Steady-state Particle Distribution Research Articles

Prospects for ultra-high-energy particle acceleration at relativistic shocks

ABSTRACT We study the acceleration of charged particles by ultra-relativistic shocks using test-particle Monte Carlo simulations. Two field configurations are considered: (i) shocks with uniform upstream magnetic field in the plane of the shock, and (ii) shocks in which the upstream magnetic field has a cylindrical geometry. Particles are assumed to diffuse in angle due to frequent non-resonant scattering on small-scale fields. The steady-state distribution of particles’ Lorentz factors is shown to approximately satisfy dN/dγ ∝ γ−2.2 provided the particle motion is scattering dominated on at least one side of the shock. For scattering dominated transport, the acceleration rate scales as tacc ∝ t1/2, though recovers Bohm scaling tacc ∝ t if particles become magnetized on one side of the shock. For uniform field configurations, a limiting energy is reached when particles are magnetized on both sides of the shock. For the cylindrical field configuration, this limit does not apply, and particles of one sign of charge will experience a curvature drift that redirects particles upstream. For the non-resonant scattering model considered, these particles preferentially escape only when they reach the confinement limit determined by the finite system size, and the distribution approaches the escapeless limit dN/dγ ∝ γ−1. The cylindrical field configuration resembles that expected for jets launched by the Blandford & Znajek mechanism, the luminous jets of active galactic nuclei and gamma-ray bursts thus provide favourable sites for the production of ultra-high-energy cosmic rays.

Ionic current driven by a viscosity gradient.

Gradients of voltage, pressure, temperature, and salinity can transport objects in micro- and nanofluidic systems by well-known mechanisms. This paper explores the dynamics of particles in a viscosity gradient with numerical simulations. The different stochastic rules used to integrate the random motion of Brownian particles affect the steady-state distribution of particles in a diffusivity gradient. Importantly, the simulations illuminate the important role that the boundary conditions play, disallowing a steady-state flux when the boundary conditions mimic those of a closed container, but allowing flux when they mimic electrodes. These results provide an interpretation for measurements of a steady ionic current flowing between electrodes separated by a nanofluidic channel with a liquid viscosity gradient.

Analysis of Smoluchowski’s Coagulation Equation with Injection

The stationary solution of Smoluchowski’s coagulation equation with injection is found analytically with different exponentially decaying source terms. The latter involve a factor in the form of a power law function that plays a decisive role in forming the steady-state particle distribution shape. An unsteady analytical solution to the coagulation equation is obtained for the exponentially decaying initial distribution without injection. An approximate unsteady solution is constructed by stitching the initial and final (steady-state) distributions. The obtained solutions are in good agreement with experimental data for the distributions of endocytosed low-density lipoproteins.

Effect of Interparticle Attraction on a Complex Plasma With a Directional Flow

In this article, the effect of attraction between particles on a 2-D complex plasma system by applying a local directional flow is investigated by Langevin molecular dynamics simulation. The structures and dynamics of the system are studied. The steady-state particle distribution and Voronoi diagram are used to represent the system structural property. The particle speed statistical distribution, mean square displacement and trajectories are used to investigate the system dynamical property. The particles' kinetic energy evolves as the time under different interparticle attractions are compared. The trajectories of individual particles by initially locating inside or outside the flow are given to show the detailed influence of attraction on different particles at different locations. These results indicate that the system properties can be influenced by different interparticle attractions by considering a directional flow.

Distribution of randomly diffusing particles in inhomogeneous media.

Diffusion can be conceptualized, at microscopic scales, as the random hopping of particles between neighboring lattice sites. In the case of diffusion in inhomogeneous media, distinct spatial domains in the system may yield distinct particle hopping rates. Starting from the master equations (MEs) governing diffusion in inhomogeneous media we derive here, for arbitrary spatial dimensions, the deterministic lattice equations (DLEs) specifying the average particle number at each lattice site for randomly diffusing particles in inhomogeneous media. We consider the case of free (Fickian) diffusion with no steric constraints on the maximum particle number per lattice site as well as the case of diffusion under steric constraints imposing a maximum particle concentration. We find, for both transient and asymptotic regimes, excellent agreement between the DLEs and kinetic Monte Carlo simulations of the MEs. The DLEs provide a computationally efficient method for predicting the (average) distribution of randomly diffusing particles in inhomogeneous media, with the number of DLEs associated with a given system being independent of the number of particles in the system. From the DLEs we obtain general analytic expressions for the steady-state particle distributions for free diffusion and, in special cases, diffusion under steric constraints in inhomogeneous media. We find that, in the steady state of the system, the average fraction of particles in a given domain is independent of most system properties, such as the arrangement and shape of domains, and only depends on the number of lattice sites in each domain, the particle hopping rates, the number of distinct particle species in the system, and the total number of particles of each particle species in the system. Our results provide general insights into the role of spatially inhomogeneous particle hopping rates in setting the particle distributions in inhomogeneous media.

Condensation transition in a conserved generalized interacting zero-range process.

A conserved generalized zero-range process is considered in which two sites interact such that particles hop from the more populated site to the other with a probability p. The steady-state particle distribution function P(n) is obtained using both analytical and numerical methods. The system goes through several phases as p is varied. In particular, a condensate phase appears for p_{l}<p<p_{c}, where the bounding values depend on the range of interaction, with p_{c}<0.5 in general. Analysis of P(n) in the condensate phase using a known scaling form shows there is universal behavior in the short-range process while the infinite range process displays nonuniversality. In the noncondensate phase above p_{c}, two distinct regions are identified: p_{c}<p≤0.5 and p>0.5; a scale emerges in the system in the latter and this feature is present for all ranges of interaction.

Density-Driven Segregation in Binary and Ternary Granular Systems

We present a first experimental study of density-induced segregation within a three-dimensional, vibrofluidised, ternary granular system. Using Positron Emission Particle Tracking (PEPT), we study the steady-state particle distributions achieved by binary and ternary granular beds under a variety of differing system parameters. In doing so, we determine the extent to which the segregative processes in ternary systems resemble their comparatively well-researched binary counterparts. We examine the influence of particle elasticity on the system, demonstrating the existence of a significant parameter range for which the effects of inelasticity-induced segregation may be safely neglected. Finally, we investigate the existence of a causal link between convective motion within a system and the segregative behaviours observed.

Losses and nonlinear steady-state particle distribution functions for fully ionized tokamak-plasmas in the collisional transport regimes

Fully ionized L-mode tokamak plasmas in the fully collisional (Pfirsch-Schluter) and in the low-collisional (banana) nonlinear transport regimes are analyzed. We derive the expressions for particles and heat losses together with the steady-state particle distribution functions in the several collisional transport regimes. The validity of the nonlinear closure equations, previously derived, has been indirectly tested by checking that the obtained particle distribution functions are indeed solutions of the nonlinear, steady-state, Vlasov-Landau gyro-kinetic equations. A quite encouraging result is the fact that, for L-mode tokamak plasmas a dissymmetry appears between the ion and electron transport coefficients: the latter submits to a nonlinear correction, which makes the radial electron coefficients much larger than the former. In particular we show that when the L-mode JET plasma is out of the linear region, the Pfirsch-Schluter electron transport coefficients are corrected by an amplification factor, which may reach values of order 10 2 . Such a correction is absent for ions. On the contrary, in the banana regime, the ion transport coefficients are increased by a factor 2 and the nonlinear corrections for electrons are negligible. These results are in line with experiments.

Precipitation strengthened high strength-conductivity copper alloys containing ZrC ceramics

Metal matrix composites (MMC) of copper and ZrC powders were prepared by the powder metallurgy route. Powders of both materials were introduced in stainless steel jag and mechanically mixed in a planetary mill under a protective atmosphere of argon during time ranging from 4 to 30 h. A characterization study of resulting powders indicated that a steady state particle distribution is developed after 12 h of milling. For this reason only these powders were consolidated by hot uniaxial pressing under 90 MPa at 923 K for 2 h. Residual porosity of these compacts was eliminated by rolling at 1073 K. Resulting materials present fine grained microstructure, tensile strength of 653 MPa and electrical conductivity up to 52% IACS. A detailed microstructural analysis by scanning and transmission electron microscopy showed not only ZrC particles but also the presence of very fine particles of carbides and oxides. WDX and EDX spectra of these particles revealed a high content of Fe and Cr for the carbides and Cu in the case of oxides.

On the application of Thomson's random flight model to the prediction of particle dispersion within a ventilated airspace

The correct prediction of mean particle concentrations within ventilated airspaces is an important practical problem. Recently, much progress has been made in understanding how to formulate correctly random flight models for particle trajectories in such inhomogeneous turbulent flows. Thomson showed that random flight models which satisfy the wellmixed condition (i.e. give the correct steady-state distribution of particles in phase space) are essentially correct. Thomson subsequently formulated a random flight model which satisfies the well-mixed condition for inhomogeneous Gaussian turbulence. Prompted by the success of this model in predicting particle dispersion in one-dimensional and two-dimensional inhomogeneous turbulent flows, an assessment is made of the ability of the model to predict correctly mean particle concentrations in a three-dimensional inhomogeneous turbulent flow within a mechanically ventilated airspace. When used in conjunction with the statistical properties of the air flow predicted by the k − ε model, Thomson's model is shown to predict mean particle concentrations in close accord with experimental findings. It is suggested that skewness of the turbulent velocity fluctuations is of secondary importance in determining particle dispersion in highly inhomogeneous turbulent flows, compared to the effects of strong mean streamline straining and large gradients in Reynolds stress. This view is supported by numerical studies undertaken using an extension of Thomson's model, which takes partial account of the skewness of velocity fluctuations. It is further suggested that when predicting particle dispersion in strongly inhomogeneous turbulent flows, it may not be necessary to devise random flight models which satisfy the well-mixed condition for more realistic (i.e. non-Gaussian) probability distribution functions of turbulent velocity fluctuations. This would allow difficulties associated with determining such probability distribution functions and the increased complexity of such models to be circumvented.

Steady-state particle distributions in shear flows of colloids and finely dispersed suspensions

On the basis of an analysis of the pseudoturbulent motion of both the suspended particles and the carrier fluid, the normal stress components in the dispersed phase are obtained for the problem of inclined confined flows of finely dispersed suspensions and colloids. These hydrodynamic pulsations are due to the shear and the work done by the average relative flow of the fluid phase on random concentration fluctuations of the disperse system because of the substantial slip of the phases of the suspension under gravity. The momentum conservation equations for the particles are obtained with allowance for the angle of inclination of the flow to the vertical and on the basis of these equations the suspension capacity of the flow as a function of the angle of inclination, particle size, Galileo number and other parameters is illustrated.

Concentration and Fractionation of Small Particles in Liquid by Ultrasound

The efficacy of the ultrasonic standing plane wave in concentrating small particles was theoretically evaluated and compared with experimental results. Acoustic energy density was estimated by measuring the ultrasonic absorption, and particle distribution was observed by dark-field microscopy. The theory predicts that diffusion is negligible in concentrating polystyrene spheres larger than 5 µ m in diameter when they are subjected to 2 J/m3 ultrasound. The half-width of the steady-state particle distribution was of the same order of magnitude as the theoretical value for the particles of 1 µ m and 2 µ m diameter. We also applied this concentrating technique to fractionation of polystyrene spheres 10 µ m in diameter, and more than 90% of the particles in the laminar flow were successfully collected.

Reflection boundary conditions for random walk models of dispersion in non-gaussian turbulence

Abstract Several methods of treating boundaries in random walk models of dispersion in flows with skew velocity distributions are considered. It is shown that only one of these methods is completely consistent with the requirement that the model should support a steady state distribution of particles which is uniform in space. This is illustrated with numerical simulations.

Random walk modelling of diffusion in inhomogeneous turbulence

AbstractThe use of random walk techniques in diffusion modelling has been very successful when the turbulence is not too inhomogeneous. In this paper the difficulties associated with inhomogeneous turbulence are investigated, and a random walk model is developed which ensures the correct steady state distribution of particles in phase space. The model requires the generation of random numbers from non‐Gaussian distributions, and this makes the model difficult to apply in cases of severe inhomogeneity. An alternative model is presented which overcomes this problem in cases where the velocity distribution is Gaussian.

ATMOSPHERIC TURBULENT DIFFUSION FROM INFINITE LINE SOURCES: AN ELECTRIC ANALOG SOLUTION

Abstract The steady-state distribution of particles emitted by an infinite line source in the presence of winds and atmospheric turbulence is determined by means of an electric analog approach. A logarithmic wind profile and a linear eddy-diffusivity profile are assumed, and a series of normalized curves relating the horizontal wind velocity, the vertical eddy diffusivity, and the source magnitude to the concentration in the horizontal and vertical direction are presented. A method for extending the technique to the treatment of finite line sources and point sources is indicated.