Table of contents

Traditional two-scale-factor universality, concerning the number of clusters within a correlation volume near the percolation threshold, is reconfirmed by a comparison of site percolation on square and triangular lattices. We also test universality at the threshold, e.g. of the ratio , where is the correlation length and L the lattice size. At the threshold, the result is sensitive to the system's shape and aspect ratio, boundary conditions, the algorithm, the detailed units measuring and L, and possibly other factors as yet unexplored.

We investigate the thermally activated escape of a particle over a potential barrier whose height fluctuates between two values. The barrier-switching process is constructed as a semi-Markov alternating process: the times at which a change of the barrier state can occcur form a general renewal process and the probability of leaving a state depends on the time the barrier resides in the state before the jump. During the interjump interval, the barrier is fixed in one of the two states and the crossing dynamics is described by a general (not necessarily exponential) decay law. We give the general formulae describing the averaged escape dynamics, where the averaging runs over all possible histories of the switching process. Using the above device of the selective residence times, the recently discussed phenomenon of resonant activation (a minimal averaged lifetime of the particle in the potential well) emerges also within the framework of the conventional exponential escape dynamics.

The problem of how to obtain quasi-classical states for quantum groups is examined. A measure of quantum indeterminacy is proposed, which involves expectation values of some natural quantum group operators. It is shown that within any finite-dimensional irreducible representation, the highest-weight vector and those unitarily related to it are the quasi-classical states.

We introduce a model of three-species two-particle diffusion-limited reactions or B, or C, and or A, with three persistence parameters (survival probabilities in reaction) of the hopping particle. We consider isotropic and anisotropic diffusion (hopping with a drift) in one dimension. We find that the particle density decays as a power law for certain choices of the persistence parameter values. In the anisotropic case, on one symmetric line in the parameter space, the decay exponent is monotonically varying between the values close to 1/3 and 1/2. On another, less symmetric line, the exponent is constant. For most parameter values, the density does not follow a power law. We also calculated various characteristic exponents for the distance of nearest particles and domain structure. Our results support the recently proposed possibility that one-dimensional diffusion-limited reactions with a drift do not fall within a limited number of distinct universality classes.

The Jacobi eigenfunctions are suitably adjusted by means of the hypervirial theorems (HVT) scheme in order to obtain, in many cases, a very satisfactory analytic approximation of the s-state eigenfunctions of the radial Gaussian potential. The results obtained show that the HVT scheme is a powerful method for obtaining (analytically) not only approximate eigenvalues but also the corresponding eigenfunctions.

We find the analytical expression for the residual entropy of the square Ising model with nearest-neighbour antiferromagnetic coupling J, in the maximum critical field , in terms of the Fibonacci matrix, which itself represents a self-similar, fractal object. The result coincides with the existing numerical data. By considering regular self-similar fractal objects rather than seemingly random transfer matrices, this approach opens the possibility of finding the corresponding solutions in more complicated cases, such as the antiferromagnets with longer than nearest-neighbour interactions and the three-dimensional antiferromagnets, as well as the possibility of unification of results pertinent to different lattices in two and three dimensions.

We study numerically and analytically the moments of the dimensionless level curvature for one-dimensional disordered rings of the circumference L pierced by a magnetic flux . The negative moments of the curvature distribution can be evaluated analytically in the extreme localization limit. The ensuing small curvature asymptotics of the corresponding distribution has a `log - normal' behaviour. Numerically studied positive moments show differences from other log - normally distributed quantities.

The path-dependent surface/time integral contribution to the topological charge in an SO(3) Yang - Mills theory is studied for paths in field space that interpolate between a background gauge field in the remote past and a gauge transform of it in the remote future. The possibility of existence of such paths along which this integral vanishes for a given initial background gauge field is related to the action of the group of gauge transformations of real, pseudo winding numbers on the physical states of the theory in the background gauge field. The analysis takes a particularly transparent form for the spherically-symmetric fields of the spherical ansatz, leading to a simple interpretation of the results.

A two-dimensional lattice gas model with nearest-neighbour attractive interaction confined in a strip of width L between two parallel boundaries at which an attractive short-range force acts is studied by Monte Carlo simulations, for cases where the system is in the wet phase near the critical wetting transition line for . We study the shift of the chemical potential of the transition in the strip as a function of L by thermodynamic integration methods, , and also obtain the thickness of the wetting film at the chemical potential at which capillary condensation occurs. In the range the data are consistent with a variation according to the Kelvin equation, , as well as with a shifted Kelvin equation, , with a constant . Thus, we find no evidence for the fluctuation correction predicted by Parry and Evans. This failure is traced back to the fact that in this range of linear dimensions there are not yet any well developed wetting layers at coexistence, and the prediction from the theory of complete wetting does not hold in this range either. Instead we empirically find a relation over the whole range of system sizes we studied.

The effect of the initial particle distribution on the one-dimensional coagulation and annihilation reaction - diffusion models is studied analytically. With this aim, an exact expression for the particle number, N(t), given explicitly in terms of the initial particle distribution, is derived. It is found that if the initial distribution has divergent first moments new decay laws occur. For the case of fractal distributions of dimension we obtain exactly the mean relative number of particles, . For long times it evolves as , with . The existence of non-fractal initial distributions that lead to new decays is also discussed. As examples are considered an interparticle probability distribution of the form which yields and a family of distributions of the form , with , which yields . These results are tested by Monte Carlo simulations.

The low-temperature series are calculated for the free energy, magnetization and susceptibility in the Q-state Potts model on the square lattice, using the improved algorithm of the finite lattice method. The series are obtained to the order of for each of Q = 5 - 50, and the result of their Padé-type analysis is compared with those of the large-Q expansion and the Monte Carlo simulations.

Within the framework of a simple model of car traffic on a one-lane highway, we study the probability of the occurrence of car accidents when drivers do not respect the safety distance between cars, and, as a result of the blockage during the time T necessary to clear the road, we determine the number of stopped cars as a function of car density. We give a simple theory in good agreement with our numerical simulations.

We generalize Bray's self-consistent screening approximation to describe the critical dynamics of the theory. In order to obtain the dynamical exponent z, we have to make an ansatz for the form of the scaling functions, which fortunately can be restricted by general arguments. Numerical values of z for d = 3, and n = 1, ..., 10 are obtained using two different ansatz, and differ by a very small amount. In particular, the value of obtained for the three-dimensional Ising model agrees well with recent Monte Carlo simulations.

The one-dimensional q-state Potts model with ferromagnetic pair interactions which decay with the distance r as is considered. We calculate, through a real-space renormalization group technique using Kadanoff blocks of length b, the critical temperature and the correlation length critical exponent as a function of for different values of q. Some of the very few known rigorous results for general q are reproduced by our approach. Several asymptotic behaviours are derived analytically for q = 2, 3 in the limit. We also obtain extrapolated critical temperatures for arbitrary values of and for q = 2, 3, 4, which we believe approximate the exact ones well, except in the region near . Furthermore, the use of another extrapolation procedure suitable only in the vicinity of led us to conjecture that the exact critical temperature is the same for any value of q. We also verify that , which is consistent with a recent conjecture of Tsallis.

We present a new method to study disordered systems in the low-temperature limit. The method uses the replicated Hamiltonian. It studies the saddle points of this Hamiltonian and shows how the various saddle-point contributions can be resummed in order to obtain the scaling behaviour at low temperatures. In a large class of strongly disordered systems, it is necessary to include saddle points of the Hamiltonian which break the replica symmetry in a vector sector, as opposed to the usual matrix sector breaking of the spin glass mean-field theory.

For a given if neurons deviating from the memorized patterns are allowed, we constructively show that if and only if all stored patterns are fixed points of the Hopfield model. If neurons are allowed with then where is the distribution function of the normal distribution. The result obtained by Amit and co-workers only formally coincides with the latter case which indicates that the replica trick approach to the capacity of the Hopfield model is only valid in the case .

We study the thermodynamical limits that are obtained from a parallel updating of ferromagnetic spins on a lattice. We investigate the relationships existing between the parallel model and the sequential model. We compare both of their free energy functions. A long-range model for which the mean-field theory is correct is also studied. The last part of the article indicates how parallel updating can be efficiently used in short-range model simulations.

We generalize a model proposed by Xu et al for ruptures in an elastic medium subject to shear stress. This model is applied to the study of earthquakes. We restrict ourselves to one-dimensional discretizations of the region on which we focus and consider the effects of disorder, degree of stress release and degree of stress nonconservation (dissipation). The one-dimensional systems display power-law cumulative size - frequency distributions over a certain range of size. The power laws cut off due to finite-size effects, i.e. the effects of the finite size of the system and the finite size of the basic unit of discretization. In addition, in the absence of disorder, there is a crossover region at small sizes and its origin is explained. The scaling properties in the absence of dissipation are characterized by exponents and as well as by a function f dependent on the parameters of the model. is associated with the cumulative size - frequency distribution in the thermodynamic limit, with the finite size of the system and f with the finite size of the basic unit of discretization. When stress dissipation is introduced into the model, a characteristic earthquake size smaller than system size appears, in contrast with the case in which stress dissipation is absent.

We discuss symbolic dynamics for the Hamiltonian describing the classical motion of hydrogen in a uniform magnetic field with zero angular momentum. For a scaled energy above a critical value , the Hamiltonian has a Cantor set repellor described by a three-lettered symbolic alphabet. For energies below it is not proven that a good symbolic dynamics description exists. We propose and investigate several methods to determine a symbolic dynamics, and conjecture that these methods label uniquely all non-escaping trajectories for and most trajectories for . We define well ordered symbols and construct the pruning front which determines the admissible orbits, the information needed for effective semiclassical resonance spectra computations.

We prove numerically for the first time that the lower part of the spectrum of the transfer matrix of the quasi-one-dimensional disordered system is strongly correlated in the neighbourhood of the critical point of the metal - insulator transition. In particular, the disorder and the system size dependence of the spectrum is governed only by one parameter which we identify with the scaling parameter of MacKinnon and Kramer. The strong correlation of the spectrum supports the idea of the universality of the random-matrix-like theory of the metal - insulator transition and provides us with a new way of calculation of the critical parameters of this transition.

An accelerated random sequential adsorption process is studied as a model of chemisorption on a line with precursor layer diffusion. In this process if the position first selected for deposition is occupied then the particle diffuses and is absorbed on the first vacant position it visits. For k-mer deposition exact results are obtained for the gap distribution function. Physically measurable quantities such as the average island size and the probabilities of island nucleation, growth and coagulation are calculated as a function of coverage and the saturation coverage is calculated as a function of k. The continuum version of this model is also considered and potential applications of the models are discussed.

The probability distribution of random walks on one-dimensional fractal structures generated by random walks (RW chains) and self-avoiding walks (SAW chains) in d-dimensional space, , is studied analytically in the case , where is the fractal dimension of the random walk, . It is shown that there exists an infinite hierarchy of critical dimensions, , with for RW chains and for SAW chains, for each term in the -expansion of , the scaling part of . Each transition is characterized by its own logarithmic correction.

Error rates of a Boolean perceptron with threshold and either spherical or Ising constraint on the weight vector are calculated for storing patterns from biased input and output distributions derived within a one-step replica symmetry breaking (RSB) treatment. For unbiased output distribution and non-zero stability of the patterns, we find a critical load, , above which two solutions to the saddlepoint equations appear; one with higher free energy and zero threshold and a dominant solution with non-zero threshold. We examine this second-order phase transition and the dependence of on the required pattern stability, , for both one-step RSB and replica symmetry (RS) in the spherical case and for one-step RSB in the Ising case.

We study the density profile and the dynamics of atoms interacting with the quantized radiation field in a trap. The density profile is evaluated by using the Bogoliubov approximation. The calculated mean number of atoms at zero temperature appears to be a function of time which shows a damped oscillation. The shape of the lines for the light scattered from Bose - Einstein condensates are also discussed.

The problem of constructing boundary conditions for nonlinear equations compatible with higher symmetries is considered. In particular, this problem is discussed for the sine - Gordon, Jiber - Shabat, Liouville and KdV equations. New results are obtained for the last two ones. The boundary condition for the KdV contains two arbitrary constants. The substitution maps it onto the boundary condition with linear dependence on t for the potentiated KdV.

After a short presentation of the toroidal moments and the necessity to introduce them in the multipole expansion of current density, the correspondent quantum operators are introduced. The toroidal momentum operator (the quantum operator corresponding to the lowest-order toroidal multipole) is analysed. A natural set of coordinates is found. Using this set of coordinates it becomes possible to find the eigenvalues and a complete orthonormal set of eigenfunctions of the projection of this operator on the Oz axis.

The structure of irreducible representations of (restricted) at roots of unity is understood within the Gelfand - Zetlin basis. The latter needs a weakened definition for non-integrable representations, where the quadratic Casimir operator of the quantum subalgebra is not completely diagonalized. This is necessary in order to take into account the indecomposable -modules that appear. The set of redefined (mixed) states has a teepee shape inside the pyramid made with the whole representation.

The functional determinant of Laplace-type operators on a three-dimensional non-compact hyperbolic manifold with invariant fundamental domain of finite volume is expressed via the Selberg zeta function related to the Picard group .

The quantization of classical theories that admit more than one Hamiltonian description is considered. This is done from a geometrical viewpoint, both at the quantization level (geometric quantization) and at the level of the dynamics of the quantum theory. A spin- system is taken as an example in which all the steps can be completed. It is shown that the geometry of the quantum theory imposes restrictions on the physically allowed non-standard quantum theories.

Given a free Abelian group of finite rank, canonical subgroup matrices are defined with respect to an arbitrary but fixed basis of , which uniquely determine all subgroups of of finite index. The use of these matrices allows the classification of all subgroups of fixed finite index. An application is made in the construction of the intersection group and the union group of two arbitrary subgroups of finite index and upper and lower bounds on the index of these groups are found. Physical applications occur for rank three in the systematic symmetry analysis of domain structures, which appear in structural phase transitions in solids.

The stability of an infinitely long magnetic fluid column of weak viscous effects is investigated. The column is subjected to a periodic azimuthal magnetic field and a rigid-body rotation. Non-axisymmetric two-dimensional perturbations are considered in this investigation. Linear analysis leads to a Mathieu equation with complex coefficients. The analytical results show that the constant magnetic field plays a stabilizing role and can be used to suppress the instability due to the rotation. When the field has been oscillating, the stabilizing role of the amplitude of the magnetic field decreases somewhat due to the applied frequency . The oscillating magnetic field plays a dual role in the stability criterion. The increase of the azimuthal wavenumber decreases the unstable region due to the increase of the column radius. A small viscosity plays a destabilizing effect due to the influence of the angular velocity in the presence of a constant or an oscillating magnetic field. A magnetic column can be stabilized at a given azimuthal wavenumber by a suitable choice of the angular velocity, the density and viscosity for the outer fluid being greater than the corresponding parameters for the inside fluid.

We discuss the differences and similarities in structure between Hamiltonian matrices of strongly chaotic time-independent systems and the evolution operator matrix for the kicked-rotor model. Since the eigenvectors of the latter are exponentially localized, we study the influence of the differences between the two systems on this property. In particular, the Wigner ensemble is used to show that, for , the effect of the slow variation of the diagonal matrix elements in the Hamiltonian matrices is restricted to the tails of the eigenvectors.

The structure of quantum wavefunctions is discussed for a billiard system with mixed classical dynamics. Using a scattering formalism, we introduce a Husimi-like distribution for S-matrix eigenvectors at arbitrary wavenumbers, k (not necessarily eigenenergies of the system). This constitutes probability density plots on the Poincaré surface of section at any k. Many eigenvectors are structured by classical objects such as invariant tori, the intermediate layer between regular and chaotic motion, and unstable periodic orbits (`scars'). We find that eigenvector structure is stable over significant energy ranges, while the associated eigenphases increase linearly with k at a rate determined by the underlying classical objects. Additionally, we observe that in the presence of time-reversal symmetry, eigenvectors scarred by non-self-retracing periodic orbits form doublets which are closely spaced in eigenphase.

Our results provide a new perspective on the quantization of structured states: quantization is determined by the linearly increasing eigenphases, and occurs whenever an eigenphase is equal to , while structure is encoded in the eigenvectors; it is present at all energies and varies only slowly. This allows one to predict many quantized eigenfunctions over an entire energy range by studying the eigenparameters of at a single intermediate energy, and identifying the underlying classical objects.

An extended Lotka - Volterra equation is considered, and a Bäcklund transformation and a nonlinear superposition formula given for it. Multiple soliton solutions are derived step by step as an application of the obtained results. Thus the N-soliton conjecture by Narita is confirmed.

The autocorrelation function of spectral determinants is proposed as a convenient tool for the characterization of spectral statistics in general, and for the study of the intimate link between quantum chaos and the random matrix theory, in particular. For this purpose, the correlation functions of spectral determinants are evaluated for various random matrix ensembles, and are compared with the corresponding semiclassical expressions. The method is demonstrated by applying it to the spectra of the quantized Sinai billiards in two and three dimensions.

The Maxwell energy-momentum tensor associated with the Liénard - Wiechert field of a point charge in arbitrary motion splits naturally into a bounded and a radiative part. It is known that the bounded part of the Maxwell tensor does not contribute to the energy-momentum balance between matter and field which is completely accounted for by its radiative part only. In this paper we show that the purely radiative part of the Maxwell tensor can be further decomposed as the sum of a term, again not contributing to the energy-momentum balance, plus a part which is completely responsible for it. Furthermore, we also manage to find that the full radiative part of the Maxwell tensor can be generated by a superpotential that has to be regarded as non-local since its definition involves integration over a finite section of the charge's world line.

A new method for the non-perturbative calculation of Green functions in quantum mechanics and quantum field theory is proposed. The method is based on an approximation of the Schwinger - Dyson equation for the generating functional by an exactly soluble equation in functional derivatives. Equations of the leading approximation and the first step are solved for the -model. At d = 1 (anharmonic oscillator) the ground-state energy is calculated. The renormalization programme is performed for the field theory at d = 2,3. At d = 4 the renormalization of the coupling involves a trivialization of the theory.

In this paper, we use a perturbative procedure due to Bender et al to solve analytically the Kadomtsev equation for a heavy atom in a very strong magnetic field.

Using a technique based on the Yangian Gelfand - Zetlin algebra and the associated Yangian Gelfand - Zetlin bases we construct an orthogonal basis of eigenvectors in the Calogero - Sutherland model with spin, and derive product-type formulae for norms of these eigenvectors.

It was shown in our previous paper that each equation in a soliton hierarchy can be factorized into two commuting x- and -finite-dimensional integrable Hamiltonian systems (FDIHSs). The separation variables for these FDIHSs are constructed by using their Lax representation. By means of the factorization and the separability of the FDIHSs we obtain the Jacobi inversion problem, which is solvable in terms of Riemann theta functions, for soliton equations. This provides a method analogous to the separation of variables for solving soliton equations.

We propose an exact method for locating the zeros of the Jost function for analytic potentials in the complex momentum plane. We further extend the method to the complex angular - momentum plane to provide the Regge trajectories. It is shown, by using several examples, that highly accurate results for extremely wide, as well as for extremely narrow, resonances with or without the presence of the Coulomb interaction can be obtained.

On page 7340, lines 9 and 10, the word `lower' should be replaced by `higher'.

Due to a production error, equation (10) of the published letter is incorrect. The correct equation is reproduced below: