Browsing by Browse by FOR 2008 "010506 Statistical Mechanics, Physical Combinatorics and Mathematical Aspects of Condensed Matter"

Triple-crystal X-ray diffractometry is widely used for the analysis of structural characteristics of single crystals and heteroepitaxial structures. Different theoretical approaches taking into account both coherent and diffuse scattering have been used to analyze the experimental data. Using the formalism of Kato statistical theory of dynamic diffraction, we developed and generalized a theoretical interpretation of different methods of diffractometry of various objects. The goal of this work is to generalize the statistical theory of X-ray diffraction for any structural defects, including both conventional defects (amorphous clusters, dislocation loops, mosaicity, et al.) and foreign nanostructures formed artificially in matrices, for example, quantum dots or quantum wires.

We propose a theory that qualitatively predicts the stability and equilibrium structure of long-lived, quasi-steady flow states in decaying two-dimensional turbulence. This theory combines a maximum entropy principal with a nonlinear parameterization of the vorticity-stream-function dependency of such long-lived states. In particular, this theory predicts unidirectional-flow states that are bistable, exhibit hysteresis, and undergo large abrupt changes in flow topology; and a vortex-pair state that undergoes continuous changes in flow topology. These qualitative predictions are confirmed in numerical simulations of the two-dimensional Navier-Stokes equation. We discuss limitations of the theory, and why a reduced quantitative theory of long-lived flow states is difficult to obtain. We also provide a partial theoretical justification for why certain sets of initial conditions go to certain long-lived flow states.

Silo honking is an acoustical emission with a fundamental frequency of several hundred Hertz and an intensity often greater than 100 dB. It occurs when a silo is discharging and is similar to the 'honk' of a lorry horn. The high amplitude of the honk makes it a significant noise pollution issue for workers at the site and for neighboring businesses and residents. This paper considers some possible excitation mechanisms that may be responsible for honking and presents measurements obtained from a full scale honking silo detailing the acoustic emissions and the associated vibration of the silo walls. Experimental results are presented which are comprised of simultaneous measurements of the three components of the wall vibrations and the acoustic pressure. The wall vibrations have an initial impulse response with a high amplitude O(100g) and subsequent oscillatory accelerations with amplitude O(10g). The frequency spectra of the acceleration and acoustic pressure measurements comprises a sharp peak at the fundamental acoustic frequency and a harmonic series of peaks at integer multiples of the fundamental frequency. It is shown that the honking is not generated by a resonance inside the silo, as in a flute or organ pipe; the sound is generated by the silo walls acting as large speakers. The interaction between the wall and the sliding pellets is considered as a possible excitation mechanism for the acoustic emissions. Laboratory friction measurements are presented using pellets from the honking silo and a wall sample. The results of these measurements show that the particles exhibit a slip-stick behavior when sheared against the wall material. This slip-stick behavior is characterized under different conditions for pellets that are known to produce honking. Particles that have not been observed to honk were also tested and did not produce slip-stick motion at the wall.

We prove that certain super-linear elliptic equations in two dimensions have many solutions when the diffusion is small. We find these solutions by constructing solutions with many sharp peaks. In three or more dimensions, this has already been proved by the authors in 'Comm. Partial Differential Equations' 30 (2005) 1331-1358. However, in two dimensions, the problem is much more difficult because there is no limit problem in the whole space. Therefore, the proof is quite different, though still a reduction argument. A direct consequence of this result is that we give a positive answer to the Lazer-McKenna conjecture for some typical nonlinearities in two dimensions.

Theoretical aspects of quantitative diffraction-enhanced imaging of weak objects are considered using the Fourier optics approach. The amplitude and phase transfer functions are introduced by analogy with the well-known case of in-line (holographic) imaging. The inverse problem of the reconstruction of the phase and amplitude of the incident wave from recorded images is solved in the case of non-absorbing objects and objects consisting of a single material and in the general case of objects with uncorrelated refraction and absorption characteristics. A comparison is given between the solutions to the inverse problem obtained using the new formalism and the geometric-optics approximation.

The rate of magnetization reversal due to the nucleation of soliton-antisoliton pairs at point-like defects is found for a uniaxial ferromagnet in an applied magnetic field. Point-like defects are considered as local variations in the magnetic anisotropy over a length scale smaller than the domain-wall width. A weak magnetic field applied along the easy axis causes the magnetization to become metastable, and the lowest activation barrier for reversal involves the nucleation of a soliton-antisoliton pair pinned to a point-like defect. Formulas are derived for the activation energy and field of reversal, and the reversal-rate prefactor is calculated using Langer's theory for the decay of a metastable state. As the applied field tends to zero, the lowest activation energy is found to be exactly half that of an unpinned soliton-antisoliton pair, and results from the formation of a spatially nonuniform metastable state when the defect strength become large. The smallest field of reversal is exactly half of the anisotropy field. The reversal-rate prefactor is found to increase with the number of point-like defects but decreases with increase in the defect strength due to a decrease in the activation entropy when translational symmetry is broken by the point-like defects, and soliton-antisoliton pairs become more strongly localized to the pinning sites.

The two-dimensional Gabor function is adapted to natural image statistics, leading to a tractable probabilistic generative model that can be used to model simple cell receptive field profiles, or generate basis functions for sparse coding applications. Learning is found to be most pronounced in three Gabor function parameters representing the size and spatial frequency of the two-dimensional Gabor function and characterized by a nonuniform probability distribution with heavy tails. All three parameters are found to be strongly correlated, resulting in a basis of multiscale Gabor functions with similar aspect ratios and size-dependent spatial frequencies. A key finding is that the distribution of receptive-field sizes is scale invariant over a wide range of values, so there is no characteristic receptive field size selected by natural image statistics. The Gabor function aspect ratio is found to be approximately conserved by the learning rules and is therefore not well determined by natural image statistics. This allows for three distinct solutions: a basis of Gabor functions with sharp orientation resolution at the expense of spatial-frequency resolution, a basis of Gabor functions with sharp spatial-frequency resolution at the expense of orientation resolution, or a basis with unit aspect ratio. Arbitrary mixtures of all three cases are also possible. Two parameters controlling the shape of the marginal distributions in a probabilistic generative model fully account for all three solutions. The best-performing probabilistic generative model for sparse coding applications is found to be a gaussian copula with Pareto marginal probability density functions.