Table of contents

The attractive Casimir force acting on a micrometer-sphere suspended in a spherical dip, close to the wall, is discussed. This set-up is in principle directly accessible to experiment. The sphere and the substrate are assumed to be made of the same perfectly conducting material.

We study the diffusion of a particle on a random lattice with fluctuating local connectivity of average value q. This model is a basic description of relaxation processes in random media with geometrical defects. We analyse here the asymptotic behaviour of the eigenvalue distribution for the Laplacian operator. We found that the localized states outside the mobility band and observed by Biroli and Monasson (1999 J. Phys. A: Math. Gen.32 L255), in a previous numerical analysis, are described by saddle-point solutions that break the rotational symmetry of the main action in the real space. The density of states is characterized asymptotically by a series of peaks with periodicity 1/q.

We determine the degree of entanglement for two indistinguishable particles based on the tensor product structure, which is a framework for emphasizing entanglement founded on observational quantities. Our theory connects the canonical entanglement and entanglement based on occupation number for two fermions and for two bosons and shows that the entanglement measure, based on linear entropy, is closely related to the correlation measure for both the bosonic and fermionic cases.

For models which exhibit a continuous phase transition in the thermodynamic limit a numerical study of small systems reveals a non-monotonic behaviour of the microcanonical specific heat as a function of the system size. This is in contrast to a treatment in the canonical ensemble where the maximum of the specific heat increases monotonically with the size of the system. A phenomenological theory is developed which permits us to describe this peculiar behaviour of the microcanonical specific heat and allows in principle the determination of microcanonical critical exponents.

We investigate the statistics of eigenstates in a weak self-affine disordered potential in one dimension, whose Gaussian fluctuations grow with distance with a positive Hurst exponent H. Typical eigenstates are superlocalized on samples much larger than a well-defined crossover length, which diverges in the weak-disorder regime. We present a parallel analytical investigation of the statistics of these superlocalized states in the discrete and the continuum formalisms. For the discrete tight-binding model, the effective localization length decays logarithmically with the sample size, and the logarithm of the transmission is marginally self-averaging. For the continuum Schrödinger equation, the superlocalization phenomenon has more drastic effects. The effective localization length decays as a power of the sample length, and the logarithm of the transmission is fully non-self-averaging.

We present a mesoscopic model for thermoelectric phenomena in terms of an interacting particle system, a lattice electron gas dynamics that is a suitable extension of the standard simple exclusion process. We concentrate on electronic heat and charge transport in different but connected metallic substances. The electrons hop between energy cells located alongside the spatial extension of the metal wire. When changing energy level, the system exchanges energy with the environment. At equilibrium the distribution satisfies the Fermi–Dirac occupation law. Installing different temperatures at two connections induces an electromotive force (Seebeck effect) and upon forcing an electric current, an additional heat flow is produced at the junctions (Peltier heat). We derive the linear response behaviour relating the Seebeck and Peltier coefficients as an application of Onsager reciprocity. We also indicate the higher order corrections. The entropy production is characterized as the anti-symmetric part under time reversal of the space–time Lagrangian.

We find conditions for existence and stability of various types of discrete breather concentrated around three central sites in a triangular lattice of one-dimensional Hamiltonian oscillators with on-site potential and nearest-neighbour coupling. In particular, we confirm that it can support non-reversible breather solutions, despite the time-reversible character of the system. They carry a net energy flux and can be called 'vortex breathers'. We prove that there are parameter regions for which they are linearly stable, for example in a lattice consisting of coupled Morse oscillators, whereas the related reversible breathers are unstable. Thus non-reversible breathers can be physically relevant.

Several order parameters have been considered to predict and characterize the transition between ordered and disordered phases in random Boolean networks, such as the Hamming distance between replicas or the stable core, which have been successfully used. In this work, we propose a natural and clear new order parameter: the temporal variance. We compute its value analytically and compare it with the results of numerical experiments. Finally, we propose a complexity measure based on the compromise between temporal and spatial variances. This new order parameter and its related complexity measure can be easily applied to other complex systems.

For a wide class of Hamiltonians, a novel method for obtaining lower and upper bounds for the lowest energy is presented. Unlike perturbative or variational techniques, this method does not involve the computation of any integral (a normalization factor or a matrix element). It just requires the determination of the absolute minimum and maximum in the whole configuration space of the local energy associated with a normalizable trial function (the calculation of the norm is not needed). After a general introduction, the method is applied to three non-integrable systems: the asymmetric annular billiard, the many-body spinless Coulombian problem, the hydrogen atom in a constant and uniform magnetic field. Being more sensitive than the variational methods to any local perturbation of the trial function, this method can be used to systematically improve the energy bounds with a local skilled analysis; an algorithm relying on this method can therefore be constructed and an explicit example for a one-dimensional problem is given.

The Laplacian functional determinants for conformal scalars and coexact one-forms are evaluated in closed form on inhomogeneous lens spaces of certain orders, including all odd primes when the essential part of the expression is given, formally, as a cyclotomic unit.

From a spectral problem and corresponding Lenard operator pairs, we derive a Dirac soliton hierarchy associated with a nonlinear Dirac system. A systematic method is proposed for constructing the N-fold Darboux transformation of the Dirac system based on its Lax pair. As an application of Darboux transformation, explicit soliton solutions of the Dirac system are given.

Lagrangian systems with nonholonomic constraints may be considered as singular differential equations defined by some constraints and some multipliers. The geometry, solutions, symmetries and constants of motion of such equations are studied within the framework of linearly singular differential equations. Some examples are given, in particular the well-known singular Lagrangian of the relativistic particle, which with the nonholonomic constraint v² = c² yields a regular system.

In this paper, we first prove the Grammian determinant solution to the discrete KP (dKP) equation by the algebraic identity of pfaffian instead of Laplace expansion for determinants. Then we present the Gram-type pfaffian solution to the pfaffianized dKP system. As an example, the N-soliton solution for the system is obtained.

In order to obtain a framework in which both non-holonomic mechanical systems and non-holonomic mechanical systems with symmetry can be described, we introduce in this paper the notion of a Lagrangian system on a subbundle of a Lie algebroid.

A pfaffianized version of the three-dimensional three-wave equation is found using Hirota and Ohta's pfaffianization procedure. In addition, n-lump solutions to the pfaffianized system are presented.

We present a simple numerical optimization procedure to search for highly entangled states of 2, 3, 4 and 5 qubits. We develop a computationally tractable entanglement measure based on the negative partial transpose criterion, which can be applied to quantum systems of an arbitrary number of qubits. The search algorithm attempts to optimize this entanglement cost function to find the maximal entanglement in a quantum system. We present highly entangled 4-qubit and 5-qubit states discovered by this search. We show that the 4-qubit state is not quite as entangled, according to two separate measures, as the conjectured maximally entangled Higuchi–Sudbery state. Using this measure, these states are more highly entangled than the 4-qubit and 5-qubit GHZ states. We also present a conjecture about the NPT measure, inspired by some of our numerical results, that the single-qubit reduced states of maximally entangled states are all totally mixed.

We generalize in this paper a theorem of Titchmarsh for the positivity of Fourier sine integrals. We then apply the theorem to derive simple conditions for the absence of positive energy bound states (bound states embedded in the continuum) for the radial Schrödinger equation with nonlocal potentials which are superpositions of a local potential and separable potentials.

We apply the asymptotic iteration method (AIM) (Ciftci, Hall and Saad 2003 J. Phys. A: Math. Gen.36 11807) to solve new classes of second-order homogeneous linear differential equation. In particular, solutions are found for a general class of eigenvalue problems which includes Schrödinger problems with Coulomb, harmonic oscillator or Pöschl–Teller potentials, as well as the special eigenproblems studied recently by Bender et al (2001 J. Phys. A: Math. Gen.34 9835) and generalized in the present paper to arbitrary dimension.

The Dirac equation in n + 1 dimensions is derived by a simple algebraic approach. The similarity in the structure of the arbitrary n-dimensional Dirac equations in a central field and their solutions is discussed.

The motion of spinning test particles along circular orbits in static vacuum spacetimes belonging to the Weyl class is discussed. Spin alignment and coupling with background parameters in the case of superimposed Weyl fields, corresponding to a single Schwarzschild black hole and a single Chazy–Curzon particle as well as to two Schwarzschild black holes and two Chazy–Curzon particles, are studied in detail for standard choices of supplementary conditions. Applications to the gravitomagnetic 'clock effect' are also discussed.

The WDVV equations of associativity arising in two-dimensional topological field theory can be represented, in the simplest nontrivial case, by a single third-order equation of the Monge–Ampère type. By investigating its Lie point symmetries, we reduce it to various nonlinear ordinary differential equations, and obtain several new explicit solutions.

Some of the formulas quoted for the case that d is an odd number should be corrected. Corrected versions of equations (34), (43) and (56) are given for this case. Please see the PDF for details.