Half-space problem of the Boltzmann equation for charged particles

For two particular collision kernels, we explicitly solve the one-dimensional stationary half-space boundary value problem of the linear Boltzmann equation including a constant external field via an extension of Case's eigenfunction technique. In the first collision model we reproduce a solution recently obtained by Cercignani; in the second model the solution of the stationary boundary value problem is presented for the first time.     

Fractional nonlinear diffusion equation,solutions and anomalous diffusion

We devote this work to investigate the solutions of a generalized diffusion equation which contains spatial fractional derivatives and nonlinear terms. The presence of external forces and absorbent terms is also considered. The solutions found here can have a compact or long tail behavior and, in particular, for the last case in the asymptotic limit, we relate these solutions to the Lévy or Tsallis distributions. In addition, from the results presented here a rich class of diffusive processes, including normal and anomalous ones, can be obtained.     

Two dimensional band-gap restricted diffusion of negatively charged excitons in rare gas solids

We present measurements of electron stimulated desorption (ESD) yields of D⁻, obtained from sub-monolayer amounts of C₂D₆ physisorbed on rare gas substrates; the latter consisting of alternating layers of Xe and Kr adsorbed at 20 K on Pt. Negatively charged excitons (NCE), for example, Xe*⁻ or Kr*⁻, which are produced by electron impact, may couple to dissociative anion states of C₂D₆, resulting in sharp resonance enhancements in the D⁻ ESD yields. Our measurements suggest that resonant diffusion of the Xe NCEs can be restricted to a single Xe layer, by placing a Kr multilayer spacer between the Xe and the metal substrate. This is attributed to the 1.85 eV insulating band-gap difference at the Xe---Kr interface.     

Chaotic diffusion and 1/f-noise of particles in two-dimensional solids

We investigate the classical nonlinear dynamics of a particle moving conservatively in a two-dimensional periodic potential. The particle exhibits diffusive motion in the absence of random forces. In a broad range of energies above the potential barrier, the diffusion process is anomalously accelerated and associated with 1/f-noise in the power spectrum of velocity fluctuations. The analysis of Poincaré surfaces of section and the distribution of free paths indicate that the phenomenon is caused by a trapping of orbits in a self-similar hierarchy of nested cantori. We describe a statistical theory for this mechanism in terms of a renewal process and a random walk on a hierarchical lattice.Work supported by Deutsche Forschungsgemeinschaft     

Space distribution and energy straggling of charged particles via Fokker-Planck equation

Summary The Fokker-Planck equation describing a beam of charged particles entering a homogeneous medium is solved here for a stationary case. Interactions are taken into account through Coulomb cross-section. Starting from the charged-particle distribution as a function of velocity and penetration depth, some important kinetic quantities are calculated, like mean velocity, range and the loss of energy per unit space. In such quantities the energy straggling is taken into account. This phenomenon is not considered in the continuous slowing-down approximation that is commonly used to obtain the range and the stopping power. Finally the well-known Bohr or Bethe formula is found as a first-order approximation of the Fokker-Planck equation. The authors of this paper have agreed to not receive the proofs for correction.     

Analytical solutions for the energy distribution of charged particles in a weak electric field

The paper deals with the stationary distribution of charged particles moving in a material medium, having scattering and absorption properties, in which a uniform electric field is present. The purpose of the work is finding analytical solutions in simplified but physically significant situations and comparing different approximations based on a spherical-harmonics expansion of the velocity distribution. Received: 28 July 1998     

The second Painlevé equation in the electrostatic probe theory: Numerical solutions for the partial absorption of charged particles by the surface

We continue the study of the second Painlevé equation within the framework of the electrostatic probe theory. The integrability conditions for the equation are found for the partial absorption of charged particles by the probe surface. A sets of solutions with the asymptotics y ∼ ν/x for x → +∞ is constructed numerically in a wide range of the free parameter ν. Also, solutions (related to those mentioned above) for half-integer and integer ν, including solutions representable in asymptotic form at x → +∞ through the Airy function y ∼ cAi(x) in the limit ν → 0, are found. The results are discussed from the standpoint of the isomonodromic deformation method.     

Solution of the one-dimensional linear Boltzmann equation for charged Maxwellian particles in an external field

The one-dimensional linear homogeneous Boltzmann equation is solved for a binary mixture of quasi-Maxwellian particles in the presence of a time-dependent external field. It is assumed that the charged particles move in a bath of neutral scatterers. The neutral scatterers are in thermal equilibrium and the concentration of the charged particles is low enough to neglect collisions between them. Two cases are considered in detail, the constant and the periodic external field. The quantities calculated are the equilibrium and the stationary distribution function, respectively, from which any desired property can be derived. The solution of the Boltzmann equation for Maxwellian particles can be reduced to the solution of the so-called cold gas equation by employing the one-dimensional variant of a convolution theorem due to Wannier. The two limiting cases, the Lorentz gas (m _A0) and the Rayleigh gas (m _A) are treated explicitly. Furthermore, by computing the central moments, the deviations from the Gaussian approximation are discussed, and in particular the large-velocity tails are evaluated.     

Self-sustained oscillations in a plasma-wall system with strongly inhomogeneous diffusion of charged particles

A simple system with a hydrogen plasma confined by a magnetic field parallel to the bounding material wall is considered. The charged particles diffuse out of the plasma, recombine on the wall and return into the plasma volume as neutrals, which are ionized by electrons. It is demonstrated that macroscopic self-sustained oscillations are an intrinsic feature of such a system if the diffusion coefficient of charged particles is strongly inhomogeneous in the plasma.     

Exact solutions of the Loretnz-Dirac equations of motion for charged particles in constant electromagnetic fields

The linearized Lorentz-Dirac equation $\text{u}\text{?}{}^{\mathrm{\mu }}\text{= F}{}^{\mathrm{\mu \nu }}\text{u}{}_{\mathrm{\nu }}\text{- κ(F}{}^{\mathrm{\mu \sigma }}\text{F}{}_{\mathrm{\lambda \sigma }}\text{u}{}^{\mathrm{\lambda }}\text{+ {F}{}^{\mathrm{\sigma \mu }}\text{u}{}_{\mathrm{\nu }}\text{F}{}_{\mathrm{\sigma \lambda }}\text{u}{}^{\mathrm{\lambda }}\text{}u}{}^{\mathrm{\mu }}\text{)}$ is solved in closed form for arbitrary constant electromagnetic fields. These solutions are also a convenient starting point to treat particle motion in the slowly varying electromagnetic fields surrounding a pulsar in the near field zone.     

Dynamical derivation of momentum diffusion coefficients at collisions of relativistic charged particles

An expression has been obtained for the diffusion tensor of particles in the momentum space on the basis of the dynamics of particles motion. The general equations have been used to determine the rms momentum spread at collisions of relativistic charged particles at times shorter than the time of randomization of particles motion and at greater times when motion is completely random.     

Exact solutions for a diffusion equation with a nonlinear external force

This work is devoted to investigate the solutions of the one-dimensional diffusion equation by taking the nonlinear external force F(x,t;ρ)=−k(t)x+K/x+κx|x|^α−1η[ρ(x,t)] into account. Our investigation is first performed by considering the case α=0 and η=1, which results in a Burgers like equation with a spatial and time dependent external force. After, we consider the case α≠0 and η=α+1 and show that the solution found may be expressed in terms of the q-exponential functions present in the Tsallis formalism. In addition, we also discuss the stationary solution for α and η arbitraries.     

General magnetized Weyl solutions: disks and motion of charged particles

We construct three families of general magnetostatic axisymmetric exact solutions of Einstein-Maxwell equations in spherical coordinates, prolate, and oblates. The solutions obtained are then presented in the system of generalized spheroidal coordinates which is a generalization of the previous systems. The method used to build such solutions is the well-known complex potential formalism proposed by Ernst, using as seed solutions vacuum solutions of the Einstein field equations. We show explicitly some particular solutions among them a magnetized Erez-Rosen solution and a magnetized Morgan-Morgan solution, which we interpret as the exterior gravitational field of a finite dislike source immersed in a magnetic field. From them we also construct using the well known “displace, cut and reflect” method exact solutions representing relativistic thin disks of infinite extension. We then analyze the motion of electrically charged test particles around these fields for equatorial circular orbits and we discuss their stability against radial perturbations. For magnetized Morgan-Morgan fields we find that inside of disk the presence of magnetic field provides the possibility of to find relativist charged particles moving in both prograde and retrograde direction.     

Kinetics of contact melting under conditions of non-steady-state diffusion

The average rate ¯v_n of contact melting under conditions of non-steady-state diffus ion was measured for 8 metallic systems using a procedure in which a crystal-liquid-crystal nonconvection system could be produced. It is shown that when the experimental data on ¯v_n are suitably reduced, it is possible to obtain the rate ¯v_S of contact melting in the steady state. v_n and v_S are calculated in the same systems, using equations derived by the authors. The calculated data on ¯v_n and on its temperature dependence agree satisfactorily with the experimental data. The coefficients of heterodiffus ion in metallic melts with large component concentrations are calculated using the experimental data for ¯v_n.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 13–17, December, 1970.