Table of contents

We examine the properties of a family of q-exponential functions, which depend on an extra parameter . These functions have a well defined meaning for both the 0 < |q| < 1 and |q| > 1 cases if only . It is shown that any two members of this family with different values of the parameter are related to each other by a Fourier - Gauss transformation.

We propose a bridge between the theory of exactly solvable models and the investigation of traffic flow. By choosing the activities in an appropriate way, the dimer configurations of the Kasteleyn model on a hexagonal lattice can be interpreted as spacetime trajectories of cars. This then allows for a calculation of the flow-density relationship (fundamental diagram). We further introduce a closely related cellular automaton model. This model can be viewed as a variant of the Nagel - Schreckenberg model in which the cars do not have a velocity memory. It is also exactly solvable and the fundamental diagram is calculated.

The least upper bound in Cirel'son's quantum generalization of the local-realistic Bell inequality is intermediate between the local-realistic and general probablistic limits. An alternative derivation of Cirel'son's inequality is presented which sheds light on the role played by complementarity in determining this bound and the fact that it falls short of the general probablistic limit.

The nonlinear version of the Fock approach, developed previously for systems with (extended) equidistant spectra (Eleonsky V M and Korolev V G 1995 J. Phys. A: Math. Gen. 28 4973), is also shown to describe systems with quadratic spectra. Some general properties of shift operators for such spectra are studied.

It is found, in particular, that families of shift operators related to the generators of the algebra su(1,1) as well as supersymmetry ladder operators arise naturally in that approach in a unified way.

In the theoretical biology framework one fundamental problem is the so-called error catastrophe in Darwinian evolution models. We re-examine Eigen's fundamental equations by mapping them into a polymer depinning transition problem in a `genotype' space represented by a unitary hypercubic lattice . The exact solution of the model shows that error catastrophe arises as a direct consequence of the equations involved and confirms some previous qualitative results.

The corrections to the Curie temperature of a ferromagnetic film consisting of N layers are calculated for for the model of D-component classical spin vectors in the limit , which is exactly soluble and close to the spherical model. The present approach accounts, however, for the magnetic anisotropy playing the crucial role in the crossover from three to two dimensions in magnetic films. In the spatially inhomogeneous case with free boundary conditions the model is non-equivalent to the standard spherical one and always leads to the diminishing of relative to the bulk.

We introduce an iteratively-defined operator sequence allowing the construction of new useful mathematical identities, involving the product of associated Laguerre polynomials. Possible physical applications as well as some methodological aspects of our approach are pointed out.

Braunstein and Caves have recently demonstrated that the Bures metric on the mixed quantum states is equivalent - up to a proportionality factor of four - to the statistical distinguishability or quantum Fisher information metric. The volume element of these metrics can then - adapting a fundamental Bayesian principle of Jeffreys to the quantum context - serve as a reparametrization-invariant prior measure over the quantum states. The implications of this line of reasoning for the two-level systems, in general, and an embedding of them into a certain set of three-level systems are investigated.

We discuss the predictability of a system that drives a chaotic system with a positive Lyapunov exponent. In the absence of feedback, the driver is regular and fully predictable. With a small feedback of strength , the state of the driver can be predicted up to a time diverging with a power of , although the total system is strongly chaotic. The exponential amplification of the uncertainty on the initial conditions of the driver coexists with very long predictability times as illustrated in a model of coupled maps and of three point vortices in a disc.

Optimal capacities of perceptrons with graded input - output relations are studied within the first-step replica-symmetry-breaking Gardner approach. Input-data errors and a limited output precision are allowed. In particular, the role of non-monotonicity in the input - output relations on the breaking and on the overall performance is determined.

The disordered structure formed by particles that are sequentially deposited onto a line is analysed. Analytical expressions of the pair correlation function, the structure factor, and the fluctuations of the number of particles are obtained for the generalized ballistic deposition model. For all integer values of the distance between particles, the pair correlation function has a discontinuity and a delta-singularity, which both result from cluster formation. A comparison is made with equilibrium configurations of sticky hard rods. It is shown that, in contrast to equilibrium systems (but as in the random sequential addition model), the correlations between particles decay faster than exponentially at large distances.

We propose a road traffic cellular automata model suitable for an urban environment. North, east, south and west car displacements are possible and road crossings are naturally implemented as rotary junctions. We consider the traffic in a Manhattan-like city and study the flow diagram and the car density profile along road segments. We observe that the length of the car queues obeys a complex dynamics and is not uniform across the network. The street length between two junctions and the turning strategies at rotaries are relevant parameters of the model. Our results are also confirmed by fully continuous traffic simulations.

We introduce a model for the fragmentation of porous random solids under the action of an external agent. In our model, the solid is represented by a bond percolation cluster on the square lattice and bonds are removed only at the external perimeter (or `hull') of the cluster. This model is shown to be related to the self-avoiding walk on the Manhattan lattice and to the disconnection events at a diffusion front. These correspondences are used to predict the leading and the first correction-to-scaling exponents for several quantities defined for hull fragmentation. Our numerical results support these predictions. In addition, the algorithm used to construct the perimeters reveals itself to be a very efficient tool for detecting subtle correlations in the pseudo-random number generator used. We present a quantitative test of two generators which supports recent results reported in more systematic studies.

The temperature-induced second-order phase transition from Bloch to linear (Ising-like) domain walls in uniaxial ferromagnets is investigated for the model of D-component classical spin vectors in the limit . This exactly solvable model is equivalent to the standard spherical model in the homogeneous case, but deviates from it and is free from unphysical behaviour in a general inhomogeneous situation. It is shown that the thermal fluctuations of the transverse magnetization in the wall (the Bloch-wall order parameter) result in the diminishing of the wall transition temperature in comparison to its mean-field value, thus favouring the existence of linear walls. For finite values of an additional anisotropy in the basis plane x,y is required; in purely uniaxial ferromagnets a domain wall behaves like a two-dimensional system with a continuous spin symmetry and does not order into the Bloch one.

Let be an arbitrary given composite-system state (statistical operator). It is shown that same-subsystem events imply each other state-dependently according to if and only if they act equally in the range of the corresponding subsystem state (reduced statistical operator). Opposite-subsystem events imply each other in the same way if and only if they are twin events, i.e. . If is a pure state, it is shown that the anti-unitary correlation operator also plays a decisive role in the latter implication.

Deposition of monodisperse hard spheres onto an inclined surface has been investigated. The process of random sequential deposition of particles in the downwards vertical direction was simulated on a computer. It was found that the solids volume fraction of the resulting aggregate is an increasing function of the angle of inclination of the surface to the horizontal.

We present results for the density of states and the statistics of the energy levels in a random tight binding matrix ensemble defined on a disordered two-dimensional Sierpinski gasket. In the absence of disorder the nearest level spacing distribution function P(S) is shown to follow the inverse power law , which defines the fractal dimension of the corresponding spectrum. In the random case P(S) approaches, instead, the Poisson law , which is consistent with localization of the corresponding eigenstates. In the presence of a random magnetic flux our results also scale towards the Poisson statistics.

Using the functional integral bosonization technique, the spinless Tomonaga - Luttinger model with impurity forward scattering is solved exactly. Explicit analytical results are given for the one- and two-particle spectral functions in terms of a convolution integral representation. Sharp structures in the frequency spectra of an ordered system are smoothed out by the disorder. The existence of a charge - density instability is predicted if the critical exponent is less than -0.5. There is no influence of the disorder on the superconducting spectral function .

We give the exact solution to the problem of a random walk on the Bethe lattice through a mapping on an asymmetric random walk on the half-line. We also study the continuous limit of this model, and discuss in detail the relation between the random walk on the Bethe lattice and Brownian motion on a space of constant negative curvature.

We investigate the limit behaviour of random walks on some non-commutative discrete groups related to knot theory. Namely, we study the connection between the limit behaviour of the Lyapunov exponent of products of non-commutative random matrices - generators of the braid group - and the asymptotics of powers of the algebraic invariants of randomly generated knots. We turn the simplest problems of knot statistics into the context of random walks on hyperbolic groups. We also consider the limit distribution of Brownian bridges on so-called locally non-commutative groups.

We study the dynamical critical behaviour of a double chain of spins where we take into account short-range four-spin interactions. This system is exactly soluble only in the thermodynamical equilibrium. We use the initial response rate of the order parameter to establish a lower bound to the dynamical critical exponent z. We show that, for the one- and two-spin-flip Glauber transition rates, the exponent z depends on the microscopic details of the Hamiltonian, that is, the dynamical critical exponent appears non-universal. This type of non-universal behaviour has already been seen in the one-dimensional Ising model with non-uniform exchange interactions.

A hierarchical froth model of the interface of a random q-state Potts ferromagnet in 2D is studied by recursive methods. A fraction p of the nearest-neighbour bonds is made inaccessible to domain walls by infinitely strong ferromagnetic couplings. Energetic and geometric scaling properties of the interface are controlled by zero-temperature fixed distributions. For , the directed percolation threshold, the interface behaves as for p = 0, and scaling supports random Ising (q = 2) critical behaviour for all q's. At three regimes are obtained for different rates of ferro versus antiferromagnetic couplings. With rates above a threshold value the interface is linear (fractal dimension ) and its energy fluctuations, scale with length as , with . When the threshold is reached the interface branches at all scales and is fractal with . Thus, at , dilution modifies both low-temperature interfacial properties and critical scaling. Below threshold the interface becomes a probe of the backbone geometry (; = backbone fractal dimension), which even controls energy fluctuations . Numerical determinations of directed percolation exponents on diamond hierarchical lattice are also presented.

Self-interacting walks and polygons on the simple cubic lattice undergo a collapse transition at the -point. We consider self-avoiding walks and polygons with an additional interaction between pairs of vertices which are unit distance apart but not joined by an edge of the walk or polygon. We prove that these walks and polygons have the same limiting free energy if the interactions between nearest-neighbour vertices are repulsive. The attractive interaction regime is investigated using Monte Carlo methods, and we find evidence that the limiting free energies are also equal here. In particular, this means that these models have the same -point, in the asymptotic limit. The dimensions and shapes of walks and polygons are also examined as a function of the interaction strength.

We discuss some examples of measures on lattice systems, which lack the property of being a Gibbs measure in a rather strong sense.

Quantization on phase spaces of general geometry devoid of any special symmetry properties is discussed on the basis of phase spaces endowed with a symplectic structure, a Riemannian geometry, and a structure. Using techniques from differential geometry, and especially exploiting the Dirac operator, we are able to offer a fully geometric quantization procedure for a wide class of symmetry free phase spaces. Our procedure leads to the conventional results in cases where the phase space is a symmetric space for which alternative quantization techniques suffice.

We demonstrate that the ground energy of a two-level system coupled with a bosonic environment can be labelled with a quantum number related to the total excitation number operator and independent of the coupling strength. Our approach is exact and is based on operator methods combining symmetry considerations with general properties of the lowest energy state. The relevance of our result in connection with the nature of the transition from the weak to strong coupling regime is briefly discussed.

A procedure is presented for finding simple approximations of the discrete eigenvalues of the Zakharov - Shabat scattering problem corresponding to the nonlinear Schrödinger equation. The approximation is in the form of an interpolation formula which combines results for small eigenvalues, obtained by a direct variational approach, and for large eigenvalues, obtained by the Bohr - Sommerfeld quantization rule.

We consider the problem of finding the ground state of a model type-II superconductor on the two-dimensional surface of a sphere, penetrated by N vortices. Numerical work shows the ground states to consist of a triangular network of the vortices with twelve five-coordinated centres. Values of N are found with particularly low-energy ground states, due to structures of high symmetry. The large-N limit is treated within elasticity theory to compare with the triangular vortex lattice that forms the ground state on an infinite flat plane. Together with numerical work this demonstrates that the thermodynamic limit of the spherical system remains different from the flat plane due to the presence of twelve disclination defects.

Wavelet packet analysis was used to measure the global scaling behaviour of homogeneous fractal signals from the slope of decay for discrete wavelet coefficients belonging to the adapted wavelet best basis. A new scaling function for the size distribution correlation between wavelet coefficient energy magnitude and position in a sorted vector listing is described in terms of a power law to estimate the Hurst exponent. Profile irregularity and long-range correlations in self-affine systems can be identified and indexed with the Hurst exponent, and synthetic one-dimensional fractional Brownian motion (fBm) type profiles are used to illustrate and test the proposed wavelet packet expansion. We also demonstrate an initial application to a biological problem concerning the spatial distribution of local enzyme concentration in fungal colonies which can be modelled as a self-affine trace or an `enzyme walk'. The robustness of the wavelet approach applied to this stochastic system is presented, and comparison is made between the wavelet packet method and the root-mean-square roughness and second-moment approaches for both examples. The wavelet packet method to estimate the global Hurst exponent appears to have similar accuracy compared with other methods, but its main advantage is the extensive choice of available analysing wavelet filter functions for characterizing periodic and oscillatory signals.

A compound tunnelling mechanism from one integrable region to another mediated by a delocalized state in an intermediate chaotic region of phase space was recently introduced to explain peculiar features of tunnelling in certain two-dimensional systems. This mechanism is known as chaos-assisted tunnelling. We study its consequences for the distribution of the level splittings and obtain a general analytical form for this distribution under the assumption that chaos assisted tunnelling is the only operative mechanism. We have checked that the analytical form we obtain agrees with splitting distributions calculated numerically for a model system in which chaos-assisted tunnelling is known to be the dominant mechanism. The distribution depends on two parameters. The first gives the scale of the splittings and is related to the magnitude of the classically forbidden processes, the second gives a measure of the efficiency of possible barriers to classical transport which may exist in the chaotic region. If these are weak, this latter parameter is irrelevant; otherwise it sets an energy scale at which the splitting distribution crosses over from one type of behaviour to another. The detailed form of the crossover is also obtained and found to be in good agreement with numerical results for models for chaos-assisted tunnelling.

We consider the problem of quantum impurity scattering for a particle on a lattice via a non-perturbative approach. We calculate the weak-coupling limit in the case of a directed lattice (tree) and we show what problems arise in the more general case due to recurrence effects. Our methods relate the problem to that of a random walk in a random environment.

We derive closed analytical expressions for the complex Berry phase of an open quantum system in a state which is a superposition of resonant states and evolves irreversibly due to the spontaneous decay of the metastable states. The codimension of an accidental degeneracy of resonances and the geometry of the energy hypersurfaces close to a crossing of resonances differ significantly from those of bound states. We discuss some of the consequences of these differences for the geometric phase factors. For example, instead of a diabolical point singularity there is a continuous closed line of singularities formally equivalent to a continuous distribution of `magnetic' charge on a diabolical circle, there are different classes of topologically inequivalent non-trivial closed paths in parameter space, the topological invariant associated with the sum of the geometric phases, dilations of the wavefunction due to the imaginary part of the Berry phase and others.

The interior - exterior duality provides a means to extract spectral information (for the interior problem) from the scattering matrix (which is relevant to the exterior problem). We study the smooth spectral counting function for the interior, and compare it to the smooth total phase shift in the exterior. To leading order in the semiclassical approximation these functions are known to coincide. Using various techniques, we study the higher-order corrections of the two functions and discuss the difference between them.

The static dielectric function of electronic liquids is studied in a wide range of thermodynamic parameters. The local-field correction to the RPA permeability is modelled to satisfy the compressibility sum rule and the short-wavelength exact relation to the zero-separation value of the radial distribution function. The latter is determined by a self-consistency procedure and is shown to verify all known asymptotic conditions.

In this paper we show that a damped harmonic oscillator (the simplest dissipative physical system) is canonically equivalent to a generalized harmonic oscillator (a conservative system) even for time-dependent parameters. As a consequence, the Hannay's angles and the adiabatic invariants of the two systems appear to be the same, modulo this equivalence. This raises the question of whether this analogy can be extended to other nonlinear dissipative systems.

Exact and approximate nonlinear self-localized modes are shown to exist in a one-dimensional chain of interacting Frenkel excitons due to exciton - exciton static attraction. Two different modes are found and their frequencies are below the exciton frequency band. These results suggest a possible new mechanism for localization of the energy of the amide - I excitons through the exciton - exciton interaction in protein molecules.

Motivated by a relation of the 1-constrained Kadomtsev - Petviashvili (KP) hierarchy with the 2-component KP hierarchy, the tau-functions of the vector k-constrained KP hierarchy are constructed by using an analogue of the Baker - Akhiezer (m + 1)-point function. These tau-functions are expressed in terms of Wronskian-type determinants.