Table of contents

The bond bending model is studied using the series expansion method on a honeycomb lattice. The elastic splay susceptibility chi _SR and the elastic compressional susceptibility chi _el are calculated up to 18th order. The elastic splay crossover exponent, zeta _SP, is found to be zeta _SP=1.31+or-0.02 which is very close to the conductivity exponent, zeta _Re, of the resistor network. From the scaling relation f_B=d nu + zeta _SP, we found that the bulk modulus exponent f_B=3.98+or-0.02 which is in excellent agreement with the result f_B=3.96+or-0.04, obtained by Zabolitzky et al. (1986) using a transfer matrix technique on the same lattice.

The magnetostatic energies and forces derived from axisymmetric models appropriate for magnetic force microscopy (MFM) of superconductors are examined. For models with a semi-infinite sample, closed form representations are obtained for arbitrary probe height. Specific boundary value problems considered are appropriate for a vortex penetrating a type-II superconductor, or for a magnetic monopole or dipole above or within a superconductor. Physically important limits such as complete flux expulsion become transparent with the new results. It is shown that previously employed approximations and numerical quadrature are unnecessary.

We present a detailed analysis for the Langevin dynamics of a spherical spin-glass model (the spherical Sherrington-Kirkpatrick model). The effects of initial conditions on the ultimate dynamical behaviour are closely examined. In addition, the effects of temperature variations in the model are studied. Somewhat surprisingly, this simple model captures some of the effects seen in laboratory spin-glasses.

We study numerically the decay of a Hamiltonian system whose transient bounded dynamics is fully chaotic but not necessarily fully hyperbolic when the phase space is initially populated by scattering experiments. We show that parabolic subsets included in the trapped orbits set are related to an algebraic tail corresponding to long times. The characteristic exponent of such a tail and that corresponding to the tail of the decay from the equilibrium population differ by one. This fact, already observed in other non-hyperbolic systems, is related to internal distributions that characterize the internal dynamics of the system.

We make a simple coupled map lattice model for simulating self-propelling particles. The interaction consists of three parts: viscous 'smearing' where the momenta are averaged over some neighbourhood, the collision where the momenta are conserved and, finally, the acceleration. As in externally forced fluids we find three regions: diffusion, convection and intermittency.

A model Fokker-Planck kinetic equation is postulated which describes coupled spin-velocity relaxation of test particles immersed in a buffer gas. The proposed model accounts for specific parts of the friction force ('Magnus'-type and 'sailing'-type) to which a spinning test particle is subjected. Special attention is paid to the 'Magnus phenomenon' in the case of spin-half particles. Some curious thermodynamic aspects are discussed.

We consider the ferromagnetic q-state Potts model on the d-dimensional lattice Z^d, d>or=2. Suppose that the Potts variables ( rho x, x in Z^d) are distributed in one of the q low-temperature phases. Suppose that n not=1, q divides q. Partitioning the single-site state space into n equal parts K₁, ..., K_n, we obtain a new random field sigma =( sigma _x, x in Z^d) by defining fuzzy variables sigma _x= alpha if rho x in K_alpha , alpha =1,...,n. We investigate the state induced on these fuzzy variables. First we look at the conditional distribution of rho x given all values sigma _y, y in Z^d. We find that below the coexistence point all versions of this conditional distribution are non-quasilocal on a set of configurations which carries positive measure. Then we look at the conditional distribution of sigma _x given all values sigma _y, y not=x. If the system is not at the coexistence point of a first-order phase transition, there exists a version of this conditional distribution that is almost surely quasilocal.

We investigate the critical relaxation of dilute Ising systems starting from a macroscopically prepared initial state with short-range correlations. Using the methods of renormalized field theory we calculate the exponent theta ' which describes the initial increase of the magnetization to second order in square root epsilon , where epsilon =4-d. Since computer simulations of the dilute Ising model have shown that a large part of the critical region is governed by crossover phenomena, we also discuss the influence of the slow crossover on the relaxation.

The dimer-dimer catalytic surface reaction of Albano's type ( 1/2 A₂+B₂ to B₂A) is studied with the aid of the cellular automaton model of Chopard and Droz (1988), as modified by Ziff, Fichthorn and Gulari (1991). A mean-field analysis is given, with results indicating that the reaction proceeds only at p=p_c (p is the partial pressure of species B₂ in the gaseous phase), when no recombination reaction of adsorbed B-species is considered, and a reaction window with B₂A production occurs for p>p_c, when the recombination mentioned above is taken into account. Our mean-field results are qualitatively in agreement with Albano's Monte Carlo simulations on a square lattice, and are valid in general, independent of the type of the lattice (square, hexagonal, etc.).

We analyse the chiral symmetry in the random +or-J XY model on a N*2 square lattice with periodic boundary conditions in the transverse direction. This 'tube' lattice may be seen as a two-dimensional lattice of which one dimension has been compactified. In the Villain formulation the discrete-valued chiralities or charges associated with the plaquettes of the lattice decouple from the continuous degrees of freedom. The difficulty of the problem lies in the fact that the chiralities interact through the long-range 'strong' one-dimensional Coulomb potential-which increases linearly with distance-as well as through an exponentially decaying 'weak' interaction. By comparing the ground-state energies for periodic, antiperiodic and reflecting boundary conditions in the longitudinal direction, we show that the chiralities and the XY spins have the same zero-T correlation length exponent, whose exact value nu _c=0.5564... we determine. The equality of these correlation lengths even in the presence of long-range chirality-chirality interactions lends support to the view that chiral-glass order cannot be sustained without simultaneous spin-glass order.

Fractional Brownian motion (FBM) is a generalization of the usual Brownian motion. A path integral representation that has recently been suggested for it is shown to be not for the FBM but for a different generalization of the Brownian motion. A new path integral representation is given and its measure has fractional derivatives of the path in it. The measure shows that the process is Gaussian but is, in general, non-Markovian, even though Brownian motion itself is Markovian. It is shown how the propagator for the motion of free FBM may be evaluated. This is somewhat more complex than for the usual path integrals, due to the occurrence of fractional derivatives. We also find the propagator in the presence of a linear absorption (potential), and for FBM on a ring.

The frequency-dependent toroid dipole polarizability gamma _l=1( omega ) which characterizes the linear response of a system to a conduction and/or displacement time-dependent external current is calculated exactly in the tridimensional charged oscillator model. Two frequencies of resonance are obtained. One of them ( omega _res=3 omega ₀, omega ₀, being the frequency of the oscillator) individualizes the toroid dipole polarizability since none of the other (usual) dipole and quadrupole electric and magnetic polarizabilities resonates at this frequency. Comparing the static result gamma _l=1^T( omega =0) with the static electric and magnetic quadrupole polarizabilities one can see that the toroid effects appear to be of the same order of magnitude as the magnetic ones and that they are very small with respect to the electric effects, but their relative importance increases proportional with h(cross) omega ₀/m₀c² (m₀ being the mass of the oscillator). For elementary particles (particularly at the subhadronic level) induced toroid moments might become predominant.

We discuss a very effective numerical method for simulating fibre-bundle models with equal load-sharing and with local load-sharing. Particular attention is paid to the case of the local load-sharing model, in which the critical load x_c is defined as the average load per fibre that causes the final complete failure. It is shown that x_c to 0 when the size of the system N to infinity . We also show analytically that the power law of the burst size distribution, D( Delta ) varies as Delta ^{- xi} , is approximately correct. The exponent xi in the local load-sharing case is not universal, since it depends on the strength distribution as well as on the size of the system.

The restricted SOS model of Andrews, Baxter and Forrester (1984) has been studied. The finite size corrections to the eigenvalue spectra of the transfer matrix of the model with a more general crossing parameter have been calculated. Therefore the conformal weights and the central charges of the non-unitary or unitary minimal conformal field have been extracted from the finite size corrections.

We investigate the possibility of constructing bicovariant differential calculi on quantum groups SO_q(N) and Sp_q(N) as a quantization of an underlying bicovariant bracket. We show that, in contrast to the GL(N) and SL(N) cases, neither of the possible graded SO and Sp bicovariant brackets (associated with a quasitriangular r-matrices) obey the Jacobi identity when the differential forms are Lie algebra-valued. The absence of a classical Poisson structure gives an indication that differential algebras describing bicovariant differential calculi on quantum orthogonal and symplectic groups are not of Poincare-Birkhoff-Witt type.

The time-dependent matrix Schrodinger equation 1/ic( delta Psi / delta t)=H(t) Psi describing two bands of an infinite number of equidistant states with different energy spacings omega _+or- in each band is studied. Both bands are linearly dependent on time t. The interaction upsilon =( square root ( omega - omega +)/ pi )tan pi s between the bands is considered to be equal for any pair of states from each band. Using the Fourier series transformation the instant eigenvalues E(t, s) are calculated which reveal the double periodicity in the energy-time plane. The corresponding eigenvalue surface in the (E, t, s)-space, apart from the triple periodicity, shows quite unexpected symmetry properties relative to the exchange of t and s, and relative to some inversions in the (E, t) plane. The latter one leads to a new equivalence between weak and strong coupling, a new kind of pseudocrossing and a new concept of antidiabatic states. The Fourier transformation reduces the problem to a 2*2 first-order differential operator. The diagonalization of H(r) for fixed t produces explicit expressions for the eigenvalues (adiabatic potential curves) and eigenstates (adiabatic basis). The time evolution operator is calculated both in the diabatic and adiabatic representations. The results are simplified for the special value of the interaction parameter.

The formula describing the isomorphism between the Moyal KP and the Sato KP hierarchies is given. We obtain the conserved densities of the dispersionless KP hierarchy by taking the limit k to 0 of the conserved quantities derived in the Sato approach.

We present a complete analysis of scale transformation of the Bloch functions and the Wannier functions in one-dimensional lattices, when a cell twice as large as the primitive cell is taken as the periodic unit. We obtain the Wannier functions for a free electron imposing an artificial periodicity and show that the Wannier functions satisfy the properties of the wavelets and wavelet packets of the multi-resolution analysis. We show that the coefficients appearing in the scale transformation of the Wannier functions for a free electron also serve as the expansion coefficients for the scale transformation of the Bloch functions and the Wannier functions in general one-dimensional lattices. Finally, we argue the importance of the translational symmetry based on the minimal primitive cell in determining the Wannier functions.

For the d-dimensional quantum mechanical harmonic oscillator with r commensurability relation between frequencies, d independent operators commutative with the oscillator Hamiltonian and all other commutative with Hamiltonian operators are constructed. These operators form a basis in the centre of the algebra of invariants (integrals of motion) for the quantum mechanical oscillator with commensurable frequencies. Not all of these d operators have classical analogues. A classical harmonic oscillator only has d-r commutative with all other invariants. These classical integrals first appeared in the Gustavson work on the Birkhoff normalization. Operators with such properties are of interest for the perturbation theory, since any of them may be (at least formally) continued to become the invariant of the perturbed Hamiltonian.

The general algebraic properties of the algebras of vector fields over the quantum linear groups GL_q(N) and SL_q(N) are studied. These quantum algebras appear to be quite similar to the classical matrix algebra. In particular, the quantum analogues of the characteristic polynomial and characteristic identity are obtained for them. The q-analogues of the Newton relations connecting two different generating sets of central elements of these algebras (the determinant-like and trace-like ones) are derived. This allows one to express the q-determinant of quantized vector fields in terms of their q-traces.

We prove that if Pi(x) and P_j(x) are two families of semi-classical orthogonal polynomials, all the linearization coefficients L_i,j,k occurring in the product of these two families satisfy a linear recurrence relation involving only the k index. This property also extends to the linearization coefficients arising from an arbitrary number of products of semi-classical orthogonal polynomials.

We study the theory of systems with constraints from the point of view of the formal theory of partial differential equations. For finite-dimensional systems we show that the Dirac algorithm completes the equations of motion to an involutive system. We discuss the implications of this identification for field theories and argue that the involution analysis is more general and flexible than the Dirac approach. We also derive intrinsic expressions for the number of degrees of freedom.