Table of contents

We present a theory of particles, obeying intermediate statistics ('anyons'), interpolating between bosons and fermions, based on the principle of detailed balance. It is demonstrated that the scattering probabilities of identical particles can be expressed in terms of the basic numbers, which arise naturally and logically in this theory. A transcendental equation determining the distribution function of anyons is obtained in terms of the statistics parameter, whose limiting values 0 and 1 correspond to bosons and fermions respectively. The distribution function is determined as a power series involving the Boltzmann factor and the statistics parameter and we also express the distribution function as an infinite continued fraction. The last form enables one to develop approximate forms for the distribution function, with the first approximant agreeing with our earlier investigation.

In this paper we show how the perturbative procedure known as stochastic limit may be useful in the analysis of the open BCS model discussed by Buffet and Martin as a spin system interacting with a fermionic reservoir. In particular we show how the same values of the critical temperature and of the order parameters can be found with a significantly simpler approach.

We focus on transport parameters in heterogeneous media with a flow modelled by an ensemble of periodic and Gaussian random fields. The parameters are determined by ensemble averages. We study to what extent these averages represent the behaviour in a single realization. We calculate the centre-of-mass velocity and the dispersion coefficient using approximations based on a perturbative expansion for the transport equation, and on the iterative solution of the Langevin equation. Compared with simulations, the perturbation theory reproduces the numerical results only poorly, whereas the iterative solution yields good results. Using these approximations, we investigate the self-averaging properties. The ensemble average of the velocity characterizes the behaviour of a realization for large times in both ensembles. The dispersion coefficient is not self-averaging in the ensemble of periodic fields. For the Gaussian ensemble the asymptotic dispersion coefficient is self-averaging. For finite times, however, the fluctuations are so large that the average does not represent the behaviour in a single realization.

We study the steady-state fluctuations of an Edwards–Wilkinson type surface with the substrate taken to be a sphere. We show that the height fluctuations on circles at a given latitude have the effective action of a perfect Gaussian 1/f noise, just as in the case of fixed radius circles on an infinite planar substrate. The effective surface tension, which is the overall coefficient of the action, does not depend on the latitude angle of the circles.

We investigate a generalization of the three-dimensional spring-pendulum system. The problem depends on two real parameters (k, a), where k is the Young modulus of the spring and a describes the nonlinearity of elastic forces. We show that this system is not integrable when k ≠ −a. We carefully investigated the case k = −a when the necessary condition for integrability given by the Morales-Ruiz–Ramis theory is satisfied. We discuss an application of the higher order variational equations for proving the non-integrability in this case.

The spatial rock-scissors-paper game (or cyclic Lotka–Volterra system) is extended to study how the spatiotemporal patterns are affected by the rewired host lattice providing uniform number of neighbours (degree) at each site. On the square lattice this system exhibits a self-organizing pattern with equal concentration of the competing strategies (species). If the quenched background is constructed by substituting random links for the nearest-neighbour bonds of a square lattice then a limit cycle occurs when the portion of random links exceeds a threshold value. This transition can also be observed if the standard link is replaced temporarily by a random one with a probability P at each step of iteration. Above a second threshold value of P the amplitude of global oscillation increases with time and finally the system reaches one of the homogeneous (absorbing) states. In this case the results of Monte Carlo simulations are compared with the predictions of the dynamical cluster technique evaluating all the configuration probabilities on one-, two-, four- and six-site clusters.

The quantum description of third harmonic generation can be formulated as an eigenvalue problem for a third-order linear differential equation. We perform a semiclassical study of this third-order equation, generalizing the familiar JWKB theory for the second-order Schrödinger equation, and deriving explicit (albeit approximate) formulas for the eigenvalues within this semiclassical context. A central role in this analysis is played by a nonlinear complex canonical transformation which permits a complete description of the classical motion (generated by a complex polynomial Hamiltonian function) in the complexified position and momentum planes.

The aim of this paper is to study quantum and affine q-Krawtchouk polynomials by means of operators of irreducible representations of the quantum algebra U_q(su₂). We diagonalize a certain operator I in such a representation and show that elements of the transition matrix from the initial (canonical) basis to the basis consisting of eigenfunctions of the operator I are expressed in terms of quantum q-Krawtchouk polynomials. Then we find an explicit form of the operator in the basis of the eigenfunctions of I, in which it has the form of a Jacobi matrix. Normalizing this basis and using the operator , we thus derive the orthogonality relations for quantum and affine q-Krawtchouk polynomials. We show that affine q-Krawtchouk polynomials are dual to quantum q-Krawtchouk polynomials. A biorthogonal system of functions (with respect to the scalar product in the representation space) is also derived.

The evaluation of thermonuclear reaction rates requires the calculation of several thermonuclear functions. These functions can be written as the Laplace transform of locally integrable functions which have an asymptotic expansion in negative rational powers of their variable. In this paper we obtain asymptotic expansions of the Laplace transform of these kinds of functions for small values of the parameter of the transformation. Error bounds are obtained at any order of the approximation for a large family of Laplace transforms which include thermonuclear functions. Then we apply this asymptotic theory to the calculation of convergent expansions of four thermonuclear functions in powers of the dimensionless Sommerfeld parameter. Some of these expansions also involve logarithmic terms in the dimensionless Sommerfeld parameter. Accurate error bounds are given at any order of the approximation.

For any one-dimensional tiling, we discuss finite-dimensional standard modules for the associated tiling bialgebra. We will note that such modules are completely reducible, and we will parametrize finite-dimensional irreducible ones, using the set of all patches. Furthermore, we will discuss the associated completed groups and Iwasawa-type decompositions. We also characterize for one-dimensional tilings to be locally nondistinguishable by the associated tiling bialgebra structures.

We study Dirac structures for generalized Courant algebroids, which are doubles of generalized Lie bialgebroids. The cases investigated include graphs of bivector fields and characteristic pairs of some sub-bundles.

Elementary Bäcklund transformations (BTs) are described for a discretization of the Zakharov–Shabat eigenvalue problem (a special case of the Ablowitz–Ladik eigenvalue problem). Elementary BTs allow the process of adding bound states to a system (i.e., the add-one-soliton BT) to be 'factorized' to solving two simpler sub-problems. They are used to determine the effect on the scattering data when bound states are added. They are shown to provide a method of calculating discrete solitons—this is achieved by constructing a lattice of intermediate potentials, with the parameters used in the calculation of the lattice simply related to the soliton scattering data. When the potentials, S_n, T_n, in the system are related by S_n = −T_n, they enable simple derivations to be obtained of the add-one-soliton BT and the nonlinear superposition formula.

Some new exact solutions of the (2 + 1)-dimensional asymmetric Nizhnik–Novikov–Veselov equation are presented using the bilinear method. The solutions to describe the interactions between two dromions, between a line soliton and a y-periodic soliton, and between two y-periodic solitons are included in our results. The detailed behaviour of the interactions is illustrated both analytically and graphically. Our analysis shows that the forms of soliton solutions and interacting properties between two solitons are related to the forms of the parameters and interaction constants.

We present a geometric description of the QRT map (which is an integrable mapping introduced by Quispel, Roberts and Thompson) in terms of the addition formula of a rational elliptic surface. By this formulation, we classify all the cases when the QRT map is periodic; and show that its period is 2, 3, 4, 5 or 6. A generalization of the QRT map which acts birationally on a pencil of K3 surfaces, or Calabi–Yau manifolds, is also presented.

We introduce a large class of holomorphic quantum states by choosing their normalization functions to be given by generalized hypergeometric functions. We call them generalized hypergeometric states in general, and generalized hypergeometric coherent states in particular, if they allow a resolution of unity. Depending on the domain of convergence of the generalized hypergeometric functions, we distinguish generalized hypergeometric states on the plane, the open unit disc and the unit circle. All states are eigenstates of suitably defined lowering operators. We then study their photon number statistics and phase properties as revealed by the Husimi and Pegg–Barnett phase distributions. On the basis of the generalized hypergeometric coherent states we introduce new analytic representations of arbitrary quantum states in Bargmann and Hardy spaces as well as generalized hypergeometric Husimi distributions and corresponding phase distributions.

Isospectral domains are non-isometric regions of space for which the spectra of the Laplace–Beltrami operator coincide. In the two-dimensional Euclidean space, instances of such domains have been given. It has been proved for these examples that the length spectrum, that is the set of the lengths of all periodic trajectories, coincides as well. However there is no one-to-one correspondence between the diffractive trajectories. It will be shown here how the diffractive contributions to the Green functions match nevertheless in a 'one-to-three' correspondence.

Uncertainty relations for particle motion in curved spaces are discussed. The relations are shown to be topologically invariant. A new coordinate system on a sphere appropriate to the problem is proposed. The case of a sphere is considered in detail. The investigation can be of interest for string and brane theory, solid state physics (quantum wires) and quantum optics.

In this paper we discuss the effect of the unpolarized state in the spin-correlation measurement of the ¹S₀ two-proton state produced in the ¹²C(d, ²He) reaction at the Kernfysisch Versneller Instituut (KVI), Groningen. We show that in the presence of the unpolarized state the maximal violation of the Clauser, Horne, Shimony and Holth (CHSH)–Bell inequality is lower than the classical limit if the purity of the state is less than ∼70%. In particular, for the KVI experiment the violation of the CHSH–Bell inequality should be corrected by a factor of ∼10% from the pure ¹S₀ state.

We depict the quantum collision between a stable helium nanodrop and an infinitely hard wall in one dimension. The scattering outcome is compared to the same event omitting the quantum pressure. Only the quantum process reflects the effect of diffraction of wave packets in space and time.

We define the oscillator and Coulomb systems on four-dimensional spaces with U(2)-invariant Kähler metric and perform their Hamiltonian reduction to the three-dimensional oscillator and Coulomb systems specified by the presence of Dirac monopoles. We find the Kähler spaces with conic singularity, where the oscillator and Coulomb systems on three-dimensional sphere and two-sheet hyperboloid originate. Then we construct the superintegrable oscillator system on three-dimensional sphere and hyperboloid, coupled to a monopole, and find their four-dimensional origins. In the latter case the metric of configuration space is a non-Kähler one. Finally, we extend these results to the family of Kähler spaces with conic singularities.

We introduce the concept of general gauge theory which includes Yang–Mills models. We use the framework of the causal approach and show that the anomalies can appear only in the vacuum sector of the identities obtained from the gauge invariance condition by applying derivatives with respect to the basic fields. For the Yang–Mills model we provide these identities in the lowest orders of the perturbation theory and prove that they are valid. The investigation of higher orders of the perturbation theory is still an open problem.

In this work we introduce a generalization of the Jauch and Rohrlich quantum Stokes operators when the arrival direction from the source is unknown a priori. We define the generalized Stokes operators as the Jordan–Schwinger map of a triplet of harmonic oscillators with the Gell–Mann and Ne'eman matrices of the SU(3) symmetry group. We show that the elements of the Jordan–Schwinger map are the constants of motion of the three-dimensional isotropic harmonic oscillator. Also, we show that the generalized Stokes operators together with the Gell–Mann and Ne'eman matrices may be used to expand the polarization matrix. By taking the expectation value of the Stokes operators in a three-mode coherent state of the electromagnetic field, we obtain the corresponding generalized classical Stokes parameters. Finally, by means of the constants of motion of the classical 3D isotropic harmonic oscillator we describe the geometrical properties of the polarization ellipse.