Table of contents

In this paper, we present a cluster algorithm for the simulation of hard spheres and related systems. In this algorithm, a copy of the configuration is rotated with respect to a randomly chosen pivot point. The two systems are then superposed, and clusters of overlapping spheres in the joint system are isolated. Each of these clusters can be `flipped` independently, a process which generates non-local moves in the original configuration. A generalization of this algorithm (which works perfectly well at small density) can be made to work successfully at densities around the solid-liquid transition point in the two-dimensional hard-sphere system.

We study the zero temperature limit of a simple model of slow relaxation without energy barriers, proposed by Ritort (1995), as well as two other closely related models with a much faster relaxation. These models can be mapped onto random walk problems, which allows for their analytic study. We analyse, in particular, a specific aspect of the former model, namely the existence of a bias leading to `entropy barriers` and to a very slow relaxation.

A complex system is often identified by the absence of a characteristic length, e.g. as in a fractal. A very large system subject to a fragmentation and/or aggregation dynamics passes through such complex configurations. We study statistically creation and maintenance of such configurations in space dimensions d=1 to 5 and find that they are easily created (maintained) for small (large) d. An intermediate d such as d=3 seems to be ideal for the creation and maintenance of complex systems. This has consequences in a statistical description of the universe.

We investigate numerically a generalized version of a model recently proposed by several authors for describing the critical behaviour of a driven interface in a random medium (the self-organized depinning model). The generalized version allows growth events with simultaneous movements of groups of connected cells of arbitrary size, and includes an external driving force. In 1+1 dimensions, the model exhibits phase transition from a phase with directed percolation exponents to one with the usual exponents ( alpha = 1/2 , beta =1/3). But at the transition point, exponents are those of Parisi models.

The exact solution of the boundary sine-Gordon model is studied in the region where the scaling dimension of the boundary field 2/3< Delta <1. The boundary contribution to the specific heat in this region scales as C~T(2 Delta ^-1-2) at small temperatures.

We consider a model of oriented, non-intersecting flux lines on the lattice Z^d, where each flux line is assigned a Boltzmann factor omega per unit length and a fugacity y. We prove the existence of free energy, both for y>0 and for y=-1, and show that it is independent of y for y>0. Using upper and lower bounds in terms of exactly solvable models, we rigorously establish that, for all y>0, the model has a phase transition at omega =1/d. For omega <1/d, we prove that the free energy and all bulk correlation functions vanish, implying the exclusion of flux lines from the bulk. In this regime, we also show that the flux line density decays at least exponentially with distance from the boundary.

Within the framework of statistical physics, we derive a cavity method for generalization by perceptrons, where the Kuhn-Tucker conditions for optimal stability are built into the cavity fields. In this way, the calculation of the generalization ability for learning processes leading to optimal stability is simplified. Within our approach, the degrees of freedom of the neurons can be rather arbitrary. For perceptrons with Ising neurons we relate our method to the traditional replica approach. New results are obtained for e-state Ports model perceptrons, including the asymptotic behaviour for alpha to infinity and general Q.

We study a model of stochastic deposition-evaporation with recombination, of three species of dimers on a line. This model is a generalization of the model recently introduced by Barma et. al (1993) to q>or=3 states per site. It has an infinite number of constants of motion, in addition to the infinity of conservation laws of the original model which are encoded as the conservation of the irreducible string. We determine the number of dynamically disconnected sectors and their sizes in this model exactly. Using the additional symmetry we construct a class of exact eigenvectors of the stochastic matrix. The autocorrelation function decays with different powers of t in different sectors. We find that the spatial correlation function has an algebraic decay with exponent 3/2, in the sector corresponding to the initial state in which all sites are in the same state. The dynamical exponent is non-trivial in this sector, and we estimate it numerically by exact diagonalization of the stochastic matrix for small sizes. We find that in this case z=2.39+or-0.05.

We develop a renormalized continuum field theory for a directed polymer interacting with a random medium and a single extended defect. The renormalization group is based on the operator algebra of the pinning potential; it has novel features due to the breakdown of hyperscaling in a random system. There is a second-order transition between a localized and a delocalized phase of the polymer; we obtain analytic results on its critical pinning strength and scaling exponents. Our results are directly related to spatially inhomogeneous Kardar-Parisi-Zhang surface growth.

The antiferromagnetic three-state Potts model on the simple-cubic lattice is studied using Monte Carlo simulations. The ordering in a medium temperature range below the critical point is investigated in detail. Two different regimes have been observed: the so-called broken sublattice-symmetry phase dominates at sufficiently low temperatures, while the phase just below the critical point is characterized by an effectively continuous order parameter and by a fully restored rotational symmetry. However, the latter phase is not the permutationally sublattice symmetric phase recently predicted by the cluster variation method.

We study the effect of removing detailed balance from the axial next-nearest-neighbour Ising (ANNNI) model on its phase diagram. Although the concepts of free energy or Gibbsian thermodynamical equilibrium no longer apply, we find numerically that the phase diagram is preserved even when the interactions are completely unidirectional. We also find that the value of the multiphase point varies with the degree of asymmetry in the rules.

Recently fractional calculus (FC) has encountered much success in the description of complex dynamics. In particular FC has proved to be a valuable tool to handle viscoelastic aspects. In this paper we construct fractional rheological constitutive equations on the basis of well known mechanical models, especially the Maxwell, the Kelvin-Voigt, the Zener and the Poynting-Thomson model. To this end we introduce a fractional element, in addition to the standard purely elastic and purely viscous elements. As we proceed to show, many of the fractional differential equations which we obtain by this construction method admit closed form, analytical solutions in terms of Fox H-functions of the Minag-Leffler type.

We look for similarity transformations which yield mappings between different one-dimensional reaction-diffusion processes. In this way results obtained for special systems can be generalized to equivalent reaction-diffusion models. The coagulation (A+A to A) or the annihilation (A+A to 0) models can be mapped onto systems in which both processes are allowed. With the help of the coagulation-decoagulation model results for some death-decoagulation and annihilation-creation systems are given. We also find a reaction-diffusion system which is equivalent to the two-species annihilation model (A+B to 0). Besides we present numerical results of Monte Carlo simulations. An accurate description of the effects of the reaction rates on the concentration in one-species diffusion-annihilation model is made. The asymptotic behaviour of the concentration in the two-species annihilation system (A+B to 0) with symmetric initial conditions is studied.

We present a complete study of boundary bound states and related boundary S-matrices for the sine-Gordon model with Dirichlet boundary conditions. Our approach is based partly on the bootstrap procedure and partly on the explicit solution of the inhomogeneous XXZ model with boundary magnetic field and solution of the boundary Thirring model. We identify boundary bound states with new `boundary strings` in the Bethe ansatz. The boundary energy is also computed.

We present a study of the multi-canonical Monte Carlo method which constructs and exploits Monte Carlo procedures that sample across an extended space of macrostates. We examine the strategies by which the sampling distribution can be constructed, showing, in particular, that a good approximation to this distribution may be generated efficiently by exploiting measurements of the transition rate between macrostates, in simulations launched from sub-dominant macrostates. We explore the utility of the method in the measurement of absolute free energies, and how it compares with traditional methods based on path integration. We present new results revealing the behaviour of the magnetization distribution of a critical finite-sized magnet, for magnetization values extending from the scaling region all the way to saturation.

We study as an example of a continuous-time random walk (CTRW) scheme under holonomic constraints the motion of a rigid triangle, moving on a plane by flips of its vertices. This interpolates between our former model of a dumbbell (two walkers joined by a fixed segment) and the Orwoll-Stockmeyer model for polymer diffusion. The jumps of the vertices follow either Poissonian or power-law waiting-time distributions, and each vertex follows its own internal clock. Numerical simulations of the triangle`s centre-of-mass motion show it to be diffusive at short and also at long times, with a broad crossover (subdiffusive) region in between. Furthermore, we provide approximate expressions for the long-time regime and generalize our findings for systems of N random walkers.

We report on a novel approach to the Deam-Edwards model (1976) for interacting polymeric networks without using replicas. Our approach utilizes the fact that a network modelled from a single non-interacting Gaussian chain of macroscopic size can be solved exactly, even for randomly distributed crosslinking junctions. We derive an exact expression for the partition function of such a generalized Gaussian structure in the presence of random external fields and for its scattering function S₀. We show that S₀ of a randomly crosslinked Gaussian network (RCGN) is a self-averaging quantity and depends only on crosslink concentration M/N, where M and N are the total numbers of crosslinks and monomers. From our derivation we find that the radius of gyration R_g of a RCGN is of the universal form R_g²=(0.26+or-0.01)a²N/M, with a being the Kuhn length. To treat the excluded volume effect in a systematic, perturbative manner, we expand the Deam-Edwards partition function in terms of density fluctuations analogous to the theory of linear polymers. For a highly crosslinked interacting network we derive an expression for the free energy of the system in terms of S₀ which has the same role in our model as the Debye function for linear polymers. Our ideas are easily generalized to crosslinked polymer blends which are treated within a modifed version of Leibler`s mean-field theory (1980) for block copolymers.

We consider the statistics of generalized Gaussian structures (GGS) exposed to a random external field. A GGS comprises N monomers connected to each other by harmonic potentials. When the spectral dimension d, of a GGS exceeds the value of two its radius of gyration R becomes independent of its mass N. The cross-over into this collapse can be treated continuously by cross-linking m precursor chains of length n in the stretched state to an object which we call a polymer bundle. We demonstrate that an external field f applied to each monomer can `unfold` such a collapsed state. In the case where every monomer has an individual, randomly distributed, charge the critical spectral dimension for the collapse is raised to four. R scales like fN^alpha with alpha =(4-d_s)/(2d_s) for d_s<4.

A generalization of the eight-vertex model by means of higher spin representations of the Sklyanin algebra is investigated by the quantum inverse scattering method and the algebraic Bethe ansatz. Under the well known string hypothesis low-lying excited states are considered and scattering phase shifts of two physical particles are calculated. The S-matrix of two-particle states is shown to be proportional to Baxter`s (1972) elliptic R-matrix with a different elliptic modulus from the original one.

The Lie algebra for the maximal contact symmetries of third-order ordinary differential equations (ODEs) is examined for type I and II hidden symmetries where the analysis of hidden symmetries for point symmetries is extended to contact symmetries. Ones invariant under the group associated with the ten-dimensional (maximal) Lie algebra may produce type I hidden symmetries for two-parameter subgroups and type II hidden symmetries for certain solvable non-Abelian three-parameter subgroups in the third-order ODEs when they are reduced in order. A new class of type II hidden symmetries is recognized in which contact symmetries transform to point symmetries in some reduction paths. Two examples of ODEs invariant under subgroups of the ten-parameter group under which y```=0 is invariant demonstrate the new class of type II hidden symmetries.

The problem of construction of the boundary conditions for the Toda lattice compatible with its higher symmetries is considered. It is demonstrated that this problem is reduced to finding the differential constraints consistent with the ZS-AKNS hierarchy. A method of their construction is offered based on the Backlund transformations. It is shown that the generalized Toda lattices corresponding to the non-exceptional Lie algebras of finite growth can be obtained by imposing one of the four simplest integrable boundary conditions on both ends of the lattice. This fact allows, in particular, the solution of the reduction problem of the series A Toda lattices into the series D lattices. Deformations of the found boundary conditions are presented which lead to the Painleve-type equations.

Here we study the singular anharmonic potentials by applying the analytic continuation method of Holubec and Stauffer (1985). In order to do that we have developed several approximations to the problem, because this method cannot be applied when the solution has essential singularities at any point of its domain. All the options here shown have the same precision, giving us the eigenvalues correct to all the decimal places provided by the computer.

A compact solution of a special biconfluent Heun equation is obtained by means of a Laplace transform. The eigenvalues and eigenfunctions of two new types of Schrodinger equation are deduced by appropriate variable transformations. The associated potential function in each case can be arranged to have a double well.

The derivation d_T on the exterior algebra of forms on a manifold M with values in the exterior algebra of forms on the tangent bundle TM is extended to multivector fields. These tangent lifts are studied with application to the theory of Poisson structures, their symplectic foliations, canonical vector fields and Poisson-Lie groups.

In a previous paper, we considered Weyl quantization of functions of the angle in phase space, in particular a phase operator Delta ( phi ) and the quantized exponentials Delta (e^{+or-1 phi} ). In this paper we consider the first and second moments of these operators with respect to the harmonic oscillator Hermite states h_n and the coherent states Phi _alpha . Taking asymptotic limits we find, for example, that var[ Delta ( phi ); h_n]= pi ²/3+O(log n/n) (n to infinity ) for the variance of Delta ( phi ) in the Hermite states. For the second moment of the phase operator in ||[ Delta ( phi )- theta ] Phi _alpha ||²=O(1/| alpha |) as | alpha | tends to infinity, amongst other results.

In a series of papers, Barnett, Pegg (1986, 1988, 1989, 1992) and various co-authors have proposed a description of quantum phase by means of a collection of s-dimensional states and operators, s>or=1. We analyse the limiting procedure they employ for large s, which is known not to be compatible with quantum mechanics in the usual sense. Further, we supply a rigorous demonstration of the asymptotic limits of the `mean` and `variance` of their system of operators in coherent states. These values had previously been given but not justified mathematically. Our analysis, based on the asymptotic analysis of certain random variables, shows that the physical deductions that can be drawn from these limits are limited. We also prove that the (s+1)-dimensional `pure phase` LHW-states they consider form a sequence of approximate eigenvectors for the Weyl-quantized angle operator Delta ( phi ) and the Toeplitz phase operator X proposed by Garrison and Wong (1970), Popov and Yarunin (1992), and others. These states X_s( theta ), to our knowledge first introduced by Lerner, Huang and Waiters (1970), can be used to construct a system of measurement as in the usual quantum theory, sensitive to certain qualities of phase, but not all. Indeed, a feature of the Barnett-Pegg method, when it gives finite answers, is the construction of associated measurement systems for different observables. We give examples of sequences of (s+1)-dimensional devices which represent measurements significantly closer to ideal for X. This serves as a model for corresponding devices for Delta ( phi ), or indeed, any observable with a continuous spectrum, contingent on its spectral decomposition being obtained explicitly.

Casimir operators for semidirect products of some semisimple groups with Heisenberg groups are computed. The analysis is carried out using dual representations on Fock space, wherein the action of the semidirect products are related to their dual groups, namely certain unitary, orthogonal, and symplectic groups. The compact symplectic group chain is also investigated; by passing to the complexification, groups `between` the symplectic groups are constructed, which are of the form of semidirect products of symplectic groups with Heisenberg groups.

Whilst many solutions have been found for the quantum Yang-Baxter equation (QYBE), there are fewer known solutions available for its higher dimensional generalizations: Zamolodchikov`s tetrahedron equation (ZTE) and Frenkel and Moore`s simplex equation (FME). In this paper, we present families of solutions to the FME which may help us to understand more about higher dimensional generalization of the QYBE.

Many scattering systems can be described as scattering of a point particle off a multicentre potential. In this paper we present a two-centre system which shows either regular or chaotic scattering depending on the kinetic energy, i.e. the velocity of the incoming particle. The transition points to chaotic scattering can be derived analytically by linearization of the Poincare map. At one of these transition velocities there is a degenerate bifurcation where the invariant set contains a parabolic surface and where the time delay statistics is algebraic with power sigma =2.

The non-conservative noisy critical height sandpile cellular automaton with open boundary conditions is studied analytically on the Bethe lattice. Using the modified method of Dhar and Majumdar (1990), the single-site probabilities, pair probabilities and the avalanche size distributions for the three versions of the automaton, with different amount of dissipated particles, are calculated.

A new method is developed for solving the conformally invariant integrals that arise in conformal field theories with a boundary. The presence of a boundary makes previous techniques for theories without a boundary less suitable. The method makes essential use of an invertible integral transform, related to the radon transform, involving integration over planes parallel to the boundary. For successful application of this method several non-trivial hypergeometric function relations are also derived.

The formalism which has been developed to give general expressions for the determinants of differential operators is extended to the physically interesting situation where these operators have a zero mode which has been extracted. In the approach adopted here, this mode is removed by a novel regularization procedure, which allows remarkably simple expressions for these determinants to be derived.

The dissipative relativistic fluid theories of divergence type are the simplest theories which are physically consistent and have a well posed-hyperbolic-initial value formulation, since they can be constructed from a single scalar function chi and a dissipation-source tensor I^ab, both of them functions of fluid variables. In this work we find the expression for this generating function for the case of a dilute gas using only the knowledge of an equilibrium fluid state, which is known from the kinetic theory of dilute gases. The generating function is obtained by imposing some conditions on the divergence theory, related to the symmetry and trace of the tensor of the fluxes. These conditions come naturally from kinetic theory, and are needed to correctly describe a dilute gas. We prove that in the neighbourhood of the equilibrium states, these divergence type equations for a dilute gas are causal for Boltzmann, Fermi or Bose equilibrium distribution functions.

The minimization problem of finding the number-phase minimum uncertainty states (MUS) is considered and its solutions are found either numerically or, under some special conditions, analytically. The phase uncertainty measure is based on the Bandilla-Paul dispersion. The problem is treated (i) in a finite-dimensional Hilbert space and (ii) for a countably infinite-dimensional Hilbert space (i.e. the standard quantum harmonic oscillator), with the constraint of a given mean photon number. The MUS relations between the photon number uncertainty and phase uncertainty are presented. Connections to some other minimization problems are discussed.

In this paper, a singularity structure analysis of some important inhomogeneous nonlinear evolution equations (NLEES) of AKNS type introduced by Burtsev, Zakharov and Mikhailov (1987) is carried out and they are shown to possess the Painleve property. The other integrability properties such as Lax pair, Backlund transformation and soliton solutions of these systems are also brought out in detail from the Painleve analysis. We also point out that the non-Painleve nature of the system of partial differential equations satisfied by the variable spectral parameter may not affect the Painleve property and hence the integrability of the associated NLEES.

It is demonstrated that the integral exact solution generation methods for the one-dimensional Schrodinger equation based on the Gelfand-Levitan formalism are in some cases equivalent to the differential ones based on the n-order Darboux transformation. Some new exact solvable potentials are generated from the effective Coulomb potential and the harmonic oscillator potential. A new form of n-soliton potential (i.e. reflectionless potential with n discrete energy levels disposed in a desirable manner) based on an explicit expression for an n-order Wronski determinant constructed from hyperbolic functions and its orthonormal discrete spectrum eigenfunctions are given.

In this paper we consider certain lattice sums which arise when a system of line charges set in a compensating jelly of opposite charge interact via the two-dimensional logarithmic potential. A lattice-limit discontinuity phenomenon, similar to that discovered by Borwein et al. (1989) in the three-dimensional case, is explored and a high-precision asymptotic method is described for the numerical computation of two-dimensional lattice limits.

Three specific problems are introduced and solved. The first is to determine the number of times the adjoint representation of SU_n occurs in the Kronecker square of a self-contragredient representation. The second is to determine the number of times the adjoint appears in the symmetric part of the square, with the third being the number of times the adjoint appears in the antisymmetric part. These problems are solved by recasting them as three problems concerning the squares of self-complementary S-functions and an equivalent adjoint S-function of a particular shape.

In a recent paper Oh and Singh determined a Hopf structure for a generalized q-oscillator algebra. We prove that under some general assumptions, the latter is, apart from some algebras isomorphic to su_q(2), su_q(1,1), or their undeformed counterparts, the only generalized deformed oscillator algebra that supports a Hopf structure. We show in addition that the latter can be equipped with a universal R-matrix, thereby making it into a quasitriangular Hopf algebra.