Differential equations for generalized Jacobi polynomials

We look for spectral type differential equations satisfied by the generalized Jacobi polynomials, which are orthogonal on the interval [-1,1] with respect to a weight function consisting of the classical Jacobi weight function together with two point masses at the endpoints of the interval of orthogonality. We show that such a differential equation is uniquely determined and we give explicit representations for the coefficients. In case of nonzero mass points the order of this differential equation is infinite, except for nonnegative integer values of (one of) the parameters. Otherwise, the finite order is explictly given in terms of the parameters.Comment: 33 pages, submitted for publicatio

Gamma-Ray Burst Afterglow: Polarization and Analytic Light Curves

GRB afterglow polarization is discussed. We find an observable, up to 10%, polarization, if the magnetic field coherence length grows at about the speed of light after the field is generated at the shock front. Detection of a polarized afterglow would show that collisionless ultrarelativistic shocks can generate strong large scale magnetic fields and confirm the synchrotron afterglow model. Non-detection, at a 1% level, would imply that either the synchrotron emission model is incorrect, or that strong magnetic fields, after they are generated in the shock, somehow manage to stay un-dissipated at ``microscopic'', skin depth, scales. Analytic lightcurves of synchrotron emission from an ultrarelativistic self-similar blast wave are obtained for an arbitrary electron distribution function, taking into account the effects of synchrotron cooling. The peak synchrotron flux and the flux at frequencies much smaller than the peak frequency are insensitive to the details of the electron distribution function; hence their observational determination would provide strong constraints on blast wave parameters.Comment: 19 pages, submitted to Ap

Attracted Diffusion-Limited Aggregation

In this paper, we present results of extensive Monte Carlo simulations of diffusion-limited aggregation (DLA) with a seed placed on an attractive plane as a simple model in connection with the electrical double layers. We compute the fractal dimension of the aggregated patterns as a function of the attraction strength \alpha. For the patterns grown in both two and three dimensions, the fractal dimension shows a significant dependence on the attraction strength for small values of \alpha, and approaches to that of the ordinary two-dimensional (2D) DLA in the limit of large \alpha. For non-attracting case with \alpha=1, our results in three dimensions reproduce the patterns of 3D ordinary DLA, while in two dimensions our model leads to formation of a compact cluster with dimension two. For intermediate \alpha, the 3D clusters have quasi-2D structure with a fractal dimension very close to that of the ordinary 2D-DLA. This allows one to control morphology of a growing cluster by tuning a single external parameter \alpha.Comment: 6 pages, 6 figures, to appear in Phys. Rev. E (2012

Free streaming in mixed dark matter

Free streaming in a \emph{mixture} of collisionless non-relativistic dark matter (DM) particles is studied by implementing methods from the theory of multicomponent plasmas. The mixture includes Fermionic, condensed and non condensed Bosonic particles decoupling in equilibrium while relativistic, heavy non-relativistic thermal relics (WIMPs), and sterile neutrinos that decouple \emph{out of equilibrium} when they are relativistic. The free-streaming length $\lambda_{fs}$ is obtained from the marginal zero of the gravitational polarization function, which separates short wavelength Landau-damped from long wavelength Jeans-unstable \emph{collective} modes. At redshift $z$ we find $\frac{1}{\lambda^2_{fs}(z)}= \frac{1}{(1+z)} \big[\frac{0.071}{\textrm{kpc}} \big]^2 \sum_{a}\nu_a g^{2/3}_{d,a}({m_a}/{\mathrm{keV}})^2 I_a$,where $0\leq \nu_a \leq 1$ are the \emph{fractions} of the respective DM components of mass $m_a$ that decouple when the effective number of ultrarelativistic degrees of freedom is $g_{d,a}$, and $I_a$ only depend on the distribution functions at decoupling, given explicitly in all cases. If sterile neutrinos produced either resonantly or non-resonantly that decouple near the QCD scale are the \emph{only} DM component,we find $\lambda_{fs}(0) \simeq 7 \mathrm{kpc} (\mathrm{keV}/m)$ (non-resonant), $\lambda_{fs}(0) \simeq 1.73 \mathrm{kpc} (\mathrm{keV}/m)$ (resonant).If WIMPs with $m_{wimp} \gtrsim 100 \mathrm{GeV}$ decoupling at $T_d \gtrsim 10 \mathrm{MeV}$ are present in the mixture with $\nu_{wimp} \gg 10^{-12}$,$\lambda_{fs}(0) \lesssim 6.5 \times 10^{-3} \mathrm{pc}$ is \emph{dominated} by CDM. If a Bose Einstein condensate is a DM component its free streaming length is consistent with CDM because of the infrared enhancement of the distribution function.Comment: 19 pages, 2 figures. More discussions same conclusions and results. Version to appear in Phys. Rev.

An Arbitrary Curvilinear Coordinate Method for Particle-In-Cell Modeling

A new approach to the kinetic simulation of plasmas in complex geometries, based on the Particle-in- Cell (PIC) simulation method, is explored. In the two dimensional (2d) electrostatic version of our method, called the Arbitrary Curvilinear Coordinate PIC (ACC-PIC) method, all essential PIC operations are carried out in 2d on a uniform grid on the unit square logical domain, and mapped to a nonuniform boundary-fitted grid on the physical domain. As the resulting logical grid equations of motion are not separable, we have developed an extension of the semi-implicit Modified Leapfrog (ML) integration technique to preserve the symplectic nature of the logical grid particle mover. A generalized, curvilinear coordinate formulation of Poisson's equations to solve for the electrostatic fields on the uniform logical grid is also developed. By our formulation, we compute the plasma charge density on the logical grid based on the particles' positions on the logical domain. That is, the plasma particles are weighted to the uniform logical grid and the self-consistent mean electrostatic fields obtained from the solution of the logical grid Poisson equation are interpolated to the particle positions on the logical grid. This process eliminates the complexity associated with the weighting and interpolation processes on the nonuniform physical grid and allows us to run the PIC method on arbitrary boundary-fitted meshes.Comment: Submitted to Computational Science & Discovery December 201

Effective dynamics of a nonabelian plasma out of equilibrium

Starting from kinetic theory, we obtain a nonlinear dissipative formalism describing the nonequilibrium evolution of scalar colored particles coupled selfconsistently to nonabelian classical gauge fields. The link between the one-particle distribution function of the kinetic description and the variables of the effective theory is determined by extremizing the entropy production. This method does not rely on the usual gradient expansion in fluid dynamic variables, and therefore the resulting effective theory can handle situations where these gradients (and hence the momentum-space anisotropies) are expected to be large. The formalism presented here, being computationally less demanding than kinetic theory, may be useful as a simplified model of the dynamics of color fields during the early stages of heavy ion collisions and in phenomena related to parton energy loss.Comment: 20 two-column pages, 2 figures. v3: minor changes. Accepted for publication in Phys. Rev.

Undamped electrostatic plasma waves

Electrostatic waves in a collision-free unmagnetized plasma of electrons with fixed ions are investigated for electron equilibrium velocity distribution functions that deviate slightly from Maxwellian. Of interest are undamped waves that are the small amplitude limit of nonlinear excitations, such as electron acoustic waves (EAWs). A deviation consisting of a small plateau, a region with zero velocity derivative over a width that is a very small fraction of the electron thermal speed, is shown to give rise to new undamped modes, which here are named {\it corner modes}. The presence of the plateau turns off Landau damping and allows oscillations with phase speeds within the plateau. These undamped waves are obtained in a wide region of the $(k,\omega_{_R})$ plane ($\omega_{_R}$ being the real part of the wave frequency and $k$ the wavenumber), away from the well-known `thumb curve' for Langmuir waves and EAWs based on the Maxwellian. Results of nonlinear Vlasov-Poisson simulations that corroborate the existence of these modes are described. It is also shown that deviations caused by fattening the tail of the distribution shift roots off of the thumb curve toward lower $k$-values and chopping the tail shifts them toward higher $k$-values. In addition, a rule of thumb is obtained for assessing how the existence of a plateau shifts roots off of the thumb curve. Suggestions are made for interpreting experimental observations of electrostatic waves, such as recent ones in nonneutral plasmas.Comment: 11 pages, 10 figure

The Effect of Neutral Atoms on Capillary Discharge Z-pinch

We study the effect of neutral atoms on the dynamics of a capillary discharge Z-pinch, in a regime for which a large soft-x-ray amplification has been demonstrated. We extended the commonly used one-fluid magneto-hydrodynamics (MHD) model by separating out the neutral atoms as a second fluid. Numerical calculations using this extended model yield new predictions for the dynamics of the pinch collapse, and better agreement with known measured data.Comment: 4 pages, 4 postscript figures, to be published in Phys. Rev. Let