Table of contents

We present a systematic method of constructing limit-quasiperiodic structures with non-crystallographic point symmetries. Such structures are different aperiodic-ordered structures from quasicrystals, and we call them 'superquasicrystals'. They are sections of higher dimensional limit-periodic structures constructed on 'super-Bravais-lattices'. We enumerate important super-Bravais-lattices. Superquasicrystals with strong self-similarities form an important subclass. The simplest example is a two-dimensional octagonal superquasicrystal.

Since the beginning of the past century many proposals for relativistic transformations in thermodynamics have been suggested. A general consensus about this matter has not been reached. In this work, we propose a scheme of thermodynamic relativistic transformations inspired by the Planck–Einstein theory, but changing the relativistic transformation of energy. This change permits the form invariance of thermodynamics. Also by means of finite-time thermodynamics we demonstrate the relativistic invariance of thermal efficiency.

In this paper we show that the existence of a primarily discrete spacetime may be a fruitful assumption from which we may develop a new approach of statistical thermodynamics in pre-relativistic conditions. The discreteness of spacetime structure is determined by a condition that mimics the Heisenberg uncertainty relations, and the motion in this spacetime model is chosen as simple as possible. From these two assumptions we define a path entropy that measures the number of closed paths associated with a given energy of the system preparation. This entropy has a dynamical character and depends on the time interval on which we count the paths. We show that there exists a like-equilibrium condition for which the path entropy corresponds exactly to the usual thermodynamic entropy and, more generally, the usual statistical thermodynamics is reobtained. This result derived without using the Gibbs-ensemble method shows that the standard thermodynamics is consistent with a motion that is time irreversible at a microscopic level. From this change of paradigm it becomes easy to derive an H-theorem. A comparison with the traditional Boltzmann approach is presented. We also show how our approach can be implemented in order to describe reversible processes. By considering a process defined simultaneously by initial and final conditions, a well-defined stochastic process is introduced and we are able to derive a Schrödinger equation, an example of time-reversible equation.

In this paper we present an alternative method to that developed by McCoy and Wu to obtain some exact results for the 2D Ising model with a general boundary magnetic field and for a finite size system. This method is a generalization of ideas from Plechko presented for the 2D Ising model in zero field, based on the representation of the Ising model using a Grassmann algebra. A Gaussian 1D action is obtained for a general configuration of the boundary magnetic field. When the magnetic field is homogeneous, we check that our results are in agreement with McCoy and Wu's previous work. This 1D action is used to compute in an efficient way the free energy in the special case of inhomogeneous boundary magnetic field. This method is useful to obtain new exact results for interesting boundary problems, such as wetting transitions.

A novel analytical approach has been developed for heat conduction in a multi-dimensional composite slab subject to time-dependent boundary changes of the first kind. Boundary temperatures are represented as Fourier series. Taking advantage of the periodic properties of boundary changes, the analytical solution is obtained and expressed explicitly. Nearly all the published works necessitate searching for associated eigenvalues in solving such a problem even for a one-dimensional composite slab. In this paper, the proposed method involves no iterative computation such as numerically searching for eigenvalues and no residue evaluation. The adopted method is simple which represents an extension of the novel analytical approach derived for the one-dimensional composite slab. Moreover, the method of 'separation of variables' employed in this paper is new. The mathematical formula for solutions is concise and straightforward. The physical parameters are clearly shown in the formula. Further comparison with numerical calculations is presented.

It is well known that a Sturmian sequence can be regarded as a rotation sequence or a balanced sequence. In this paper, by a rotation sequence, we first construct a series of sequences with complexity kn + 1, then, from these sequences, we reconstruct other Sturmian sequences and discuss their relationships.

When using the one-centre two-range expansion method to evaluate multicentre integrals over Slater type orbitals (STOs), it may become necessary to compute numerical values of the corresponding Fourier coefficients, also known as Barnett–Coulson/Löwdin Functions (BCLFs) (Bouferguene and Jones 1998 J. Chem. Phys.109 5718). To carry out this task, it is crucial to not only have a stable numerical procedure but also a fast algorithm. In previous work (Bouferguene and Rinaldi 1994 Int. J. Quantum Chem.50 21), BCLFs were represented by a double integral which led to a numerically stable algorithm but this turned out to be disappointingly time consuming. The present work aims at exploring another path in which BCLFs are represented either by an infinite series involving modified Bessel functions or by an integral whose integrand is a smooth function. Both of these representations have the advantage of being symmetrical with respect to the cusp parameter a and the radial variable r. As a consequence, it is no longer necessary to split the integrals over r [0, + ∞) into several components with a different analytical form in each of these. A numerical study is also carried out to help select the most appropriate method to be used in practice.

For the Kowalevski gyrostat, a change of variables similar to that for the Kowalevski top is done. We establish one-to-one correspondence between solutions of the Kowalevski gyrostat and the Clebsch system and demonstrate that Kowalevski variables for the gyrostat practically coincide with elliptic coordinates on a sphere for the Clebsch case.

Known shape-invariant potentials for the constant-mass Schrödinger equation are taken as effective potentials in a position-dependent effective mass (PDEM) one. The corresponding shape-invariance condition turns out to be deformed. Its solvability imposes the form of both the deformed superpotential and the PDEM. A lot of new exactly solvable potentials associated with a PDEM background are generated in this way. A novel and important condition restricting the existence of bound states whenever the PDEM vanishes at an end point of the interval is identified. In some cases, the bound-state spectrum results from a smooth deformation of that of the conventional shape-invariant potential used in the construction. In others, one observes a generation or suppression of bound states, depending on the mass-parameter values. The corresponding wavefunctions are given in terms of some deformed classical orthogonal polynomials.

I discuss in this paper the behaviour of the solutions of the so-called q-hyperbolic potentials, i.e. Pöschl–Teller-like and conditionally solvable potentials, in terms of the path integral formalism. The differences in comparison to the usual Pöschl–Teller-like potentials are investigated, including the discrete energy spectra and the bound-state wavefunctions. We also point out the relation of the q-deformation with curvature on hyperboloids.

The well-known two-slit interference is understood as a special relation between an observable (localization at the slits) and a state (being on both slits). The relation between an observable and a quantum state is investigated in the general case. It is assumed that the amount of coherence equals that of the incompatibility between an observable and a state. On these grounds, an argument is presented that leads to a natural quantum measure of coherence, called 'coherence or incompatibility information'. Its properties are studied in detail, making use of 'the mixing property of relative entropy' derived in this paper. A precise relation between the measure of coherence of an observable and that of its coarsening is obtained and discussed from the intuitive point of view. Convexity of the measure is proved, and thus the fact that it is an information entity is established. A few more detailed properties of the coherence information are derived with a view to investigating the final-state entanglement in general repeatable measurement, and, more importantly, general bipartite entanglement in follow-ups of this study.

We consider the entanglement in the ground state of the XY model of an infinite chain. Following Bennett, Bernstein, Popescu and Schumacher, we use the entropy of a sub-system as a measure of entanglement. Vidal, Latorre, Rico and Kitaev have conjectured that the von Neumann entropy of a large block of neighbouring spins approaches a constant as the size of the block increases. We evaluate this limiting entropy as a function of anisotropy and transverse magnetic field. We use the methods based on the integrable Fredholm operators and the Riemann–Hilbert approach. It is shown how the entropy becomes singular at the phase transition points.

The survival probability of a quantum state encodes essential information concerning the decay rate of quantum particles and is the primary object for investigating the time–energy uncertainty relations and the occurrence of the quantum Zeno effect. The purpose of this article is to uncover some curious properties concerning the relations between the values of the survival probability of a quantum state at different times. These relations put surprising restrictions on the evolution pictures of quantum states, and also illustrate their peculiar intricacies.

We study a fluid model of an infinitesimally thin plasma sheet occupying the xy plane, loosely imitating a single base plane from graphite. In terms of the fluid charge e/a² and mass m/a² per unit area, the crucial parameters are q 2πe²/mc²a², a Debye-type cutoff on surface-parallel normal-mode wavenumbers k, and X K/q. The cohesive energy β per unit area is determined from the zero-point energies of the exact normal modes of the plasma coupled to the Maxwell field, namely TE and TM photon modes, plus bound modes decaying exponentially with |z|. Odd-parity modes (with E_x,y(z = 0) = 0) are unaffected by the sheet except for their overall phases, and are irrelevant to β, although the following paper shows that they are essential to the fields (e.g. to their vacuum expectation values), and to the stresses on the sheet. Realistically one has X ≫ 1, the result β ∼ ℏcq^1/2K^5/2 is nonrelativistic, and it comes from the surface modes. By contrast, X ≪ 1 (nearing the limit of perfect reflection) would entail β ∼ −ℏcqK²log(1/X): contrary to folklore, the surface energy of perfect reflectors is divergent rather than zero. An appendix spells out the relation, for given k, between bound modes and photon phase-shifts. It is very different from Levinson's theorem for 1D potential theory: cursory analogies between TM and potential scattering are apt to mislead.

We study the self-stresses experienced by the single plasma sheet modelled in the preceding paper, and determine the exact mean-squared Maxwell fields in vacuum around it. These are effects that probe the physics of such systems further than do the ground-state eigenvalues responsible for the cohesive energy β; in particular, unlike β they depend not only on the collective properties but also on the self-fields of the charge carriers. The classical part of the interaction between the sheet and a slowly moving charged particle follows as a byproduct. The main object is to illustrate, in simple closed or almost closed form, the consequences of imperfect (dispersive) reflectivity. The largely artificial limit of perfect reflection reduces all the results to those long familiar outside a half-space taken to reflect perfectly from the outset; but a careful examination of the approach to this limit is needed in order to resolve paradoxes associated with the surface energy, and with the mechanism which, in the limit, disjoins the two flanking half-spaces both electromagnetically and quantally.

The scaling limit of the two-dimensional Ising model above the critical temperature is considered as an example for relativistic quantum theories on two-dimensional Minkowski space exhibiting a factorizing S-matrix. In this model, a recently proposed criterion for the existence of local quantum field theories with a prescribed factorizing scattering matrix is verified, thereby establishing a new constructive approach to two-dimensional quantum field theory in a particular example. The existence proof is accomplished by analysing the nuclearity properties of certain specific subsets of fermionic Fock spaces, and yields as a byproduct also a verification of the energy nuclearity condition of Buchholz and Wichmann in models of free fermions in four spacetime dimensions.

The structure of a two-dimensional dusty plasma in an external magnetic field has been investigated in detail by molecular dynamics simulation. The pair correlation function, mean square displacement, the static structure factor and the bond angle correlation function have been calculated to characterize the structural properties of the dusty plasma for different values of coupling constant and magnetic field strength. The results show that the dusty plasma system with a fixed coupling constant can have a phase transition from gas to fluid, and from fluid to a solid state, when the external magnetic field strength is increased to a critical value. The critical magnetic field strength generally decreases with increasing coupling constant in the system.