Electron Transfer through a One-Dimensional Fractal Structure

An adequate model of electron tunneling through a self-similar fractal potential (SFP) defined on a Cantor set is extended to a generalized Cantor set. It is demonstrated that, as in a specific case, the Schrödinger equation for the SFP is reduced to a functional equation for the transfer matrix which admits solutions of three types. Two of them are single-parameter solutions corresponding to SFP barriers and lacunas with arbitrary powers. In both cases, the transfer matrices are nonanalytic in the long-wavelength region and have fractal dimensionalities there. The third solution type includes a unique solution corresponding to the SFP barrier with fixed power for a given barrier width. The corresponding transfer matrix is analytic at the point k = 0. It is shown that generally the SFP possesses only the property of approximate scale invariance on the generalized Cantor set in the long- and short-wavelength regions. Only the limiting SFP, whose fractal dimensionality is equal to unity, possesses the property of rigorous scale invariance irrespective of its power. It is shown that SFPs with identical fractal dimensionalities but different lacunas are described by different transfer matrices.     

Fractal spectra in generalized Fibonacci one-dimensional magnonic quasicrystals

In this work we carry out a theoretical analysis of the spectra of magnons in quasiperiodic magnonic crystals arranged in accordance with generalized Fibonacci sequences in the exchange regime, by using a model based on a transfer-matrix method together random-phase approximation (RPA). The generalized Fibonacci sequences are characterized by an irrational parameter σ(p,q)$\sigma (p,q)$, which rules the physical properties of the system. We discussed the magnonic fractal spectra for first three generalizations, i.e., silver, bronze and nickel mean. By varying the generation number, we have found that the fragmentation process of allowed bands makes possible the emergence of new allowed magnonic bulk bands in spectra regions that were magnonic band gaps before, such as which occurs in doped semiconductor devices. This interesting property arises in one-dimensional magnonic quasicrystals fabricated in accordance to quasiperiodic sequences, without the need to introduce some deferent atomic layer or defect in the system. We also make a qualitative and quantitative investigations on these magnonic spectra by analyzing the distribution and magnitude of allowed bulk bands in function of the generalized Fibonacci number F_n and as well as how they scale as a function of the number of generations of the sequences, respectively.     

On the Connections between the Non-linearity of Equations of Motion in Quantum Theory,Asymptotic Behavior and Renormalization

It is shown that the ordinary perturbation expressions used in quantum mechanics lead to the wrong asymptotic behavior of the Heisenberg observables as function of time. This difficulty is traced to the non-linearity of the Heisenberg equations of motion and is studied in the context of a one-dimensional non-linear oscillator problem. It is found that the correct asymptotic behavior can be obtained by a process of renormalization analogous to renormalization theory in quantum field theory. It turns out that the renormalized parameters analogous to mass and wave-function renormalization are not c-numbers but are instead q-numbers. It is suggested that the renormalization parameters of quantum field theory are also q-numbers.     

The Waring formula and fusion rings

The potential that generates the cohomology ring of the Grassmannian is given in terms of the elementary symmetric functions using the Waring formula that computes the power sum of roots of an algebraic equation in terms of its coefficients. As a consequence, the fusion potential for su(N)_K is obtained. This potential is the explicit Chebyshev polynomial in several variables of the first kind. We also derive the fusion potential for sp(N)_K from a reciprocal algebraic equation. This potential is identified with another Chebyshev polynomial in several variables. We display a connection between these fusion potentials and generalized Fibonacci and Lucas numbers. In the case of su(N)_K the generating function for the generalized Fibonacci numbers obtained are in agreement with Lascoux using the theory of symmetric functions. For sp(N)_K, however, the generalized Fibonacci numbers are obtained from new sequences.     

Applying the finite-thickness model to designs of cylindrical microlenses with small -numbers

The finite-thickness model (FTM) is applied to the design of cylindrical microlenses based on the wave-front interference principle, rather than the existing zero-thickness model (ZTM). This design method is very simple in physics and highly efficient in computations. For cylindrical lenses with different f-numbers (from f/1.5 to f/0.6), the detailed designs by using both the FTM and the ZTM are carried out. To show the superiority of the FTM to the ZTM, we investigate the focal performance of all the designed lenses based on rigorous electromagnetic theory and the boundary element method. Numerical results reveal that the constructed cylindrical lenses by the FTM are witnessed to exhibit much better focusing performance than those by the ZTM, especially for the small f-numbers.     

Fractal model of the atom and some properties of the matter through an extended model of scale relativity

The scale relativity model was extended for the motions on fractal curves of fractal dimension D _F and third order terms in the equation of motion of a complex speed field. It results that, in a fractal fluid, the convection, dissipation and dispersion are compensating at any scale (differentiable or non-differentiable), whereas a generalized Schrödinger type equation is obtained for an irrotational movement of the fractal fluid. For D _F = 2 and the dissipative approximation of the motions, the fractal model of atom is build: the real part of the complex speed field describes the electron motion on stationary orbits according to a quantification condition, while the imaginary part of the complex speed field gives the electron energy quantification. For D _F = 3 and the dispersive approximation of motions, some properties of the matter are explained: at the differentiable scale the flowing regimes (non-quasi-autonomous and quasi-autonomous) of the fractal fluids are separated by the experimental “0.7 structure”, while for the non-differentiable scale the fractal potential acts as an energy accumulator and controls through coherence the transport phenomena. Moreover, the compatibility between the differentiable and non-differentiable scales implies a Cantor space-time, and consequently a fractal at any scale. Thus, some properties of the matter (the anomaly of nano-fluids thermal conductivity, the superconductivity etc.) can be explained by this model.     

Generalized Forms, Connections, and Gauge Theories

Generalized differential forms of type N = 2, and flat generalized connections are used to describe the SO(p, q) form of Cartan's structure equations for metric geometries, source-free Yang-Mills fields, and the Einstein–Yang-Mills equations in four dimensions. Maxwell's equations for type N = 2 forms are also constructed.     

Transfer Matrix for the Cantor Staircase Fractal Potential

The precision analytical method developed previously for studying nonrelativistic particle tunneling through the self-similar fractal potential is used to derive the transfer matrix for the Cantor staircase fractal potential. A generalized functional equation for the transfer matrix is derived and a solution is retrieved for the case in which the fractal dimensionality of the corresponding Cantor set is close to unity. The tunneling parameters are calculated including the transmission coefficient and phases of scattered waves.     

Transport properties of Fibonacci heterostructures: a nonparabolic approach

A fourth order hamiltonian is used to explore transport properties of semiconductor Fibonacci heterostructures. The tunneling current and time delay are obtained for different Fibonacci sequences constructed withGaAsandAl_xGa_1 − xAs. Energy minibands are calculated to study the fractal dimension and critical electronic states in quasi-periodic arrays. Results show that nonparabolic corrections produce changes in the tunneling current, time delay and fractal dimension, and a low voltage shift of the current peaks compared with the parabolic theory. The electronic states preserve their critical nature in the presence of nonparabolic effects.     

Effects of ramification and connectivity degree on site percolation threshold on regular lattices and fractal networks

This Letter is focused on the impact of network topology on the site percolation. Specifically, we study how the site percolation threshold depends on the network dimensions (topological d and fractal D), degree of connectivity (quantified by the mean coordination number $Z_{\infty }$), and arrangement of bonds (characterized by the connectivity index Q also called the ramification exponent). Using the Fisher's containment principle, we established exact inequalities between percolation thresholds on fractal networks contained in the square lattice. The values of site percolation thresholds on some fractal lattices were found by numerical simulations. Our findings suggest that the most relevant parameters to describe properly the values of site percolation thresholds on fractal networks contained in square lattice (Sierpiński carpets and Cantor tartans) and based on the square lattice (weighted planar stochastic fractal and Cantor lattices) are the mean coordination number and ramification exponent, but not the fractal dimension. Accordingly, we propose an empirical formula providing a good approximation for the site percolation thresholds on these networks. We also put forward an empirical formula for the site percolation thresholds on d-dimensional simple hypercubic lattices.     

The influence of fractality on the time evolution of the diffusion process

In the literature, the deviations from standard behaviors of the solutions of the kinetic equation and the analogous diffusion equation are put forward by investigations which are carried out in the frame of fractional mathematics and nonextensive physics. On the other hand, the physical origins of the order of derivative namely α in fractional mathematics and the entropy index q in nonextensive physics are a topic of interest in scientific media. In this study, the solutions of the diffusion equation which have been obtained in the framework of fractional mathematics and nonextensive physics are revised. The diffusion equation is solved by the cumulative diminuation/growth method which has been developed by two of the present authors and physical nature of the parameters α and q are enlightened in connection with fractality of space and the memory effect. It has been emphasized that the mathematical basis of deviations from standard behavior in the distribution functions could be established by fractional mathematics where as the physical mechanism could be revealed using the cumulative diminuation/growth method.     

A dissipative network model with neighboring activation

We study the properties of spectrum and eigenstates of the Google matrix of a directed network formed by the procedure calls in the Linux Kernel. Our results obtained for various versions of the Linux Kernel show that the spectrum is characterized by the fractal Weyl law established recently for systems of quantum chaotic scattering and the Perron-Frobenius operators of dynamical maps. The fractal Weyl exponent is found to be ν ≈ 0.65 that corresponds to the fractal dimension of the network d ≈ 1.3. An independent computation of the fractal dimension by the cluster growing method, generalized for directed networks, gives a close value d ≈ 1.4. The eigenmodes of the Google matrix of Linux Kernel are localized on certain principal nodes. We argue that the fractal Weyl law should be generic for directed networks with the fractal dimension d < 2.     

On the phase transition to sheet percolation in random Cantor sets

Thed-dimensional random Cantor set is a generalization of the classical middle-thirds Cantor set. Starting with the unit cube [0, 1] ^d , at every stage of the construction we divide each cube remaining intoM ^d equal subcubes, and select each of these at random with probabilityp. The resulting limit set is a random fractal, which may be crossed by paths or (d–1)-dimensional sheets. We examine the critical probabilityp _s(M, d) marking the existence of these sheet crossings, and show that p_s(M,d)1–p_c(M ^d) asM, where p_c(M ^d) is the critical probability of site percolation on the lattice (M ^d) obtained by adding the diagonal edges to the hypercubic lattice ^d. This result is then used to show that, at least for sufficiently large values ofM, the phases corresponding to the existence of path and sheet crossings are distinct.     

Scaling behavior of surface irregularity in the molecular domain: From adsorption studies to fractal catalysts

For an unexpected variety of solids, the surface topography from a few up to as many as a thousand angstroms is very well described by fractal dimension,D. This follows from measurements of the number of molecules in surface monolayers, as function of adsorbate or adsorbent particle size. As an illustration, we present a first case, amorphous silica gel, whereD has been measured independently by each of the two methods. (The agreement, 3.02±0.06 and 3.04±0.05, is excellent, and the result is modeled by a heavy generalized Menger sponge.) The examples as a whole divide into amorphous and crystalline materials, but presumably all of them are to be modeled as random fractal surfaces. The observedD values exhaust the whole range between 2 and 3, suggesting that there are a number of different mechanisms by which such statistically self-similar surfaces form. We show that fractal surface dimension entails interfacial power laws much beyond what is the source of theseD values. Examples are reactive scattering events when neutrons of variable flux pass the surface (this is of interest for locating fractal substrates that may support adlayer phase transitions); the rate of diffusion-controlled chemical reactions at fractal surfaces; and the fractal implementation of the traditional idea that the active sites of a catalyst are edge and apex sites on the surface.     

Space-charge analysis for the admittance of semiconductor junctions with deep impurity levels

The space charge analysis within depletion layers in semiconductors containing deep trap levels is reconsidered. A simple approach to the frequency dependence of the admittance ofp ⁺/n junctions is properly generalized in order to deal with the effect of interface states at heterojunctions and Schottky barriers, as well as with a special case for the space distribution of the trap density. The density of interface states atp-Ge/CdS heterojunctions is so derived. Approximate analytical solutions accounting for a spatially distributed time constant are obtained for the admittance ofp ⁺/n junctions with a single trap state. A comparison with experimental data is given and discussed.     

An exact renormalization-group approach for local phonon properties of single-atom and double-atom generalized Fibonacci systems

We present an exact renormalization-group approach to study the local phonon properties of the double-atom generalized Fibonacci systems in which the masses and spring constants are all associated with the aperiodic sequences constructed by the inflation rule: {A, B}{A ⁿ B^m, A}.n(m+1) basic transformations and 2n(n+m–1)+1 basic decimations are introduced. By applying the combinations of derived transformations and decimations, we can determine exactly the renormalized local environment, up to infinite order, of any given site in the double-atom systems. Both the single-atom and double-atom phonon models are employed, and the local density of states in several sites of some generalized Fibonacci systems are numerically calculated.     

Synthesis, structure, and dielectric properties of KH2PO4 fractal aggregates

Macroscopic fractal aggregates of KH₂PH₄ (KDP) measuring up to 500 μm have been obtained. The fractal structure forms as a result of the precipitation of KDP particles from a supersaturated aqueous solution in the presence of a temperature gradient followed by a diffusioncontrolled mechanism of aggregation. The electron-microscopic analysis performed has shown that the fractals are formed predominantly from crystallites of the tetragonal modification measuring ∼1 μm. The dielectric constant (ɛ) of fractal KH₂PO₄ has been measured in the temperature range 80–300 K. A characteristic anomaly has been discovered on the ɛ(T) curve in the vicinity of 122 K, which attests to a ferroelectric phase transition. The absolute value of ɛ is significantly smaller than the components ɛ ₁₁ and ɛ ₃₃ for KH₂PO₄. Fiz. Tverd. Tela (St. Petersburg) 41, 2059–2061 (November 1999)     

p-adic quantum mechanics

An extension of the formalism of quantum mechanics to the case where the canonical variables are valued in a field ofp-adic numbers is considered. In particular the free particle and the harmonic oscillator are considered. In classicalp-adic mechanics we consider time as ap-adic variable and coordinates and momentum orp-adic or real. For the case ofp-adic coordinates and momentum quantum mechanics with complex amplitudes is constructed. It is shown that the Weyl representation is an adequate formulation in this case. For harmonic oscillator the evolution operator is constructed in an explicit form. For primesp of the form 4l+1 generalized vacuum states are constructed. The spectra of the evolution operator have been investigated. Thep-adic quantum mechanics is also formulated by means of probability measures over the space of generalized functions. This theory obeys an unusual property: the propagator of a massive particle has power decay at infinity, but no exponential one.     

Apollonian packings as physical fractals

The Apollonian packings (APs) of spheres are fractals that result from a space-filling procedure. We discuss the finite size effects for finite intervals s?∈?[s _min,?s _max] between the largest and the smallest sizes of the filling spheres. We derive a simple analytical generalization of the scale-free laws, which allows a quantitative study of such physical fractals. To test our result, a new efficient space-filling algorithm has been developed which generates random APs of spheres with a finite range of diameters: the correct asymptotic limit s _min/s _max?→?0 and the known APs' fractal dimensions are recovered and an excellent agreement with the generalized analytic laws is proved within the overall range of sizes.     

Fractal axicons

Cantor rings are rotational symmetric pupils that are generated from a Cantor set of a given level of growth. These pupils have certain fractal properties. For example, it is known that when illuminated by a general spherical wavefront they provide self-similar patterns at transverse planes in the Fraunhofer region. In this paper, we study the response of Cantor rings when illuminated by a Bessel light beam conforming what we call fractal axicons. It is shown that, with this kind of illumination a close replica of the radial profile of the pupil is obtained along the optical axis, i.e., we show that the axial behaviour of these pupils has self-similarity properties that can be correlated to those of the diffracting aperture. The influence of several construction parameters is numerically investigated.