Efficient numerical solution of the time fractional diffusion equation by mapping from its Brownian counterpart

The solution of a Caputo time fractional diffusion equation of order $0<\alpha<1$ is expressed in terms of the solution of a corresponding integer order diffusion equation. We demonstrate a linear time mapping between these solutions that allows for accelerated computation of the solution of the fractional order problem. In the context of an $N$-point finite difference time discretisation, the mapping allows for an improvement in time computational complexity from $O\left(N^2\right)$ to $O\left(N^\alpha\right)$, given a precomputation of $O\left(N^{1+\alpha}\ln N\right)$. The mapping is applied successfully to the least-squares fitting of a fractional advection diffusion model for the current in a time-of-flight experiment, resulting in a computational speed up in the range of one to three orders of magnitude for realistic problem sizes.Comment: 9 pages, 5 figures; added references for section

Generalised balance equations for charged particle transport via localised and delocalised states: Mobility, generalised Einstein relations and fractional transport

A generalised phase-space kinetic Boltzmann equation for highly non-equilibrium charged particle transport via localised and delocalised states is used to develop continuity, momentum and energy balance equations, accounting explicitly for scattering, trapping/detrapping and recombination loss processes. Analytic expressions detail the effect of these microscopic processes on the mobility and diffusivity. Generalised Einstein relations (GER) are developed that enable the anisotropic nature of diffusion to be determined in terms of the measured field-dependence of the mobility. Interesting phenomena such as negative differential conductivity and recombination heating/cooling are shown to arise from recombination loss processes and the localised and delocalised nature of transport. Fractional transport emerges naturally within this framework through the appropriate choice of divergent mean waiting time distributions for localised states, and fractional generalisations of the GER and mobility are presented. Signature impacts on time-of-flight current transients of recombination loss processes via both localised and delocalised states are presented.Comment: 21 pages, 4 figure

Strong lensing optical depths in a \LambdaCDM universe

We investigate strong gravitational lensing in the concordance $\Lambda$CDM cosmology by carrying out ray-tracing along past light cones through the Millennium Simulation, the largest simulation of cosmic structure formation ever carried out. We extend previous ray-tracing methods in order to take full advantage of the large volume and the excellent spatial and mass resolution of the simulation. As a function of source redshift we evaluate the probability that an image will be highly magnified, will be highly elongated or will be one of a set of multiple images. We show that such strong lensing events can almost always be traced to a single dominant lensing object and we study the mass and redshift distribution of these primary lenses. We fit analytic models to the simulated dark halos in order to study how our optical depth measurements are affected by the limited resolution of the simulation and of the lensing planes that we construct from it. We conclude that such effects lead us to underestimate total strong-lensing cross sections by about 15 percent. This is smaller than the effects expected from our neglect of the baryonic components of galaxies. Finally we investigate whether strong lensing is enhanced by material in front of or behind the primary lens. Although strong lensing lines-of-sight are indeed biased towards higher than average mean densities, this additional matter typically contributes only a few percent of the total surface density.Comment: version accepted for publicatio

Third-order transport coefficients for localised and delocalised charged-particle transport

We derive third order transport coefficients of skewness for a phase-space kinetic model that considers the processes of scattering collisions, trapping, detrapping and recombination losses. The resulting expression for the skewness tensor provides an extension to Fick's law which is in turn applied to yield a corresponding generalised advection-diffusion-skewness equation. A physical interpretation of trap-induced skewness is presented and used to describe an observed negative skewness due to traps. A relationship between skewness, diffusion, mobility and temperature is formed by analogy with Einstein's relation. Fractional transport is explored and its effects on the flux transport coefficients are also outlined.Comment: 11 pages, 3 figure

Evolution of X-ray cluster scaling relations in simulations with radiative cooling and non-gravitational heating

We investigate the redshift dependence of X-ray cluster scaling relations drawn from three hydrodynamic simulations of the LCDM cosmology: a Radiative model that incorporates radiative cooling of the gas, a Preheating model that additionally heats the gas uniformly at high redshift, and a Feedback model that self-consistently heats cold gas in proportion to its local star-formation rate. While all three models are capable of reproducing the observed local Lx-Tx relation, they predict substantially different results at high redshift (to z=1.5), with the Radiative, Preheating and Feedback models predicting strongly positive, mildly positive and mildly negative evolution, respectively. The physical explanation for these differences lies in the structure of the intracluster medium. All three models predict significant temperature fluctuations at any given radius due to the presence of cool subclumps and, in the case of the Feedback simulation, reheated gas. The mean gas temperature lies above the dynamical temperature of the halo for all models at z=0, but differs between models at higher redshift with the Radiative model having the lowest mean gas temperature at z=1.5. We have not attempted to model the scaling relations in a manner that mimics the observational selection effects, nor has a consistent observational picture yet emerged. Nevertheless, evolution of the scaling relations promises to be a powerful probe of the physics of entropy generation in clusters. First indications are that early, widespread heating is favored over an extended period of heating that is associated with galaxy formation.Comment: Accepted for publication in ApJ. Minor changes following referee's comment

Salicylaldehyde hydrazones: buttressing of outer sphere hydrogen-bonding and copper-extraction properties

Salicylaldehyde hydrazones are weaker copper extractants than their oxime derivatives, which are used in hydrometallurgical processes to recover ~20 % of the world’s copper. Their strength, based on the extraction equilibrium constant Ke, can be increased by nearly three orders of magnitude by incorporating electron-withdrawing or hydrogen-bond acceptor groups (X) ortho to the phenolic OH group of the salicylaldehyde unit. Density functional theory calculations suggest that the effects of the 3-X substituents arise from a combination of their influence on the acidity of the phenol in the pH-dependent equilibrium, Cu2+ + 2Lorg ⇌ [Cu(L–H)2]org + 2H+, and on their ability to ‘buttress’ interligand hydrogen bonding by interacting with the hydrazone N–H donor group. X-ray crystal structure determination and computed structures indicate that in both the solid state and the gas phase, coordinated hydrazone groups are less planar than coordinated oximes and this has an adverse effect on intramolecular hydrogen-bond formation to the neighbouring phenolate oxygen atoms

Numerical renormalization group study of the correlation functions of the antiferromagnetic spin-12\frac{1}{2} Heisenberg chain

We use the density-matrix renormalization group technique developed by White \cite{white} to calculate the spin correlation functions $=(-1)^l \omega(l,N)$ for isotropic Heisenberg rings up to $N=70$ sites. The correlation functions for large $l$ and $N$ are found to obey the scaling relation $\omega(l,N)=\omega(l,\infty)f_{XY}^{\alpha} (l/N)$ proposed by Kaplan et al. \cite{horsch} , which is used to determine $\omega(l,\infty)$. The asymptotic correlation function $\omega(l,\infty)$ and the magnetic structure factor $S(q=\pi)$ show logarithmic corrections consistent with $\omega(l,\infty)\sim a\sqrt{\ln{cl}}/l$, where $c$ is related to the cut-off dependent coupling constant $g_{eff}(l_0)=1/\ln(cl_0)$, as predicted by field theoretical treatments.Comment: Accepted in Phys. Rev. B. 4 pages of text in Latex + 5 figures in uuencoded form containing the 5 postscripts (mailed separately

Hydrodynamical Simulations of the Lyman Alpha Forest: Model Comparisons

We investigate the properties of the Lyman alpha forest as predicted by numerical simulations for a range of currently viable cosmological models. This is done in order to understand the dependencies of the forest on cosmological parameters. Focusing on the redshift range from two to four, we show that: (1) most of the evolution in the distributions of optical depth, flux and column density can be understood by simple scaling relations, (2) the shape of optical depth distribution is a sensitive probe of the amplitude of density fluctuations on scales of a few hundred kpc, (3) the mean of the b distribution (a measure of the width of the absorption lines) is also very sensitive to fluctuations on these scales, and decreases as they increase. We perform a preliminary comparison to observations, where available. A number of other properties are also examined, including the evolution in the number of lines, the two-point flux distribution and the HeII opacity.Comment: 37 pages, 21 figures, submitted to Ap

Dark-Matter Decays and Self-Gravitating Halos

We consider models in which a dark-matter particle decays to a slightly less massive daughter particle and a noninteracting massless particle. The decay gives the daughter particle a small velocity kick. Self-gravitating dark-matter halos that have a virial velocity smaller than this velocity kick may be disrupted by these particle decays, while those with larger virial velocities will be heated. We use numerical simulations to follow the detailed evolution of the total mass and density profile of self-gravitating systems composed of particles that undergo such velocity kicks as a function of the kick speed (relative to the virial velocity) and the decay time (relative to the dynamical time). We show how these decays will affect the halo mass-concentration relation and mass function. Using measurements of the halo mass-concentration relation and galaxy-cluster mass function to constrain the lifetime--kick-velocity parameter space for decaying dark matter, we find roughly that the observations rule out the combination of kick velocities greater than 100 km/s and decay times less than a few times the age of the Universe.Comment: 17 pages, 10 figures, replaced with published versio