturbation expansions.

The Dirac equation in Kerr spacetime is separated using the rotating tetrad formalism. This allows solutions of the Dirac equation, in flat spacetime, written in oblate and prolate spheroidal coordinates to be extracted. The usual MIT bag boundary condition, -i gamma ^mu n_v Psi = Psi is then found to be incompatible with a non-vanishing separated wavefunction except in the spherical limit. However, it is shown that an alternative boundary condition exists that is physically motivated and allows for a non-trivial solution.

Ground and excited state energy levels of the Schrodinger equation for various model potentials, in two-dimensional space are calculated, using inner product and renormalized series techniques, over wide ranges of relevant perturbation parameters. Mixed parity potentials are treated, whereas earlier works have treated only even parity perturbations.

The authors report results of a Monte Carlo study of the irreversible dimer-dimer reaction (A₂+B₂) on a square lattice. They study the three possibilities for which the probability of formation of AB (adsorbed) is greater than, equal to or less than the formation of AB₂ (gas). For the entire range of partial pressures (p) of B₂ in the gas phase the lattice is poisoned by a combination of A and AB. At p=²/₃ an irreversible phase transition is observed. For 0<p<²/₃ the coverages depend on the choice of formation probabilities of AB and AB₂. The results are compared to those of Albano (ibid., vol.25, p.2557, (1992)).