Table of contents

We show how the maximum entropy formalism can be applied in nonlinear chaotic systems. In particular we show how one can use a few moments of the time evolution of a dynamical variable to infer the probability density and hence the Lyapunov exponent.

A method to reveal and estimate the fractal scaling properties of positive and negative increments in one-valued functions that describe natural processes is proposed. Structural functions, which were introduced by Kolmogorov (1941) for analysing the scaling properties of small-scale turbulence, provide the basis of the method. Examples are given to illustrate the application of the proposed method for analysing the trace of one of the coordinates of Brownian motion, simulated asymmetric wave forms, internal waves, sand waves in one-directional streams and river turbulence.

We investigate the effect of fluctuations in the effective Hamiltonian that describes the microphase separation in random copolymers. This Hamiltonian is previously studied on the level of mean-field theory where a phase transition to a phase with periodic microdomain structure was predicted. It is shown that the one-loop treatment of fluctuations is exact in the thermodynamic limit and that the phase predicted by the mean field is unstable within the framework of the studied Hamiltonian.

A huge number of computers connected together form an international network (Internet). As a result of their communication, trains of data packages travel through this network; due to the limited transfer rate and the behaviour of data source agents, the traffic fluctuates. Our measurements show that the power spectrum of the round-trip time between two points on Internet is 1/f-like over a broad range. Models of collective phenomena, such as highway traffic models could be appropriate for describing this behaviour.

We present numerical and analytical results for a special kind of one-dimensional probabilistic cellular automaton, the so-called Domany-Kinzel automaton. It is shown that the phase boundary separating the active and the recently found chaotic phase exhibits re-entrant behaviour. Furthermore exact results for the p₂=0 line are discussed.

Energy in a static magnetic field can be converted to other forms such as mechanical energy and heat. This is done by perturbing the magnetostatic energy by an electromagnetic wave-generating what has been described earlier as a 'companion wave'. When energy is drawn from this latter wave, the magnetic field decays.

A mod jm) spin state in an adiabatically-cycled magnetic field acquires a geometric phase of m times the solid angle described by B, so that for m=0 states the geometric phase vanishes. However, if B is not cycled, but is made to reverse direction, an m=0 state returns to itself and in so doing acquires a geometric phase factor of (-1)^j. This phase is of a topological character; parameter space is the real projective plane, in which the phase distinguishes trivial from non-trivial cycles.

Complex potentials that absorb the incoming wave in a finite distance without reflection or transmission are found by a simple inversion technique, both for stationary and wavepacket scattering in one dimension.

Let group generators having finite-dimensional representation be realized as Hermitian linear differential operators without inhomogeneous terms as takes place, for example, for the SO(n) group. Then corresponding group Hamiltonians containing terms linear in generators (along with quadratic ones) give rise to quasi-exactly solvable models with a magnetic field in a curved space. In particular, for the SO(4) group Hamiltonian with isotropic quadratic part, the manifold within which a quantum particle moves has the geometry of the Einstein universe.

We consider the quantum model of a two-level system linearly coupled to M harmonic oscillators. Using symmetry arguments we show the existence of new constants of motion for this physical system. Starting with this knowledge, we develop a systematic treatment by which an effective canonical decoupling of different degrees of freedom is obtained.

The spin model of Coolen et al. (1993), involving slow dynamical laws for the couplings linking fast spins, is considered as a spin-glass model having explicit quenched disorder. The thermodynamic behaviour predicted is reminiscent of a Sherrington-Kirkpatrick spin-glass with one-step replica-symmetry breaking-generalizing the parallels of the simplest form of this model, which was entirely free of frozen disorder. However, even though the slow evolution of the couplings allows some balancing between achieving unfrustrated spin configurations and adopting those favoured by the quenched disorder, it is seen that this is not generally sufficient to avoid replica-symmetry breaking throughout the frozen phases. Moreover, there are seen to be two distinct types of spin-glass phase, each of which has both ergodic (replica-symmetric) and ergodicity-broken regimes.

We study the effect of an external magnetic field on magnetic long-range order (MLRO) in the global ground states of the quantum XY model and the isotropic Heisenberg model on the simple cubic lattices. We shall rigorously prove that, while an external magnetic field (staggered in the antiferromagnetic case) which favours MLRO in a specific spin direction, say the x-direction, is turned on, it completely suppresses the MLRO in the perpendicular spin directions in the global ground states of these models.

The Kardar-Parisi-Zhang (KPZ) equation describing kinetic roughening is solved numerically for (3+1)-dimensional systems in the strong-coupling phase. By massive use of supercomputing tools we calculate for the first time the exponent beta (=0.181+or-0.007) with an accuracy comparable with that of lattice growth model simulations. A possible source of errors in parallel Monte Carlo calculations is pointed out.

We use a transfer matrix with suitably defined vertex weights to algebraically enumerate n-step self-avoiding walks confined to cross an L*M rectangle on the square lattice. We construct the exact generating functions for self-avoiding walks from the south-west to south-east corners for L=1, 2, 3, 4, 5 and infinite height M corresponding to a half-strip. We also consider the number of n-sided polygons rooted to the south-west corner of the half-strip and give a formulation to treat self-avoiding walks across the full strip. In each case, the exact generating functions are ratios of polynomials in the step fugacity. We investigate the singularity structure of the generating functions along with the finite-size scaling in M of the singularity in the analogue of the heat capacity. We find the critical exponents gamma =1 and gamma =2 for the half- and open-strip, along with nu = 1/2 . These results are indicative of the one-dimensional or Gaussian nature of self-avoiding walks in infinitely long, but finitely wide strips.

We present improved estimates of critical exponents for 3D polymers near a plane adsorbing surface. The improvements were possible by simulating rather long chains (up to N=2000) by means of a new algorithm. Apart from minor but significant adjustments in the location of the tricritical point and in the various gamma -exponents, our main result is that the crossover exponent for critical adsorption is compatible with phi = 1/2 . This is much lower than previous estimates and suggests that this exponent might be superuniversal.

Annealed random-bond Ising models with frustration for the square lattice in two dimensions are considered. Three special models are solved exactly by mapping them to their dual models. Thermodynamic quantities are calculated analytically for the three models.

The effect of randomly distributed synaptic background activity on the states of self-sustained firing in a model neural network with shunting is investigated. Using mean field theory, the steady state of the network is expressed in terms of an ensemble-averaged single-neuron Green's function. This Green's function is shown to satisfy a matrix equation identical in form to that found in the tight-binding-alloy model of excitations on a one-dimensional disordered lattice. The ensemble averaging is then performed using a coherent potential approximation thus allowing the steady-state firing rate of the network to be determined. The firing rate is found to decrease as the mean level of background activity across the network is increased; a uniform background (zero variance) leads to a greater reduction than a randomly distributed one (non-zero variance).

The time evolution of the distance between two random initial configurations subjected to the same thermal noise is used to study dynamical phase transitions in attractor neural networks trained by the Hebb rule. Numerical results are given for fully connected architectures, whereas, in the dilute case, both analytical and numerical outcomes are provided and a good agreement is shown to exist between the two sets of results.

We describe all contractions of the affine Kac-Moody algebra A₁⁽¹) which preserve various gradings by the cyclic groups Z₂ and Z₃ as well as a grading by Z₂*Z₃, together with the contractions of simultaneously graded representations of A₁⁽¹) and their tensor products. An application is given to the branching rules for A₁⁽¹) contains/implies A₁⁽¹) for a number of distinct embeddings each associated with a particular grading.

We find all (four) first integrals for two-dimensional Ermakov systems.

We apply Ramanujan's sum in q-analysis to evaluate the one-point functions of the XXZ model.

The supersymmetric eigenvalue model has been proposed as the analogue for two-dimensional supergravity of the matrix model. The double-scaling limit exhibits an integrable fermionic hierarchy which is related to the Korteweg-de Vries hierarchy. In this paper, a discrete analogue of this integrable structure, related to the Toda lattice, is found in the supersymmetric eigenvalue model.

We develop the operator formalism to show how systematically the fractional Fourier transformation of a wavefunction, recently introduced by Namias (1980), can be derived from the rotation of the corresponding Wigner distribution function in phase space. In this formalism, the phase factor obtained by McBride and Kerr (1987) is seen to come from the caustics of the harmonic oscillator Green function. Then the idea is generalized to the case of an arbitrary area-preserving linear transformation in phase space, and a concept of the special affine Fourier transformation (SAFT) is introduced. An explicit form of the integral representation of the SAFT is given, and some simple examples are presented.

The bremsstrahlung of a system of classically fast charged particles which do not interact with each other but which do undergo multiple elastic scattering by randomly positioned atoms of a medium is studied. We derived the spectrum of the bremsstrahlung of such particles through a systematic kinetic analysis of the radiation process in the medium. It is shown that the spectral distribution of the emission energy of the bremsstrahlung depends significantly on both the characteristics of the scattering of the particles in the medium and the parameters characterizing the initial set of the particles.

It is proven how the post-gelation behaviour originally suggested by Flory(1993) can be obtained as a result of a limiting process, passing from a finite to an infinite system. In a previous paper by the authors it was shown how the post-gelation behaviour first suggested by Stockmayer(1943) can be obtained by passing to the limit of an infinite system in a different way. It is thus demonstrated that different post-gelation solutions of Smoluchowski's coagulation equation can be obtained by different limiting processes.

The expansion of the Laughlin ansatz for describing the ground-state wavefunction for the fractional quantum Hall effect as a linear combination of Slater determinantal wavefunctions for N particles is discussed in terms of the corresponding expansion of even powers of the Vandermonde alternant into Schur functions. Two new algorithms for computing the coefficients of the complete expansion are given. They appear to be substantially more efficient than other methods and avoid any use of symmetric group characters. A number of examples are given and the results obtained for N=7, 8 and 9 reviewed. The separate calculation of individual coefficients is also discussed.

Quantum mechanics is enlarged into a contextual subquantum model, with the purpose of describing causally individual quantum systems. This model is based on the spectra of the quantum-mechanical observables. A C*-algebraic generalization of this model is presented. Various physically interesting aspects of the constructed subquantum model are discussed.

Methods of constructing random matrices typical of circular unitary and circular orthogonal ensembles are presented. We generate numerically random unitary matrices and show that the statistical properties of their spectra (level-spacing distribution, number variance) and eigenvectors (entropy, participation ratio, eigenvector statistics) confer to the predictions of the random-matrix theory, for both CUE and COE.

We discuss quantum dynamics of a particle in a time and space periodic field. We consider a soluble model of a particle in a plane wave, and reduce the quantum problem to a random Riccati equation. We show that solutions of the Riccati equation experience an abrupt change from a periodic to non-periodic behaviour under a variation of classical parameters with respect to the Planck constant. As a consequence the classical results on particle trapping by an electromagnetic wave need a quantum correction.

Using a continuous unitary transformation recently proposed by Wegner(1994) together with an approximation that neglects irrelevant contributions, we obtain flow equations for Hamiltonians. These flow equations yield a diagonal or almost diagonal Hamiltonian. As an example we investigate the Anderson Hamiltonian for dilute magnetic alloys. We study the different fixed points of the flow equations and the corresponding relevant, marginal or irrelevant contributions. Our results are consistent with results obtained by a numerical renormalization-group method, but our approach is considerably simpler.

Explicit formulae are worked out for the eigenvalue multiplicity of a system of n independent quantum harmonic oscillators in the general case of 1<or=s<or=n-1 resonance relations among the frequencies omega ₁... omega _n. As a particular case we prove that, even though the quantum numbers are always less than the degrees of freedom, the eigenvalues are, in general, intrinsically degenerate only in the completely resonant case s=n-1.

We analyse transition potentials, i.e. potentials exhibiting limiting inverse-square behaviour V(r) to ^{r approximately 0} alpha r^-2 in non-relativistic quantum mechanics using the techniques of supersymmetry. For the range - 1/4 <or= alpha < 3/4 , the eigenvalue problem becomes ill defined (since it is not possible to choose a unique eigenfunction based on square integrability and boundary conditions). It is shown that supersymmetric quantum mechanics (SUSYQM) provides a natural prescription for a unique determination of the spectrum. Interestingly, our SUSYQM-based approach picks out the same 'less singular' wavefunctions as the conventional approach and thus provides a simple justification for the usual practice in the literature. Two examples (the Poschl-Teller II potential and a two-anyon system on the plane) have been worked out for illustrative purposes.

In view of the current interest in relativistic spin-1 systems and the recent work on the Dirac oscillator, we introduce the Duffin-Kemmer-Petiau (DKP) equation obtained by using an external potential linear in r. Since the spin-0 representation leads to a harmonic oscillator in the non-relativistic limit and becomes an harmonic oscillator with a spin-orbit coupling of the Thomas form for vector bosons, we call the equation the DKP oscillator. This oscillator is a relativistic generalization of the quantum harmonic oscillator for scalar and vector bosons. We show that it conserves total angular momentum and that it is exactly solvable for both scalar and vector DKP bosons. We calculate and discuss the eigenvalues and eigenstates of the DKP oscillator in the spin-0 and spin-1 representations.

We consider two- and three-dimensional quantum billiards with discrete symmetries. The boundary condition is either Dirichlet or Neumann. We derive the first terms of the Weyl expansion for the level density projected onto the irreducible representations of the symmetry group. The formulae require only knowledge of the character table of the group and the geometrical properties (such as surface, perimeter etc....) of sub-parts of the billiard invariant under a group transformation. As an illustration, the method is applied to the icosahedral billiard.

A classical version of the Magnus expansion well suited to studying adiabatic time-evolution is built up. The method improves the adiabatic approximation while being symplectic in character. It is shown that the first-order approximation is already accurate enough even far from the adiabatic limit. An analysis of the changes suffered by the adiabatic invariant of a linear Hamiltonian system along its time-evolution illustrates part of the above results. Asymptotic formulae for such changes are also obtained with explicit computation of pre-exponential factors.