Table of contents

For a class of Hamiltonian systems, naturally arising in the modern theory of separation of variables, we establish their maximal superintegrability by explicitly constructing the additional integrals of motion.

It is commonly believed that photon polarization entanglement can only be obtained via pair creation within the same source or via postselective measurements on photons that overlapped within their coherence time inside a linear optics setup. In contrast to this, we show here that polarization entanglement can also be produced by distant single photon sources in free space and without the photons ever having to meet, if the detection of a photon does not reveal its origin—the which way information. In the case of two sources, the entanglement arises under the condition of two emissions in certain spatial directions and leaves the dipoles in a maximally entangled state.

We give statistical definitions of the length, l, of a loose prime knot tied into a long, fluctuating ring macromolecule. Monte Carlo results for the equilibrium, good solvent regime show that ⟨l⟩ ∼ N^t, where N is the ring length and t ≃ 0.75 is independent of the knot type. In the collapsed regime below the theta temperature, length determinations based on the entropic competition of different knots within the same ring show knot delocalization (t ≃ 1).

We study the dynamics of a particle in continuous time and space, the displacement of which is governed by an internal degree of freedom (spin). In one definite limit, the so-called quantum random walk is recovered but, although quite simple, the model possesses a rich variety of dynamics and goes far beyond this problem. Generally speaking, our framework can describe the motion of an electron in a magnetic sea near the Fermi level when linearization of the dispersion law is possible, coupled to a transverse magnetic field. Quite unexpected behaviours are obtained. In particular, we find that when the initial wave packet is fully localized in space, the J_z angular momentum component is frozen; this is an interesting example of an observable which, although it is not a constant of motion, has a constant expectation value. For a non-completely localized wave packet, the effect still occurs although less pronounced, and the spin keeps for ever memory of its initial state. Generally speaking, as time goes on, the spatial density profile looks rather complex, as a consequence of the competition between drift and precession, and displays various shapes according to the ratio between the Larmor period and the characteristic time of flight. The density profile gradually changes from a multimodal quickly moving distribution when the scattering rate is small, to a unimodal standing but flattening distribution in the opposite case.

We study single-flip dynamics in sets of three-dimensional rhombus tilings with fixed polyhedral boundaries. This dynamics is likely to be slowed down by so-called 'cycles': such structures arise when tilings are encoded via the 'partition-on-tiling' method and are susceptible to break connectivity by flips or at least ergodicity, because they locally suppress a significant amount of flip degrees of freedom. We first address the so-far open question of the connectivity of tiling sets by elementary flips. We prove exactly that sets of tilings of codimension 1 and 2 are connected for any dimension and tiling size. For higher codimension tilings of dimension 3, the answer depends on the precise choice of the edge orientations, which is a non-trivial issue. In most cases, we can prove connectivity despite the existence of cycles. In the few remaining cases, among which one is the icosahedral symmetry, the question remains open. We also study numerically flip-assisted diffusion to explore the possible effects of the previously mentioned cycles. Cycles do not seem to slow down significantly the dynamics, at least as far as self-diffusion is concerned.

The spectra of previously constructed auxiliary matrices for the six-vertex model at roots of unity are investigated for spin-chains of even and odd length. The two cases show remarkable differences. In particular, it is shown that for even roots of unity and an odd number of sites the eigenvalues contain two linear independent solutions to Baxter's TQ-equation corresponding to the Bethe ansatz equations above and below the equator. In contrast, one finds for even spin-chains only one linear independent solution and complete strings. The other main result is the proof of a previous conjecture on the degeneracies of the six-vertex model at roots of unity. The proof rests on the derivation of a functional equation for the auxiliary matrices which is closely related to a functional equation for the eight-vertex model conjectured by Fabricius and McCoy.

It is considered that the cycle of a quantum heat engine using many non-interacting spin-1/2 systems as the working substance is composed of two adiabatic and two isomagnetic field processes and is referred to as a spin quantum Brayton engine cycle. Based on the quantum master equation and semi-group approach, expressions for the efficiency and power output of the cycle are derived. By using numerical solutions, the power output of the heat engine subject to finite cycle duration is optimized. The maximum power output and the corresponding parameters are calculated numerically. The optimal region of the efficiency and the optimal ranges of temperatures of the working substance and times spent on the two isomagnetic field processes are determined, so that the general optimum performance characteristics of the cycle are revealed. Moreover, the optimal performance of the cycle in the high-temperature limit is also analysed in detail. The results obtained here are further generalized, so that they may be directly used to describe the performance of a quantum Brayton heat engine using spin-J systems as the working substance.

In this paper, a novel analytical approach to heat conduction in a composite slab subject to periodic temperature changes has been developed. Taking advantage of the periodic properties of boundary changes, the corresponding analytical solution is obtained and expressed explicitly. Unlike most of the traditional methods, it involves no residue evaluation and no iterative computation such as a numerical search for eigenvalues. The adopted method is simple and elegant. The physical parameters are clearly shown in the mathematical formula of the solution. Furthermore, comparison of the method with numerical calculations demonstrates the applicability and accuracy of the method.

Searches through biological databases provide the primary motivation for studying sequence alignment statistics. Other motivations include physical models of annealing processes or mathematical similarities to, e.g., first-passage percolation and interacting particle systems. Here, we investigate sequence alignment statistics, partly to explore two general mathematical methods. First, we model the global alignment of random sequences heuristically with Markov additive processes. In sequence alignment, the heuristic suggests a numerical acceleration scheme for simulating an important asymptotic parameter (the Gumbel scale parameter λ). The heuristic might apply to similar mathematical theories. Second, we extract the asymptotic parameter λ from simulation data with the statistical technique of robust regression. Robust regression is admirably suited to 'asymptotic regression' and deserves to be better known for it.

We study the mean time for a random walk to traverse between two arbitrary sites of the Erdős–Renyi random graph. We develop an effective medium approximation that predicts that the mean first-passage time between pairs of nodes, as well as all moments of this first-passage time, are insensitive to the fraction p of occupied links. This prediction qualitatively agrees with numerical simulations away from the percolation threshold. Near the percolation threshold, the statistically meaningful quantity is the mean transit rate, namely, the inverse of the first-passage time. This rate varies non-monotonically with p near the percolation transition. Much of this behaviour can be understood by simple heuristic arguments.

In high-T_c superconductors dopant atoms supply holes or excess electrons. Electric conduction happens in the neighbourhood of dopants within a circle several lattice constants wide. Percolation of these conducting areas leads to global conduction. Diffusing d-electrons in these areas can destroy antiferromagnetism: the Néel temperatures decrease with doping. Based on an Ising model with antiferromagnetic interactions acting in the part of the lattice not covered by conducting areas, the specific heat, the staggered susceptibility and the spin correlation lengths show very broad peaks even for low dopant concentrations. In doped cuprates, due to the small size granularity comparable with the sizes of our simulated systems, possible peak height singularities are always suppressed.

We consider the trapping reaction, A + B → B, where A and B particles have a diffusive dynamics characterized by diffusion constants D_A and D_B. The interaction with B particles can be formally incorporated in an effective dynamics for one A particle as was recently shown by Bray et al (2003 Phys. Rev. E 67 060102). We use this method to compute, in space dimension d = 1, the asymptotic behaviour of the spatial fluctuation, ⟨z²(t)⟩^1/2, for a surviving A particle in the perturbative regime, D_A/D_B ≪ 1, for the case of an initially uniform distribution of B particles. We show that, for t ≫ 1, ⟨z²(t)⟩^1/2 ∝ t^ϕ with ϕ = 1/4. By contrast, the fluctuations of paths constrained to return to their starting point at time t grow with the larger exponent 1/3. Numerical tests are consistent with these predictions.

The effect of internal noise on phase synchronization of two coupled chemical chaotic oscillators is investigated numerically using chemical Langevin equations. It is found that internal noise can enhance the phase synchronization, and there exists an optimal internal noise level such that the best phase synchronization is achieved. Since the magnitude of the internal noise is changed via the variation of the system size, these phenomena also indicate the existence of an optimal system size.

We argue that one can factorize the difference equation of hypergeometric type on non-uniform lattices in the general case. It is shown that in the most cases of q-linear spectrum of the eigenvalues, this directly leads to the dynamical symmetry algebra su_q(1, 1), whose generators are explicitly constructed in terms of the difference operators, obtained in the process of factorization. Thus all models with the q-linear spectrum (some of them, but not all, previously considered in a number of publications) can be treated in a unified form.

In the framework of geometrical optics we consider the inverse problem consisting of obtaining refractive indexes n = n(x, y) of a two-dimensional transparent heterogeneous isotropic (dispersive or not) medium from a known (observed or given) family f(x, y) = c₀ of planar light rays of a definite colour. We establish a first-order linear partial differential equation relating the assigned family of light rays with all possible refractive indexes compatible with this family. Using this equation we derive certain criteria to check whether a given family of rays can be traced in the presence of a refractive index, which we assume in advance to be either radial or homogeneous of any degree m. We give appropriate examples for the two special cases and also an example for the general case.

A criterion is formulated for existence and another for the non-existence of complex eigenvalues for a class of non-self-adjoint operators in Hilbert space invariant under a particular discrete symmetry. Applications to the PT-symmetric Schrödinger operators are discussed.

A symmetric (2+1)-dimensional Lotka–Volterra equation is proposed. By means of a dependent variable transformation, the equation is firstly transformed into multilinear form and further decoupled into bilinear form by introducing auxiliary independent variables. A bilinear Bäcklund transformation is found and then the corresponding Lax pair is derived. Explicit solutions expressed in terms of pfaffian solutions of the bilinear form of the symmetric (2+1)-dimensional Lotka–Volterra equation are given. As a special case of the pfaffian solutions, we obtain soliton solutions and dromions.

We use a novel parametrization of the flowing Hamiltonian to show that the flow equations based on continuous unitary transformations, as proposed by Wegner, can be implemented through a nonlinear partial differential equation involving one flow parameter and two system specific auxiliary variables. The implementation is non-perturbative as the partial differential equation involves a systematic expansion in fluctuations, controlled by the size of the system, rather than the coupling constant. The method is applied to the Lipkin model to construct a mapping which maps the non-interacting spectrum onto the interacting spectrum to a very high accuracy. This function is universal in the sense that the full spectrum for any (large) number of particles can be obtained from it. In a similar way expectation values for a large class of operators can be obtained, which also makes it possible to probe the structure of the eigenstates.

The Laplace operator ∇² on N-dimensional Euclidean space in spherical coordinates is . The free-particle Schrödinger time evolution operator may be constructed by exponentiation, . Denoting a central finite difference approximation of ▵_r by , the matrix , with , is investigated and explicitly evaluated for N = 3. provides an approximation of the leading term of the radial component of the kinetic energy contribution to the evolution operator. An unconditionally stable numerical algorithm for quantum mechanical scattering is proposed based on this approximation when N = 3.

Transmission and reflection of a quantal particle by a single-hump potential barrier are analysed by means of an amplitude-phase decomposition of the wavefunction on both sides of the barrier. The amplitude-phase analysis of the wavefunction provides a particular invariant of the Ermakov–Lewis type, which originates in the matching process. The transmission and reflection coefficients turn out to be simple functions of this invariant. Numerical calculations of the invariant for an Eckart–Epstein potential barrier provide very accurate results.

We re-examine in detail a canonical quantization method à la Gupta–Bleuler in which the Fock space is built over a so-called Krein space. This method has already been successfully applied to the massless minimally coupled scalar field in de Sitter spacetime for which it preserves covariance. Here, it is formulated in a more general context. An interesting feature of the theory is that, although the field is obtained by canonical quantization, it is independent of Bogoliubov transformations. Moreover, no infinite term appears in the computation of T^μν mean values and the vacuum energy of the free field vanishes: ⟨0∣T⁰⁰∣0⟩ = 0. We also investigate the behaviour of the Krein quantization in Minkowski space for a theory with interaction. We show that one can recover the usual theory with the exception that the vacuum energy of the free theory is zero.

We analyse local field redefinition and duality for gauge field theories in three dimensions. We find that both Maxwell–Chern–Simons and the self-dual models admit the same field redefinition. Maxwell–Proca action and its dual also share this property. We show explicitly that a gauge-fixing term has no influence on duality and field redefinition.

A linear theory is developed for the temporal instability of a viscous liquid microjet of Newtonian fluid with a spatially periodic variation of surface tension imposed along its length. The variation of surface tension induces Marangoni flow within the jet that leads to breakup and drop formation. An analytical expression is derived for the behaviour of the free surface of the microjet. This expression is useful for parametric analysis of jet instability and breakup as a function of jet radius, wavelength and fluid properties.

Using techniques of effective field theory, we consider the thermodynamical properties of a dilute two-dimensional plasma interacting via a 1/r potential. The first one-loop correction to the partition function is already logarithmically divergent in the effective theory. The finite part of the one-loop induced couplings is then explicitly evaluated by matching the density–density correlator in the effective theory and in the full quantum theory. This task is performed from the formulation of the Coulomb problem in momentum space by projecting the two-dimensional momentum space onto the surface of a three-dimensional sphere. We also report some higher order results that, in the case of the one-component plasma immersed in a uniform neutralizing background, are resummed to obtain the complete leading-log partition function.