From finite to infinite crystals: Some electronic properties of BCC clusters

Using results obtained within the framework of the tight binding nearest neighbour approximation (Bílek O.,Skála L., Czech. J. Phys.B 28 (1978), 1003) the analytical formulae for the band width, binding energy per atom, charge orders and bond orders of bcc clusters having a rectangular parallelepiped geometry are given. Numerical results for the number of atoms in the cluster in the range 9-2×10⁶ illustrate the convergence of these properties. The asymptotical form of the formulae yields further detailed information. The band width and bond orders converge more quickly than the binding energy while the charge orders are equal to unity independently of the size of the cluster.One of the authors (L. K.) should like to thank the Faculty of Mathematics and Physics of Charles University in Prague for its kind hospitality.     

From finite to infinite volumes: removal of boundaries in diffuse wave imaging

In this Letter, we present a method that removes the contribution of the boundaries on the measurements from highly scattering media, transforming the signals captured from a bounded medium to measurements that would have been obtained if no boundary were present. This approach opens new possibilities in tomographic imaging in diffuse media as it eliminates the need for explicitly modeling boundaries and significantly simplifies reconstruction requirements.     

Superradiance of polaritons in finite one-dimensional crystals of increasing length: Passage from a dimer to an infinite crystal

Solutions of the dispersion equations for polariton states in finite one-dimensional crystals of arbitrary length are obtained. The appearance and evolution of the radiative and nonradiative polariton branches are traced as the length varies from two monomers to limiting values, above which the spectrum no longer undergoes significant changes. The dependences of the frequencies and radiative widths on the polariton wave vector are found for various orientations of the dipole moment of the quantum transition. The evolution of superradiance as the length of the crystal increases is traced. Some previously unknown significant features of the polariton spectrum are noted particularly the damping of the branch traditionally termed nonradiative as a consequence of emission from the end faces. Fiz. Tverd. Tela (St. Petersburg) 40, 2136–2140 (November 1998)     

Exact solution of finite size Mean Field Percolation and application to nuclear fragmentation

Random Graphs and Mean Field Percolation are two names given to the most general mathematical model of systems composed of a set of connected entities. It has many applications in the study of real life networks as well as physical systems. The model shows a precisely described phase transition, but its solution for finite systems was yet unresolved. However, atomic nuclei, as well as other mesoscopic objects (e.g. molecules, nano-structures), cannot be considered as infinite and their fragmentation does not necessarily occur close to the transition point. Here, we derive for the first time the exact solution of Mean Field Percolation for systems of any size, as well as provide important information on the internal structure of Random Graphs. We show how these equations can be used as a basis to select non-trivial correlations in systems and thus to provide evidence for physical phenomena.     

Analysis of arbitrary index profile planar optical waveguides and multilayer nonlinear structures: a simple finite difference algorithm

we present here a simple numerical method to obtain the mode effective indices as well as field distributions of modes of any arbitrary profile planar optical waveguide. The method is based on the solutions of scalar and semivectorial Helmoltz’s equation by finite difference algorithm and devised with a field convergence technique. This approach is quite general and can be applied straightforwardly to calculate the guided as well as quasi- or leaky modes of any arbitrary planar structure without the need to solve any eigenvalue equation or any complex matrix formalism. Besides providing the ease of application, the algorithm is particularly useful for waveguides with any graded index profile or with irregular multilayered structure and multilayered waveguides with a localised arbitrary nonlinear film. The performance of our method is verified against typical problems with analytical solutions or having results known otherwise, and is shown to yield results with good accuracy.     

From the eden model to the kinetic growth walk: A generalized growth model with a finite lifetime of growth sites

We propose a new class of cluster growth models where growth sites have a finite lifetime , which contains as special cases the Eden model ( = ) and the kinetic growth walk ( = 1). For finite but large values the growth process can be characterized by a crossover timet _X; for times belowt _X an Eden-type cluster is formed, while for times abovet _X the growth process belongs to the universality class of the self-avoiding random walk. The crossover time increases monotonically with . We develop a scaling theory for the time evolution of the mean end-to-end distance between the seed and the last-added site, and for the average number of growth sites by which the kinetics of the growth process can be characterized. We test this scaling theory by extensive Monte Carlo simulations. We also extend our results to inhomogeneous media (percolation systems).     

Semiempirical model calculations for chemisorption on transition metals: I. Exact solution and discussion of a simple model

A specially parameterized model Hamiltonian for chemisorption is proposed, which treats the interaction between adsorbate and metal electrons in some detail. The exact solution is obtained for the case of an electron band with zero width. It is identical with the Hartree-Fock solution. There are no relaxation effects on the ionization energies within this model. A formalism for performing semiempirical calculations is shown to yield a good approximate solution to the model Hamiltonian for the simple case treated in this paper. Qualitative features of wavefunctions, charge transfer, adsorption energy and ionization energies are discussed.     

The exact solution of the mutator model with directed mutations,linear fitness function,and finite genome length

In this study, we considered the model by Beckman and Loeb [Proc. Natl. Acad. Sci. U.S.A. 103 (2006) 14140] for the mutator phenomena. We construct an infinite population Crow-Kimura model with a mutator gene, directed mutations, a linear fitness function, and a finite genome length. We solved analytically the dynamics of the model using the generating function method. Such models provide realistic predictions for finite population sizes and have been widely discussed recently. The analytical formulas provided can be used to calculate the advantage of the mutator mechanism for the accumulation of mutations in cancer biology.     

Dynamics and thermodynamics of a simple model similar to self-gravitating systems: the HMF model

We discuss the dynamics and thermodynamics of the Hamiltonian Mean Field model (HMF) which is a prototypical system with long-range interactions. The HMF model can be seen as the one Fourier component of a one-dimensional self-gravitating system. Interestingly, it exhibits many features of real self-gravitating systems (violent relaxation, persistence of metaequilibrium states, slow collisional dynamics, phase transitions,...) while avoiding complicated problems posed by the singularity of the gravitational potential at short distances and by the absence of a large-scale confinement. We stress the deep analogy between the HMF model and self-gravitating systems by developing a complete parallel between these two systems. This allows us to apply many technics introduced in plasma physics and astrophysics to a new problem and to see how the results depend on the dimension of space and on the form of the potential of interaction. This comparative study brings new light in the statistical mechanics of self-gravitating systems. We also mention simple astrophysical applications of the HMF model in relation with the formation of bars in spiral galaxies.     

Surface diffusion of Pb clusters on Cu(1 1 1) influenced by size dependent cluster-substrate misfit: Monte Carlo tight binding simulation model

The migration of two-dimensional Pb clusters on Cu(1 1 1) surface is studied in the framework of three-dimensional (3D) continuum space tight binding (TB) computational model. Monte Carlo simulations based on many-body TB interactions reveal significant influence of cluster-substrate misfit on diffusion behavior of two-dimensional (2D) clusters. The analysis of pair distribution function (PDF) demonstrates that cluster lattice constant depends on the number of atoms N for 1 < N < 10. The observed effect is more pronounced for heteroepitaxial than homoepitaxial systems. It can be explained in the framework of model accounting for enhanced charge transfer in heteroepitaxy and strong influence of surface potential on cluster atomic arrangement. The variation of lattice constant leads to variable misfit which affects the island migration behavior. The results are discussed in a physical model that implies cluster diffusion with size dependent cluster-substrate misfit.     

Contribution of particle-particle,hole-hole and particle-hole ring diagrams to the binding energies of finite nuclei

Two classes of diagrams, namely particle-particle, hole-hole (pp, hh) and particle-hole (ph) ring diagrams are summed for the nuclei ¹⁶O and ⁴⁰Ca, and their contributions to the ground-state energy shift ΔE of these nuclei is calculated. We find that hh and mixed diagrams (involving both pp and hh interactions) are not less important than the usual pp ladder diagrams which are summed in the standard Brueckner approach. We also study the convergence of these two classes of diagrams as the dimension of the model space involved is increased, and as a function of the residual interaction used. In evaluating these diagrams a transition-amplitude method is used. This is compared to the quasi-boson correlation expression for the ground-state energy due to particle-hole excitations and to an analogous correlation expression resulting from particle-particle and hole-hole excitations. Additionally we derive expressions for, and evaluate a subclass of these diagrams namely “TDA” ring diagrams, where unlike the usual pp, hh and ph diagrams, backward-folding graphs are excluded. We find that the backward-folding graphs are negligible for pp, hh ring diagrams and small for ph graphs. In the smallest model space considered for ⁴⁰Ca we also obtained the TDA ring diagram contributions via matrix inversion techniques which additionally allow us to study the relative importance of ph exchange graphs neglected in the ring-diagram formalism, and of cross TDA diagrams (i.e. TDA ring diagrams where both pp, hh, and ph interactions are allowed). Finally we study the uncertainties spurious effects introduce in ring-diagram calculations.     

Scaling and renormalization group in replica-symmetry-breaking space: evidence for a simple analytical solution of the Sherrington-Kirkpatrick model at zero temperature

Using numerical self-consistent solutions of a sequence of finite replica symmetry breakings (RSB) and Wilson's renormalization group but with the number of RSB steps playing a role of decimation scales, we report evidence for a nontrivial T-->0 limit of the Parisi order function q(x) for the Sherrington-Kirkpatrick spin glass. Supported by scaling in RSB space, the fixed point order function is conjectured to be q*(a)=sqrt[pi]/2 a/xi erf(xi/a) on 0 a at T =0 and xi approximately 1.13+/-0.01. Xi plays the role of a correlation length in a-space. q*(a) may be viewed as the solution of an effective 1D field theory.     

Quantum theory of sticking: equivalence of various approaches and application to a simple model

The basic formulae of four different quantum mechanical approaches to the calculation of the sticking coefficient of an atom on a cold solid are re-examined in order to discuss the connections between them. All approaches pursued exactly give the same result, and this is exhibited for a one-dimensional model which is exactly solvable to provide closed-form formulae for the sticking coefficient. Numerical results, which reproduce the qualitative behaviour of atomic sticking, are presented for different assumed densities of substrate excitations responsible for the energy loss, including the physically important cases of low-energy single phonons, and low-energy electron-hole pairs in a metal.     

A geometric solution to the coincidence problem, and the size of the landscape as the origin of hierarchy

Weinberg's seminal prediction of the cosmological constant relied on a provisional method for regulating eternal inflation which has since been put aside. We show that a modern regulator, the causal patch, improves agreement with observation, removes many limiting assumptions, and yields additional powerful results. Without assuming necessary conditions for observers such as galaxies or entropy production, the causal patch measure predicts the coincidence of vacuum energy and present matter density. Their common scale, and thus the enormous size of the visible Universe, originates in the number of metastable vacua in the landscape.     

Bubble population phenomena in sonochemical reactor: II. Estimation of bubble size distribution and its number density by simple coalescence model calculation

A simple bubble population model, with emphasis on the bubble–bubble coalescence, is proposed. In this model, the bubble size distribution is simulated through the iteration of fundamental bubble population process: generation, dissolution, and coalescence. With this simple modelling, the bubble size distribution experimentally observed by the pulsed laser diffraction method and the void rate obtained by the capillary system at 443 kHz were successfully simulated. The experimental results on the bubble population growth by the repetitive pulsed sonication and the effect of pulse width on the bubble population were recreated by the numerical simulation in a semi-quantitative manner. The importance of coalescence of bubbles especially for the effect of addition of surfactant is demonstrated. By decreasing the coalescence frequency by one order of magnitude in the simulation, both the drastic decrease in the total bubble volume as well as the depression of bubble size distribution centring from a few tens of microns in water to a few microns in a dilute surfactant solution can be simultaneously derived.     

Observables in a noncommutative approach to the unification of quanta and gravity: a finite model

We further develop a noncommutative model unifying quantum mechanics and general relativity proposed in Gen. Rel. Grav. (36, 111–126 (2004)). Generalized symmetries of the model are defined by a groupoid given by the action of a finite group on a space E. The geometry of the model is constructed in terms of suitable (noncommutative) algebras on . We investigate observables of the model, especially its position and momentum observables. This is not a trivial thing since the model is based on a noncommutative geometry and has strong nonlocal properties. We show that, in the position representation of the model, the position observable is a coderivation of a corresponding coalgebra, coparallelly to the well-known fact that the momentum observable is a derivation of the algebra. We also study the momentum representation of the model. It turns out that, in the case of the algebra of smooth, quickly decreasing functions on , the model in its quantum sector is nonlocal, i.e., there are no nontrivial coderivations of the corresponding coalgebra, whereas in its gravity sector such coderivations do exist. They are investigated.This revised version was published online in April 2005. The publishing date was inserted.