Table of contents

We discuss the spectrum and the eigenfunctions of a 2D Wannier-Stark system in both the tight-binding and single-band approximations. We show that the unfamiliar infinite-variable Bessel functions play a crucial role in these considerations. Furthermore, a closed formula for the eigenfunction for an arbitrary tight-binding ansatz in terms of an expansion in Wannier states is derived. Finally we give a closed form solution for the Wannier-Stark states in the scope of the single-band approximation.

Analytical arguments and dynamic Monte Carlo simulations show that the microscopic structure of field-driven solid-on-solid interfaces depends strongly on the details of the dynamics. For non-conservative dynamics with transition rates that factorize into parts dependent only on the changes in interaction energy and field energy, respectively (soft dynamics), the intrinsic interface width is field independent. For non-factorizing rates, such as the standard Glauber and Metropolis algorithms (hard dynamics), it increases with the field. Consequences for the interface velocity and its anisotropy are discussed.

We introduce a new method, allowing one to describe slowly time-dependent Langevin equations through the behaviour of individual paths. This approach yields considerably more information than the computation of the probability density. In particular, scaling laws can be obtained easily. The main idea is to show that for sufficiently small noise intensity and slow time dependence, the vast majority of paths remain in small space-time sets, typically in the neighbourhood of potential wells. The size of these sets often has a power-law dependence on the small parameters, with universal exponents. The overall probability of exceptional paths is exponentially small, with an exponent also showing power-law behaviour. The results cover time spans up to the maximal Kramers time of the system. We apply our method to three phenomena characteristic for bistable systems: stochastic resonance, dynamical hysteresis and bifurcation delay, where it yields precise bounds on transition probabilities, and the distribution of hysteresis areas and first-exit times. We also discuss the effect of coloured noise.

We study supervised learning and generalization in coupled perceptrons trained on-line using two learning scenarios. In the first scenario the teacher and the student are independent networks and both are represented by an Ashkin–Teller perceptron. In the second scenario the student and the teacher are simple perceptrons but are coupled by an Ashkin–Teller-type four-neuron interaction term. Expressions for the generalization error and the learning curves are derived for various learning algorithms. The analytical results find excellent confirmation in numerical simulations.

A fully standard quantum model of a particle interacting with a single-mode phonon system under the influence of a thermodynamic bath is considered. Numerically exact solution shows that, for very specific values of parameters involved, the phonon mode cooperating with the particle becomes able to respond to particle hops and thus to suppress the back particle transfer. The particle becomes free at the end of the process, during which it can be transferred prevailingly in one direction only, even going uphill in energy, at the cost of just the thermal energy of the single bath. This behaviour is due to the fact that both the particle and the particle + oscillator density matrices differ, in the stationary situations and for at least intermediate oscillator coupling to the bath, from the respective canonical forms.

Rephrasing the backbone of two-dimensional percolation as a monochromatic path crossing problem, we investigate the latter by a transfer matrix approach. Conformal invariance links the backbone dimension D_b to the highest eigenvalue of the transfer matrix T, and we obtain the result D_b = 1.6431 ± 0.0006. For a strip of width L, T is roughly of size 2^3L, but we manage to reduce it to ∼L!. We find that the value of D_b is stable with respect to inclusion of additional 'blobs' tangent to the backbone in a finite number of points.

In this paper, we investigate a common traffic phenomenon—the merging of a platoon of moving cars at one density into another platoon moving at a different density. A new macroscopic model, the 'speed gradient (SG) model,' is used in the simulations. It is shown that different traffic patterns appear according to different upstream and downstream densities. Generally, these patterns can be classified into stable traffic and unstable traffic. The transition from unstable pattern to stable pattern is discussed in detail.

Membrane fusion, protein folding and macromolecular assembly are a few of the many processes in which the interaction of near-neutral and semi-permeable fluid surfaces plays an important role. The electrostatic force between membranes is solved from Coulomb's law by first casting the expression for charge by way of the Fredholm integral equation, and then integrating the effect of the charge distribution to obtain the expression for force. The surface charge density is conveniently described by a Langevin type expression which suggests a saturation type behaviour describing a transition from `soft' to `hard' sphere where increasing electrolyte strength and particle size modify the pair-interaction force.

Motivated by recent investigations of transport properties of strongly correlated 1D models and thermal conductivity measurements of quasi 1D magnetic systems, we present results for the integrable spin-½ XXZ chain. The thermal conductivity κ(ω) of this model has ℜ κ(ω) = δ(ω), i.e. it is infinite for zero frequency ω. The weight of the delta peak is calculated exactly by a lattice path integral formulation. Numerical results for wide ranges of temperature and anisotropy are presented. The low- and high-temperature limits are studied analytically.

We propose a technique called dimensional descent to show that Wigner's little group for massless particles, which acts as a generator of gauge transformation for usual Maxwell theory, has an identical role even for topologically massive gauge theories. The examples of B ∧ F theory and Maxwell–Chern–Simons theory are analysed in detail.

An analysis of CPN models is given in terms of general coordinates or arbitrary interpolating fields. Only closed expressions made from simple functions are involved. Special attention is given to CP2 and CP4. In the first of these the retrieval of stereographic coordinates reveals the Hermitian form of the metric. A similar analysis for the latter case allows comparison with the Fubini-Study metric.

In this paper the Casimir effect arising from the case of Dirichlet boundary conditions confining the massless scalar field at finite temperature is reexamined for a (D-1)-dimensional rectangular cavity with equal or unequal finite p edges and different spacetime dimensions D. We derive an expression for the Casimir energy for a p-dimensional cavity at nonzero temperature. We show that the sign of the Casimir energy remains positive irrespective of whether p is odd or even if the thermal corrections to the standard Casimir effect are sufficiently large. Furthermore, we also find the temperature influences on choosing edges which lead to the Casimir energy being positive or negative.

A relationship between two old mathematical subjects is observed: the theory of hypergeometric functions and the separability in classical mechanics. Separable potential perturbations of integrable billiard systems and the Jacobi problem for geodesics on an ellipsoid are expressed through the Appell hypergeometric functions F₄ of two variables. Even when the number of degrees of freedom increases, if an ellipsoid is symmetric, the number of variables in the hypergeometric functions does not increase. Wider classes of separable potentials are given by the obtained new formulae automatically.

The quantum mechanical hypervirial theorems (HVT) technique is used in treating two basic inequalities relating the ground-state mean square radius of the orbit of a particle in a central potential and its kinetic energy, respectively, to the spacing of the two lowest energy levels ΔE. For quite a wide class of those potentials, the parameters of which also lead to a sufficiently small dimensionless quantity s, the difference between the two sides of each inequality is of order s³ and higher (while ΔE is of order s and higher), and thus it is expected in general to be quite small.

Elementary function representations of Schlömilch series introduced by Twersky (Twersky V 1961 Arch. Ration. Mech. Anal.8 323-32) are used to construct the exact analytical expressions for the classical electromagnetic problem of transverse electric multiple scattering by an infinite array of insulating dielectric circular cylinders at oblique incidence.

Different symmetry formalisms for difference equations on lattices are reviewed and applied to perform symmetry reduction for both linear and nonlinear partial difference equations. Both Lie point symmetries and generalized symmetries are considered and applied to the discrete heat equation and to the integrable discrete time Toda lattice.

Using the Matsubara formalism, we consider the massive (λφ⁴)_D vector N-component model in the large-N limit, the system being confined between two infinite parallel planes. We investigate the behaviour of the coupling constant as a function of the separation L between the planes. For the Wick-ordered model in D = 3 we are able to give an exact formula to the L dependence of the coupling constant. For the non-Wick-ordered model we indicate how expressions for the coupling constant and the mass can be obtained for arbitrary dimension D in the small-L regime. Closed exact formulae for the L-dependent renormalized coupling constant and mass are obtained in D = 3 and their behaviours as functions of L are displayed. We are also able to obtain, in generic dimension D, an equation for the critical value of L corresponding to a second-order phase transition in terms of the Riemann zeta-function. In D = 3 an implicit formula for the critical L is given.

Closed form representations in terms of well known special functions are deduced for the Mellin transform of a product of two Fox-Wright psi functions ₁Ψ₁. An application concerned with the recent analysis of the critical behavior of a variety of distinct physical systems is provided.

We show that the Landau quantum systems (or integer quantum Hall effect systems) in a plane, sphere or a hyperboloid, can be explained in a complete and meaningful way by group-theoretical considerations concerning the symmetry group of the corresponding configuration space. The crucial point in our development is the role played by locality and its appropriate mathematical framework, the fibre bundles. In this way the Landau levels can be understood as the local equivalence classes of the symmetry group. We develop a unified treatment that supplies the correct geometric way to recover the planar case as a limit of the spherical or the hyperbolic quantum systems when the curvature goes to zero. This is an interesting case where a contraction procedure gives rise to nontrivial cohomology starting from a trivial one. We show how to reduce the quantum hyperbolic Landau problem to a Morse system using horocyclic coordinates. An algebraic analysis of the eigenvalue equation allows us to build ladder operators which can help in solving the spectrum under different boundary conditions.

We consider dynamical systems with discrete time (maps) that possess one or more integrals depending upon parameters. We show that integrals can be used to replace parameters in the original map so as to construct a different map with different integrals. We also highlight a process of reparametrization that can be used to increase the number of parameters in the original map prior to using integrals to replace them. Properties of the original map and the new map are compared. The theory is motivated by, and illustrated with, examples of a three-dimensional trace map and some four-dimensional maps previously shown to be integrable.

The problem of identifying the dynamical Lie algebras of finite-level quantum systems subject to external control is considered, with special emphasis on systems that are not completely controllable. In particular, it is shown that the dynamical Lie algebra for an N-level system with symmetrically coupled transitions, such as a system with equally spaced energy levels and uniform transition dipole moments, is a subalgebra of so(N) if N = 2ℓ + 1, and a subalgebra of sp(ℓ) if N = 2ℓ. General criteria for obtaining either so(2ℓ + 1) or sp(ℓ) are established.

Hermitian supersymmetric partnership between singular potentials V(q) = q² + G/q² breaks down and can only be restored on certain ad hoc subspaces (Das A and Pernice S 1999 Nucl. Phys. B 561 357). We show that within extended, -symmetric quantum mechanics, the supersymmetry between singular oscillators can be completely re-established in a way which is continuous near G = 0 and leads to a new form of the bosonic creation and annihilation operators.