Fractional Dimensionality Research Articles

The long and short of random media: Acoustic Anderson localization

Wave propagation in random media relates to a large number of important current problems in physics and acoustics. For long wavelengths the wave equation is locally approximated by the Laplace equation, and this together with the complex boundary conditions provided by the random medium describes many interesting phenomena involving percolation, fractional dimensionality, etc. An important puzzle in the long wavelength regime was solved just recently; a homework problem once assigned by I. Rudnick provided the key to the solution. In the short wavelength regime, where wavelengths are on the order of the variations in the random medium, one may observe effects of Anderson localization, which is currently receiving much attention in the study of quantum phenomena in disordered solids. Recent experiments on acoustic Anderson localization, including nonlinear effects, will be reported. [Work supported by ONR.]

Solitons in a system with broken translational invariance: Anomaly in the specific heat of the dilute [formula omitted] anisotropic XY chain

The thermodynamic properties of the dilute S = 1 2 anisotropic XY chain are calculated. At low temperatures the dominant contribution comes from a low energy “complex mode”, which assumes zero energy when translational symmetry is restored in the pure system. The specific heat shows the anomalous behaviour C( T) ∝ T d where d is a fractional spectral dimensionality related to a scaling property of the density of states of the “complex modes”.

Neutron scattering from fractals

The scattering formalism for fractal structures is presented. Volume fractals are exemplified by silica particle clusters formed either from colloidal suspensions or by flame hydrolysis. The determination of the fractional dimensionality through scattering experiments is reviewed, and recent small-angle neutron scattering studies of the variaion with aggregation rate are presented. These results allow a very detailed comparison to be made with the theoretical scattering curves. Preliminary incoherent inelastic data on the low-frequency dynamics of hydroxylated silica particle aggregates show a clear deviation from a Debye spectrum and give the first experimental estimates for the spectral dimensionality.

Solitons and fractional spectral dimensionality in the dilute S=(1/2) anisotropic XY chain.

The thermodynamic properties of a soliton-bearing system without translation invariance---the dilute $S=\frac{1}{2}$ anisotropic $\mathrm{XY}$ chain---are calculated. At low temperatures the dominant contribution comes from a low-energy complex mode, which assumes zero energy when translational symmetry is restored in the pure system. The density of states of the complex modes is found to exhibit a scaling property with a fractional spectral dimensionality $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}=\frac{2\mathrm{ln}c}{\mathrm{ln}\ensuremath{\lambda}}$, with $c$ the concentration of magnetic sites and $\ensuremath{\lambda}$ related to the anisotropy. Consequently, the specific heat shows the anomalous behavior $C(T)\ensuremath{\propto}{T}^{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{d}}$.

Fractional dimensionality of localization and quantum Hall current relaxation

Abstract In a strong magnetic field the dimensionality γ of localization may vary from 0 to 91/48. The relaxation of the quantized Hall current may allow γ to he determined experimentally.

Fractal Enhancement of Cartographic Line Detail

In plane geometry curves have a dimension of exactly 1 and no width. In nature, all curvilinear features have width, and most have dimension greater than 1, but less than 2. Many phenomena, such as coastlines, have the same look, even when viewed at greatly varying scales. The former property is called fractional dimensionality, and the latter is called self similarity. Curves digitized from maps may be analyzed to obtain measures of these properties, and knowledge of them can be used to manipulate the shape of cartographic objects. An algorithm is described which enhances the detail of digitized curves by altering their dimensionality in parametrically controlled, self-similar fashion. Illustrations show boundaries processed by the algorithm.