Table of contents

We explore an intriguing connection between the Fermi–Dirac and Bose–Einstein statistics and the thermal baths obtained from a vacuum radiation of coherent states of zero modes in a second quantized (many-particle) theory on the compact O(3) and noncompact O(2, 1) isometry subgroups of the de Sitter and anti-de Sitter spaces, respectively. The high frequency limit is retrieved as a (zero-curvature) group contraction to the Newton–Hooke (harmonic oscillator) group. We also make some comments on the vacuum energy density and the cosmological constant problem.

For a lattice Λ with n vertices and dimension d equal to or higher than 2, the number of spanning trees N_ST(Λ) increases asymptotically as exp(nz_Λ) in the thermodynamic limit. We present exact integral expressions for the asymptotic growth constant z_Λ for spanning trees on several lattices. By taking different unit cells in the calculation, many integral identities can be obtained. We also give z_Λ(p) on the homeomorphic expansion of k-regular lattices with p vertices inserted on each edge.

We calculate exactly the partition function Z(G, Q, v) of the Q-state Potts model with temperature-like Boltzmann variable v for strip graphs G of the square and triangular lattices of various widths L_y and arbitrarily great lengths L_x, with a variety of boundary conditions, and with Q and v restricted to satisfy conditions corresponding to the ferromagnetic phase transition on the associated two-dimensional lattices. From these calculations, in the limit L_x → ∞, we determine the continuous accumulation loci of the partition function zeros in the v and Q planes. Strips of the honeycomb lattice are also considered. We discuss some general features of these loci.

At the free-fermion point, the six-vertex model with domain wall boundary conditions (DWBC) can be related to the Aztec diamond, a domino tiling problem. We study the mapping on the level of complete statistics for general domains and boundary conditions. This is obtained by associating with both models a set of non-intersecting lines in the Lindström–Gessel–Viennot (LGV) scheme. One of the consequences for DWBC is that the boundaries of the ordered phases are described by the Airy process in the thermodynamic limit.

For many stochastic processes there is an underlying coordinate space, V, with the process moving from point to point in V or on variables (such as spin configurations) defined with respect to V. There is a matrix of transition probabilities (whether between points in V or between variables defined on V) and we focus on its 'slow' eigenvectors, those with eigenvalues closest to that of the stationary eigenvector. These eigenvectors are the 'observables', and can be used to recover geometrical features of V.

Mechanics of fluid membranes may be described in terms of the concepts of mechanical deformations and stresses or in terms of mechanical free-energy functions. In this paper, each of the two descriptions is developed by viewing a membrane from two perspectives: a microscopic perspective, in which the membrane appears as a thin layer of finite thickness and with highly inhomogeneous material and force distributions in its transverse direction, and an effective, two-dimensional perspective, in which the membrane is treated as an infinitely thin surface, with effective material and mechanical properties. A connection between these two perspectives is then established. Moreover, the functional dependence of the variation in the mechanical free energy of the membrane on its mechanical deformations is first studied in the microscopic perspective. The result is then used to examine to what extent different, effective mechanical stresses and forces can be derived from a given, effective functional of the mechanical free energy.

We discuss the collective behaviour of a set of operators and variables that constitute a program and the emergence of meaningful computational properties in the language of statistical mechanics. This is done by appropriately modifying available Monte Carlo methods to deal with hierarchical structures. The study suggests, in analogy with simulated annealing, a method to automatically design programs. Reasonable solutions can be found, at low temperatures, when the method is applied to simple toy problems such as finding an algorithm that determines the roots of a function or one that makes a nonlinear regression. Peaks in the specific heat are interpreted as signalling phase transitions which separate regions where different algorithmic strategies are used to solve the problem.

The diagonal case of the sl(2) Richardson–Gaudin quantum pairing model is known to be solvable as an Abel–Jacobi inversion problem. The effect of random time-dependent perturbations of the single-particle spectrum on the exact solution is an open question of considerable physical relevance. Weak perturbations introduce a new, slow time scale, while preserving the nonlinear character of the dynamics. In this paper, such perturbations are considered. It is shown that the long-time asymptotics can be obtained by a deformation of the original integrable system, equivalent to phase averaging over the fast time scale.

This paper presents the Euler–Lagrange equations and the transversality conditions for fractional variational problems. The fractional derivatives are defined in the sense of Riemann–Liouville and Caputo. The connection between the transversality conditions and the natural boundary conditions necessary to solve a fractional differential equation is examined. It is demonstrated that fractional boundary conditions may be necessary even when the problem is defined in terms of the Caputo derivative. Furthermore, both fractional derivatives (the Riemann–Liouville and the Caputo) arise in the formulations, even when the fractional variational problem is defined in terms of one fractional derivative only. Examples are presented to demonstrate the applications of the formulations.

With the basic Clifford units being identified as mirrors, it is demonstrated how proper and improper symmetry operations of point groups in spaces of arbitrary dimensions can be parametrized. In such an approach consistency with parametrizations for groups in three dimensions can be achieved even if double groups are considered. The conversion of Clifford parameters into Cartesian matrices and vice versa is discussed and, for rotations in , also the parametrization in terms of pairs of rotations in . The formalism is illustrated by a number of examples.

It is shown that the ground-state energy of heavy atoms is, to leading order, given by the non-relativistic Thomas–Fermi energy. The proof is based on the relativistic Hamiltonian of Brown and Ravenhall which is derived from quantum electrodynamics yielding energy levels correctly up to order α²Ry.

We have constructed a Heisenberg-type algebra generated by the Hamiltonian, the step operators and an auxiliary operator. This algebra describes quantum systems having eigenvalues of the Hamiltonian depending on the eigenvalues of the two previous levels. This happens, for example, for systems having the energy spectrum given by a Fibonacci sequence. Moreover, the algebraic structure depends on the two functions f(x) and g(x). When these two functions are linear we classify, analysing the stability of the fixed points of the functions, the possible representations for this algebra.

The derivation of dispersion relations for linear optical constants is considered starting from the representation of an optical property as a Herglotz function. One form of the Kramers–Kronig relations is determined directly and the second is obtained using elementary properties of the Hilbert transform. Application to the complex refractive index is considered.

In this work the propagation of nonlinear electromagnetic short waves in a ferromagnetic medium is discussed. It is shown that such waves propagate perpendicular to the magnetization density. The evolution of the wave under the influence of perturbations in one transverse dimension is considered; the asymptotic model equation governing the dynamics is a (2+1) generalization of the well-known sine-Gordon model. We exhibit the line-soliton solution and study its transverse stability. A numerical study of the model corroborates our analytical predictions.

We study twisted Jacobi manifolds, a concept that we had introduced in a previous note. Twisted Jacobi manifolds can be characterized using twisted Dirac–Jacobi, which are sub-bundles of Courant–Jacobi algebroids. We show that each twisted Jacobi manifold has an associated Lie algebroid with a 1-cocycle. We introduce the notion of quasi-Jacobi bialgebroid and prove that each twisted Jacobi manifold has a quasi-Jacobi bialgebroid canonically associated. Moreover, the double of a quasi-Jacobi bialgebroid is a Courant–Jacobi algebroid. Several examples of twisted Jacobi manifolds and twisted Dirac–Jacobi structures are presented.

We study a family of self-adjoint partial differential operators H_ω, where ω is a large parameter. In the simplest case each operator acts in as under the boundary conditions of a certain type. We are interested in the behaviour of the eigenvalues λ_n(H_ω) as ω → ∞. Let Λ₀ stand for the lowest eigenvalue of the Schrödinger operator −∂²_y + Q(y) in . Under some assumptions about the data we show that the numbers λ_n(H_ω) − ωΛ₀ converge to the eigenvalues of the boundary value problem −ψ'' = λψ on (a, b), under some boundary conditions induced by those for the original operators. Possible generalizations are also discussed.

We discuss a recent application of the asymptotic iteration method (AIM) to a perturbed Coulomb model. Contrary to what was argued before we show that the AIM converges and yields accurate energies for that model. We also consider alternative perturbation approaches and show that one of them is equivalent to that recently proposed by another author.

This work is concerned with a quantization of the Pais–Uhlenbeck oscillators from the point of view of their multi-Hamiltonian structures. It is shown that the 2nth-order oscillator with a simple spectrum can be quantized as the usual anisotropic n-dimensional oscillator.

One-dimensional electronic conduction is investigated in a special case usually referred to as the harmonic crystal, meaning essentially that atoms are assumed to move like coupled harmonic oscillators within the Born–Oppenheimer approximation. We recall their dispersion relation and derive a WKB system approximately satisfied by any electron's wavefunction inside a given energy band. This is then numerically solved according to the method of K-branch solutions. Numerical results are presented in the case where atoms move with one- or two-modes vibrations; finally, we include the case where the Poisson self-interaction potential also influences the electrons' dynamics.

We investigate the quantum entanglement of the ground state and the mixed states at finite temperatures for a two-qubit system within the framework of an anisotropic Heisenberg XYZ model in the presence of a nonuniform magnetic field. As a measure of the entanglement, the concurrence of the two-qubit states is calculated and is analysed in detail as a function of the coupling constants, magnetic field and temperature. Consequently, we show that the combined influence of the anisotropic interaction and the nonuniformity of the magnetic fields predicts much pronounced entanglement properties.

The d-dimensional generalization of the point canonical transformation for a quantum particle endowed with a position-dependent mass in the Schrödinger equation is described. Illustrative examples including the harmonic oscillator, Coulomb, spiked harmonic, Kratzer, Morse oscillator, Pőschl–Teller and Hulthén potentials are used as reference potentials to obtain exact energy eigenvalues and eigenfunctions for target potentials at different position-dependent mass settings.

Consistent couplings between a massless tensor field with the mixed symmetry of the Riemann tensor and a massless vector field are analysed in the framework of Lagrangian BRST cohomology. Under the assumptions on smoothness, locality, Lorentz covariance and Poincaré invariance of the deformations, combined with the requirement that the interacting Lagrangian is at most second-order derivative, it is proved that there are no consistent cross-interactions between a single massless tensor field with the mixed symmetry of the Riemann tensor and one massless vector field.

By including a potential into the flat metric, we study the interaction of the sine-Gordon soliton with different potentials. We will show numerically that while the soliton–barrier system shows fully classical behaviour, the soliton–well system demonstrates non-classical behaviour. In particular, solitons with low velocities are trapped in the well and radiate energy. Also for narrow windows of initial velocity, a soliton reflects back from a potential well.

We derive together the exact local, covariant, continuous and off-shell nilpotent Becchi–Rouet–Stora–Tyutin (BRST) and anti-BRST symmetry transformations for the U(1) gauge field (A_μ), the (anti-)ghost fields and the Dirac fields of the Lagrangian density of a four (3 + 1)-dimensional QED by exploiting a single restriction on the six (4, 2)-dimensional supermanifold. A set of four even spacetime coordinates x^μ (μ = 0, 1, 2, 3) and two odd Grassmannian variables θ and parametrize this six-dimensional supermanifold. The new gauge invariant restriction on the above supermanifold owes its origin to the (super) covariant derivatives and their intimate relations with the (super) 2-form curvatures constructed with the help of 1-form (super)gauge connections and (super) exterior derivatives . The results obtained by exploiting (i) the horizontality condition, and (ii) one of its consistent extensions, are shown to be a simple consequence of this new single restriction on the above supermanifold. Thus, our present endeavour provides an alternative to (and, in some sense, generalization of) the horizontality condition of the usual superfield formalism applied to the derivation of BRST symmetries.

We derive and demonstrate a practical and accurate algorithm designed to impose Dirichlet boundary conditions (specified boundary velocity) on the edge nodes of a lattice Boltzmann fluid simulation space. The current algorithm models the lattice fluid on boundary and bulk nodes to identical accuracy and in a demonstrably equivalent manner. Whilst this new method applies to rectangular geometries, it adds to our previous method [2] (i) the condition of mass conservation and (ii) the ability to treat with unimpaired accuracy both internal and external corners.