Green function for the diffusion limit of one-dimensional continuous time random walks

It is shown how the fractional probability density diffusion equation for the diffusion limit of one-dimensional continuous time random walks may be derived from a generalized Markovian Chapman-Kolmogorov equation. The non-Markovian behaviour is incorporated into the Markovian Chapman-Kolmogorov equation by postulating a Levy like distribution of waiting times as a kernel. The Chapman-Kolmogorov equation so generalised then takes on the form of a convolution integral. The dependence on the initial conditions typical of a non-Markovian process is treated by adding a time dependent term involving the survival probability to the convolution integral. In the diffusion limit these two assumptions about the past history of the process are sufficient to reproduce anomalous diffusion and relaxation behaviour of the Cole-Cole type. The Green function in the diffusion limit is calculated using the fact that the characteristic function is the Mittag-Leffler function. Fourier inversion of the characteristic function yields the Green function in terms of a Wright function. The moments of the distribution function are evaluated from the Mittag-Leffler function using the properties of characteristic functions and a relation between the powers of the second moment and higher order even moments is derived. (C) 2004 Elsevier B.V. All rights reserved.

Finite difference time domain modeling of phase grating diffusion

In this paper, a method for modeling diffusion caused by non-smooth boundary surfaces in simulations of room acoustics using finite difference time domain (FDTD) technique is investigated. The proposed approach adopts the well-known theory of phase grating diffusers to efficiently model sound scattering from rough surfaces. The variation of diffuser well-depths is attained by nesting allpass filters within the reflection filters from which the digital impedance filters used in the boundary implementation are obtained. The presented technique is appropriate for modeling diffusion at high frequencies caused by small surface roughness and generally diffusers that have narrow wells and infinitely thin separators. The diffusion coefficient was measured with numerical experiments for a range of fractional Brownian diffusers.

White matter fractional anisotrophy predicts balance performance in older adults

Aging is characterized by brain structural changes that may compromise motor functions. In the context of postural control, white matter integrity is crucial for the efficient transfer of visual, proprioceptive and vestibular feedback in the brain. To determine the role of age-related white matter decline as a function of the sensory feedback necessary to correct posture, we acquired diffusion weighted images in young and old subjects. A force platform was used to measure changes in body posture under conditions of compromised proprioceptive and/or visual feedback. In the young group, no significant brain structure-balance relations were found. In the elderly however, the integrity of a cluster in the frontal forceps explained 21% of the variance in postural control when proprioceptive information was compromised. Additionally, when only the vestibular system supplied reliable information, the occipital forceps was the best predictor of balance performance (42%). Age-related white matter decline may thus be predictive of balance performance in the elderly when sensory systems start to degrade.

MRI diffusion tractography study in individuals with schizotypal features: A pilot study

Diffusion tensor imaging (DTI) studies have identified changes in white matter tracts in schizophrenia patients and those at high risk of transition. Schizotypal samples represent a group on the schizophrenia continuum that share some aetiological risk factors but without the confounds of illness. The aim of the current study was to compare tract microstructural coherence as measured by
fractional anisotropy (FA) between 12 psychometrically defined schizotypes and controls. We investigated bilaterally the uncinate and arcuate fasciculi (UF and AF) via a probabilistic tractography algorithm (PICo), with FA values compared between groups. Partial correlations were also examined between measures of subclinical hallucinatory/delusional experiences and FA values. High schizotypes
were found to have significantly higher FA values in bilateral UF only, but failed to reach significance in each hemisphere. In the whole sample there was a positive correlation between increasing FA values and measures of hallucinatory experience in the right AF. These findings suggest subtle changes in microstructural coherence are present in schizotypes. Correlations between mild hallucinatory experience and increasing FA values could indicate increasing coherence could be associated with symptom formation.

The role of Mittag-Leffler functions in anomalous relaxation

The inertia-corrected Debye model of rotational Brownian motion of polar molecules was generalized by Coffey et al. [Phys. Rev. E, 65, 32 102 (2002)] to describe fractional dynamics and anomalous rotational diffusion. The linear- response theory of the normalized complex susceptibility was given in terms of a Laplace transform and as a function of frequency. The angular-velocity correlation function was parametrized via fractal Mittag-Leffler functions. Here we apply the latter method and complex-contour integral- representation methods to determine the original time-dependent amplitude as an inverse Laplace transform using both analytical and numerical approaches, as appropriate. (C) 2004 Elsevier B.V. All rights reserved.

Microscopic models for dielectric relaxation in disordered systems

It is shown how the Debye rotational diffusion model of dielectric relaxation of polar molecules (which may be described in microscopic fashion as the diffusion limit of a discrete time random walk on the surface of the unit sphere) may be extended to yield the empirical Havriliak-Negami (HN) equation of anomalous dielectric relaxation from a microscopic model based on a kinetic equation just as in the Debye model. This kinetic equation is obtained by means of a generalization of the noninertial Fokker-Planck equation of conventional Brownian motion (generally known as the Smoluchowski equation) to fractional kinetics governed by the HN relaxation mechanism. For the simple case of noninteracting dipoles it may be solved by Fourier transform techniques to yield the Green function and the complex dielectric susceptibility corresponding to the HN anomalous relaxation mechanism.

Fixed-node diffusion Monte Carlo computations for closed-shell jellium spheres

Fixed-node diffusion Monte Carlo computations are used to determine the ground state energy and electron density for jellium spheres with up to N = 106 electrons and background densities corresponding to the electron gas parameter 1 less than or equal to r(s)less than or equal to5.62. We analyze the density and size dependence of the surface energy, and we extrapolate our data to the thermodynamic limit. The results agree well with the predictions of density functional computations using the local density approximation. In the case of N = 20, we extend our computation to higher densities and identify a transition between atomic- and jelliumlike nodal structures occurring at the background density corresponding to r(s)=0.13. In this case the local density approximation is unable to reproduce the changes in the correlation energy due to the discontinuous transition in the ground state nodal structure. We discuss the relevance of our results for nonlocal approximations to density functional theory.