On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree

In our recent works [R. Szmytkowski, J. Phys. A 39 (2006) 15147; corrigendum: 40 (2007) 7819; addendum: 40 (2007) 14887], we have investigated the derivative of the Legendre function of the first kind, $P_{\nu}(z)$, with respect to its degree $\nu$. In the present work, we extend these studies and construct several representations of the derivative of the associated Legendre function of the first kind, $P_{\nu}^{\pm m}(z)$, with respect to the degree $\nu$, for $m\in\mathbb{N}$. At first, we establish several contour-integral representations of $\partial P_{\nu}^{\pm m}(z)/\partial\nu$. They are then used to derive Rodrigues-type formulas for $[\partial P_{\nu}^{\pm m}(z)/\partial\nu]_{\nu=n}$ with $n\in\mathbb{N}$. Next, some closed-form expressions for $[\partial P_{\nu}^{\pm m}(z)/\partial\nu]_{\nu=n}$ are obtained. These results are applied to find several representations, both explicit and of the Rodrigues type, for the associated Legendre function of the second kind of integer degree and order, $Q_{n}^{\pm m}(z)$; the explicit representations are suitable for use for numerical purposes in various regions of the complex $z$-plane. Finally, the derivatives $[\partial^{2}P_{\nu}^{m}(z)/\partial\nu^{2}]_{\nu=n}$, $[\partial Q_{\nu}^{m}(z)/\partial\nu]_{\nu=n}$ and $[\partial Q_{\nu}^{m}(z)/\partial\nu]_{\nu=-n-1}$, all with $m>n$, are evaluated in terms of $[\partial P_{\nu}^{-m}(\pm z)/\partial\nu]_{\nu=n}$.Comment: LateX, 40 pages, 1 figure, extensive referencin

Dissolution in a field

We study the dissolution of a solid by continuous injection of reactive ``acid'' particles at a single point, with the reactive particles undergoing biased diffusion in the dissolved region. When acid encounters the substrate material, both an acid particle and a unit of the material disappear. We find that the lengths of the dissolved cavity parallel and perpendicular to the bias grow as t^{2/(d+1)} and t^{1/(d+1)}, respectively, in d-dimensions, while the number of reactive particles within the cavity grows as t^{2/(d+1)}. We also obtain the exact density profile of the reactive particles and the relation between this profile and the motion of the dissolution boundary. The extension to variable acid strength is also discussed.Comment: 6 pages, 6 figures, 2-column format, for submission to PR

Anomalous diffusion with absorption: Exact time-dependent solutions

Recently, analytical solutions of a nonlinear Fokker-Planck equation describing anomalous diffusion with an external linear force were found using a non extensive thermostatistical Ansatz. We have extended these solutions to the case when an homogeneous absorption process is also present. Some peculiar aspects of the interrelation between the deterministic force, the nonlinear diffusion and the absorption process are discussed.Comment: RevTex, 16 pgs, 4 figures. Accepted in Physical Review

Directed polymers in high dimensions

We study directed polymers subject to a quenched random potential in d transversal dimensions. This system is closely related to the Kardar-Parisi-Zhang equation of nonlinear stochastic growth. By a careful analysis of the perturbation theory we show that physical quantities develop singular behavior for d to 4. For example, the universal finite size amplitude of the free energy at the roughening transition is proportional to (4-d)^(1/2). This shows that the dimension d=4 plays a special role for this system and points towards d=4 as the upper critical dimension of the Kardar-Parisi-Zhang problem.Comment: 37 pages REVTEX including 4 PostScript figure

Revisiting Brownian motion as a description of animal movement: a comparison to experimental movement data

1) Characterization of patterns of animal movement is a major challenge in ecology with applications to conservation, biological invasions and pest monitoring. Brownian random walks, and diffusive flux as their mean field counterpart, provide one framework in which to consider this problem. However, it remains subject to debate and controversy. This study presents a test of the diffusion framework using movement data obtained from controlled experiments. 2) Walking beetles (Tenebrio molitor) were released in an open circular arena with a central hole and the number of individuals falling from the arena edges was monitored over time. These boundary counts were compared, using curve fitting, to the predictions of a diffusion model. The diffusion model is solved precisely, without using numerical simulations. 3) We find that the shape of the curves derived from the diffusion model is a close match to those found experimentally. Furthermore, in general, estimates of the total population obtained from the relevant solution of the diffusion equation are in excellent agreement with the experimental population. Estimates of the dispersal rate of individuals depend on how accurately the initial release distribution is incorporated into the model. 4) We therefore show that diffusive flux is a very good approximation to the movement of a population of Tenebrio molitor beetles. As such, we suggest that it is an adequate theoretical/modelling framework for ecological studies that account for insect movement, although it can be context specific. An immediate practical application of this can be found in the interpretation of trap counts, in particular for the purpose of pest monitoring

Application of the Convection–Dispersion Equation to Modelling Oral Drug Absorption

Models of systemic drug absorption after oral administration are frequently based on a direct or a delayed first-order rate process. In practice, the use of the first-order approach to predict drug concentrations in blood plasma frequently yields a considerable mismatch between predicted and measured concentration profiles. This is particularly true for the upswing of the plasma concentration after oral administration. The current investigation explores an alternative model to describe the absorption rate based on the convection–dispersion equation describing the transport of chemicals through the GI tract. This equation is governed by two parameters, transport velocity and dispersion coefficient. One solution of this equation for a specific set of initial and boundary conditions was used to model absorption of paracetamol in a 22-year-old man after oral administration. The GI-tract passage rate in this subject was influenced by co-administration of drugs that stimulate or delay gastric emptying. The transport-limited absorption function is more accurate in describing the plasma concentration versus time curve after oral administration than the first-order model. Additionally, it provides a mechanistic explanation for the observed curve through the differences in GI-tract passage rate

Energetic particle influence on the Earth's atmosphere

This manuscript gives an up-to-date and comprehensive overview of the effects of energetic particle precipitation (EPP) onto the whole atmosphere, from the lower thermosphere/mesosphere through the stratosphere and troposphere, to the surface. The paper summarizes the different sources and energies of particles, principally galactic cosmic rays (GCRs), solar energetic particles (SEPs) and energetic electron precipitation (EEP). All the proposed mechanisms by which EPP can affect the atmosphere are discussed, including chemical changes in the upper atmosphere and lower thermosphere, chemistry-dynamics feedbacks, the global electric circuit and cloud formation. The role of energetic particles in Earth’s atmosphere is a multi-disciplinary problem that requires expertise from a range of scientific backgrounds. To assist with this synergy, summary tables are provided, which are intended to evaluate the level of current knowledge of the effects of energetic particles on processes in the entire atmosphere