Table of contents

We semiclassically derive the leading off-diagonal correction to the spectral form factor of quantum systems with a chaotic classical counterpart. To this end we present a phase space generalization of a recent approach for uniformly hyperbolic systems (Sieber M and Richter K 2001 Phys. Scr. T 90 128, Sieber M 2002 J. Phys. A: Math. Gen.35 L613). Our results coincide with corresponding random matrix predictions. Furthermore we study the transition from the Gaussian orthogonal to the Gaussian unitary ensemble.

A thorough review of acoustic and electromagnetic wavelets is given, including a first account of recent progress in understanding their sources. These physical wavelets, introduced in 1994, are families of 'small' solutions of the wave and Maxwell equations generated from a single member by group operations including translations, Lorentz transformations, and scaling. They are parametrized by complex spacetime points z = x − iy, where x gives the centre of their region of origin and y gives the extension and orientation of this region in spacetime. They are thus pulsed beams whose origin, direction and focus are all governed by z and which give, by superposition, 'wavelet representations' of acoustic and electromagnetic waves. Recently this idea has been developed substantially by the rigorous understanding of the source distributions required to launch and absorb the wavelets, defined as extended delta functions. The unexpected simplicity and complex structure of the sources in the Fourier domain suggests their potential use in the construction of fast algorithms for the analysis and synthesis of acoustic and electromagnetic waves. The review begins with a brief account of the physical wavelets associated with massive (Klein–Gordon and Dirac) fields, which are relativistic coherent states.

We analyse some aspects of the third law of thermodynamics. We first review both the entropic version (N) and the unattainability version (U) and the relation occurring between them. Then, we heuristically interpret (N) as a continuity boundary condition for thermodynamics at the boundary T = 0 of the thermodynamic domain. On a rigorous mathematical footing, we discuss the third law both in Carathéodory's approach and in Gibbs' one. Carathéodory's approach is fundamental in order to understand the nature of the surface T = 0. In fact, in this approach, under suitable mathematical conditions, T = 0 appears as a leaf of the foliation of the thermodynamic manifold associated with the non-singular integrable Pfaffian form δQ_rev. Being a leaf, it cannot intersect any other leaf S = const of the foliation. We show that (N) is equivalent to the requirement that T = 0 is a leaf. In Gibbs' approach, the peculiar nature of T = 0 appears to be less evident because the existence of the entropy is a postulate; nevertheless, it is still possible to conclude that the lowest value of the entropy S has to be attained at the boundary of the convex set where S is defined.

We develop further our analysis of the third law of thermodynamics; in particular, we discuss further conditions ensuring the validity of the third law of thermodynamics in its entropic form (N). The introduction in standard homogeneous thermodynamics of the framework in which the absolute temperature T appears as an independent coordinate for the entropy S is followed by the introduction of a more general framework in which Gibbs thermodynamic space, where only extensive independent coordinates appear, is suitably generalized. General properties of S are also discussed. An analysis of the differential conditions which can ensure the validity of (N) follows. Then, we introduce a condition involving the behaviour of generalized heat capacities along curves leaving the surface T = 0 and we show that, under suitable mathematical conditions, it is equivalent to (N). The physical meaning of this condition is also clarified, and amounts to the impossibility for a system to leave a state at T = 0 without heat absorption. Then, we show that a condition of minimum entropy at T = 0 is again equivalent to (N) under suitable conditions. Some notes about (N) when one allows deformation coordinates to be divergent as T → 0⁺ and about phase coexistence and mixtures also appear.

The time-dependent pair correlation functions for a degenerate ideal quantum gas of charged particles in a uniform magnetic field are studied on the basis of equilibrium statistics. In particular, the influence of a flat hard wall on the correlations is investigated, both for a perpendicular and a parallel orientation of the wall with respect to the field. The coherent and incoherent parts of the time-dependent structure function in position space are determined from an expansion in terms of the eigenfunctions of the one-particle Hamiltonian. For the bulk of the system, the intermediate scattering function and the dynamical structure factor are derived by taking successive Fourier transforms. In the vicinity of the wall the time-dependent coherent structure function is found to decay faster than in the bulk. For coinciding positions near the wall the form of the structure function turns out to be independent of the orientation of the wall. Numerical results are shown to corroborate these findings.

The formation and mechanical properties of a polymer network on and between two flat parallel surfaces are investigated. Most treatments of surface-attached polymers have been limited to scaling theory. In the present investigation we probe the physics of the system by means of a mathematical description of the random crosslinking of ideal (or phantom) chains. We modify an existing bulk model of network formation by Deam and Edwards, with polymer–polymer crosslinks, to include surfaces and polymer–surface crosslinks. We investigate two variations of this model: in the first place, the polymer–surface links are formed anywhere along the contours of the long, ideal polymer chains. In the second brush network model, the surface links are restricted to one endpoint of each macromolecule. Within the framework of replica theory, we compute statistical averages and the elastic properties of the systems such as the stress–strain relationship. In both cases the elastic modulus of the bulk network is altered, and has a characteristic form due to the confinement. Furthermore, we find that the stress–strain relationship depends on the manner of crosslinking.

Energy level statistics of Hermitian random matrices Ĥ with Gaussian independent random entries H_i≥j is studied for a generic ensemble of almost diagonal random matrices with ⟨|H_ii|²⟩ ∼ 1 and ⟨|H_i≠j|²⟩ = b(|i − j|) ≪ 1. We perform a regular expansion of the spectral form-factor K(τ) = 1 + bK₁(τ) + b²K₂(τ) + ⋯ in powers of b ≪ 1 with the coefficients K_m(τ) that take into account interaction of (m + 1) energy levels. To calculate K_m(τ), we develop a diagrammatic technique which is based on the Trotter formula and on the combinatorial problem of graph edges colouring with (m + 1) colours. Expressions for K₁(τ) and K₂(τ) in terms of infinite series are found for a generic function (|i − j|) in the Gaussian orthogonal ensemble (GOE), the Gaussian unitary ensemble (GUE) and in the crossover between them (the almost unitary Gaussian ensemble). The Rosenzweig–Porter and power-law banded matrix ensembles are considered as examples.

We consider the effect of memory-dependent transport on the survivability of a population dispersing in a closed domain surrounded by a hostile environment. The model we use combines memory-dependent diffusion with Malthusian growth. The former introduces an additional parameter, the correlation time for memory effects, that must be taken into account in determining the critical length below which survival does not occur. Results are obtained for all possible parametric conditions, and the relevance to non-linear growth conditions is discussed.

We reexamine the problem of having nonconservative equations of motion arise from the use of a variational principle. In particular, a formalism is developed that allows the inclusion of fractional derivatives. This is done within the Lagrangian framework by treating the action as a Volterra series. It is then possible to derive two equations of motion, one of these is an advanced equation and the other is retarded.

In this paper we obtain a complete classification of all possible non-trivial similarity solutions of the integro–differential fragmentation equation with continuous mass loss rate. These solutions include the effects of both continuous and discrete mass loss rates. The similarity solutions are compared with the solutions found earlier. The comparison shows that some previous solutions are obtained as special cases from our solutions. The results reported here provide further evidence of the usefulness of the Lie group method for obtaining similarity solutions for either differential or integro–differential equations.

We present a generalization to three qubits of the standard Bloch sphere representation for a single qubit and of the seven-dimensional sphere representation for two qubits presented in Mosseri et al (Mosseri R and Dandoloff R 2001 J. Phys. A: Math. Gen.34 10243). The Hilbert space of the three-qubit system is the 15-dimensional sphere S¹⁵, which allows for a natural (last) Hopf fibration with S⁸ as base and S⁷ as fibre. A striking feature is, as in the case of one and two qubits, that the map is entanglement sensitive, and the two distinct ways of un-entangling three qubits are naturally related to the Hopf map. We define a quantity that measures the degree of entanglement of the three-qubit state. Conjectures on the possibility of generalizing the construction for higher qubit states are also discussed.

We construct a new geometrical framework for Yang–Mills Lagrangian field theories as an appropriate quotient space of the standard first jet-bundle and investigate the geometrical properties of the resulting mathematical setting. We deduce the field equations from a variational problem formulated through a regular Poincaré–Cartan form, thus ensuring the kinematical admissibility of critical sections. We state a generalized Nöther theorem and explicitly consider the case of the free Yang–Mills field.

The authors of this paper would like to correct line 13 on page 6502 and equations (19), (32) and (39). Please see pdf for details.