Electron microscopy study of aggregation of microclusters of sulphur

A study of aggregation of sulphur particles in colloidal suspension of sulphur in water-methanol mixture using TEM and electron diffraction is reported. From the micrographs the aggregates formed have been found to be random and tenuous indicating a fractal structure. The electron diffraction patterns of the aggregates are used to study the mechanism of diffusion and reaction limited aggregation.     

Critical Behavior of Two Interacting Linear Polymer Chains in a Good Solvent

A model of two interacting (chemically different) linear polymer chains is solved exactly using the real-space renormalization group transformation on a family of Sierpinski gasket type fractals and on a truncated 4-simplex lattice. The members of the family of the Sierpinski gasket-type fractals are characterized by an integer scale factorb which runs from 2 to ∞. The Hausdorff dimensiond _F of these fractals tends to 2 from below asb → ∞. We calculate the contact exponenty for the transition from the State of segregation to a State in which the two chains are entangled forb = 2-5. Using arguments based on the finite-size scaling theory, we show that forb→∞, y = 2 - v(b) d _F, wherev is the end-toend distance exponent of a chain. For a truncated 4-simplex lattice it is shown that the system of two chains either remains in a State in which these chains are intermingled in such a way that they cannot be told apart, in the sense that the chemical difference between the polymer chains completely drop out of the thermodynamics of the system, or in a State in which they are either zipped or entangled. We show the region of existence of these different phases separated by tricritical lines. The value of the contact exponenty is calculated at the tricritical points.     

Harmonic mappings of the Sierpinski gasket to the circle

Harmonic mappings from the Sierpinski gasket to the circle are described explicitly in terms of boundary values and topological data. In particular, all such mappings minimize energy within a given homotopy class. Explicit formulas are also given for the energy of the mapping and its normal derivatives at boundary points.

     

Computer Simulation of the Structure of Porous Electrodes with an Immobilized Enzyme and Nanosized Support Particles

The efficiency of the operation of a porous electrode with an immobilized enzyme is defined, in particular, by a lucky structure of its active layer, which can contain nanosized particles of the support. The composites of such a kind are prepared with the aid of methods of colloidal chemistry. The aim of this particular investigation is to perform a computer simulation of processes of coagulation of particles of the support and their possible heterocoagulation with molecules of the enzyme. Algorithms of the formation of nanocomposite structures in solution are suggested. Calculations show that the concentration of the enzyme molecules in the nanocomposite structures cannot exceed a certain critical value. On the other hand, at a fixed value of the concentration of the enzyme molecules, the concentration of the support particles must not fall below a certain threshold quantity, which provides for the passing of current through the active layer. In order for all the enzyme molecules, rather than for a fraction of these, in the composite to take part in the process of bioelectrocatalysis, the concentration of support particles must be increased even higher, to an optimum value.__________Translated from Elektrokhimiya, Vol. 41, No. 6, 2005, pp. 738–747.Original Russian Text Copyright © 2005 by Chirkov, Rostokin.     

Quantum chaos and fractals with atoms in cavities

We study the coupled translational, electronic, and field dynamics of the combined system “a two-level atom + a single-mode quantized field + a standing-wave ideal cavity”. In the semiclassical approximation with a point-like atom, interacting with the classical field, the dynamics is described by the Heisenberg equations for the atomic and field expectation values which are known to produce semiclassical chaos under appropriate conditions. We derive Hamilton–Schrödinger equations for probability amplitudes and averaged position and momentum of a point-like atom interacting with the quantized field in a standing-wave cavity. They constitute, in general, an infinite-dimensional set of equations with an infinite number of integrals of motion which may be reduced to a dynamical system with four degrees of freedom if the quantized field is supposed to be initially prepared in a Fock state. This system is found to produce semiquantum chaos with positive values of the maximal Lyapunov exponent. At exact resonance, the semiquantum dynamics is regular. At large values of detuning |δ|1, the Rabi atomic oscillations are usually shallow, and the dynamics is found to be almost regular. The Doppler–Rabi resonance, deep Rabi oscillations that may occur at any large value of |δ| to be equal to |αp₀|, is found numerically and described analytically (with α to be the normalized recoil frequency and p₀ the initial atomic momentum). Two gedanken experiments are proposed to detect manifestations of semiquantum chaos in real experiments. It is shown that in the chaotic regime values of the population inversion z_out, measured with atoms after transversing a cavity, are so sensitive to small changes in the initial inversion z_in that the probability of detecting any value of z_out in the admissible interval [−1,1] becomes almost unity in a short time. Chaotic wandering of a two-level atom in a quantized Fock field is shown to be fractal. Fractal-like structures, typical with chaotic scattering, are numerically found in the dependence of the time of exit of atoms from the cavity on their initial momenta.     

Wavelet transforms and order-two densities of fractals

We highlight a correspondence between order-two densities and wavelet-like transforms of certain fractal measures. We use a variant of the ergodic theorem to demonstrate that these densities and transforms are well-behaved for a large class of quasi-self-similar fractals. We show that parallel ideas can be used to study the local behavior of certain fractal functions.     

Propagation and trapping of excitations on percolation clusters

A study is presented of migration of optical or magnetic excitations on percolation clusters which terminates upon reaching a trapping site. The theory is based on the extension of results from the theory of random walks to systems without translational invariance, together with the use of scaling concepts. For the case of an excitation which resides on one type of atom in a randomly mixed crystal near the percolation threshold, new power laws for the time and concentration dependences of the mean number of sites visited at timet of the kinetics of arrival at traps are obtained. Some of these results are also tested for the first time by numerical simulations.     

Quantum Fractals on <Emphasis Type="Italic">n</Emphasis>–Spheres. Clifford Algebra Approach

Using the Clifford algebra formalism we extend the quantum jumps algorithm of the Event Enhanced Quantum Theory (EEQT) to convex state figures other than those stemming from convex hulls of complex projective spaces that form the basis for the standard quantum theory. We study quantum jumps on n-dimensional spheres, jumps that are induced by symmetric configurations of non-commuting state monitoring detectors. The detectors cause quantum jumps via geometrically induced conformal maps (M?bius transformations) and realize iterated function systems (IFS) with fractal attractors located on n-dimensional spheres. We also extend the formalism to mixed states, represented by “density matrices” in the standard formalism, (the n-balls), but such an extension does not lead to new results, as there is a natural mechanism of purification of states. As a numerical illustration we study quantum fractals on the circle (one-dimensional sphere and pentagon), two–sphere (octahedron), and on three-dimensional sphere (hypercubetesseract, 24 cell, 600 cell, and 120 cell). The attractor, and the invariant measure on the attractor, are approximated by the powers of the Markov operator. In the appendices we calculate the Radon-Nikodym derivative of the SO(n + 1) invariant measure on Sⁿ under SO(1, n + 1) transformations and discuss the Hamilton’s “icossian calculus” as well as its application to quaternionic realization of the binary icosahedral group that is at the basis of the 600 cell and its dual, the 120 cell. As a by-product of this work we obtain several Clifford algebraic results, such as a characterization of positive elements in a Clifford algebra as generalized Lorentz “spin–boosts”, and their action as M?bius transformation on n-sphere, and a decomposition of any element of Spin⁺(1, n + 1) into a spin–boost and a spin–rotation, including the explicit formula for the pullback of the SO(n + 1) invariant Riemannian metric with respect to the associated M?bius transformation.