Collisionless System Research Articles

Nonplanar ion-acoustic shock waves in degenerate plasmas with positively charged heavy ions

The theoretical and numerical study on the nonlinear propagation of heavy-ion-acoustic (HIA) shock waves has been carried out in an unmagnetized, collisionless dense plasma system (containing degenerate electron and inertial light ion fluids, and positively charged static heavy ions). The normal mode analyse is used to investigate the linear wave properties. Reductive perturbation technique is used to derive the Burgers equation which admits a localized wave solution for the shock profile. It is seen that the shock wave characteristics have been influenced significantly for the non-relativistic as well as for the ultra-relativistic limits. It has also been found that the effect of degenerate pressure and number density of electron and inertial light ion fluids, and positively charged static heavy ions significantly modify the basic features (speed, amplitude, width, etc.) of HIA shock waves. The relevance of our results in astrophysical objects (like white dwarfs and neutron stars), which are of scientific interest, is briefly discussed.

Cylindrical and Spherical Ion-Acoustic Shock Waves in a Relativistic Degenerate Multi-Ion Plasma

A rigorous theoretical investigation has been made to study the existence and basic features of the ion-acoustic (IA) shock structures in an unmagnetized, collisionless multi-ion plasma system (containing degenerate electron fluids, inertial positively as well as negatively charged ions, and arbitrarily charged static heavy ions). This investigation is valid for both non-relativistic and ultra-relativistic limits. The reductive perturbation technique has been employed to derive the modified Burgers equation. The solution of this equation has been numerically examined to study the basic properties of shock structures. The basic features (speed, amplitude, width, etc.) of these electrostatic shock structures have been briefly discussed. The basic properties of the IA shock waves are found to be significantly modified by the effects of arbitrarily charged static heavy ions and the plasma particle number densities. The implications of our results in space and interstellar compact objects like white dwarfs, neutron stars, black holes, and so on have been briefly discussed.

Roles of positively charged heavy ions and degenerate plasma pressure on cylindrical and spherical ion acoustic solitary waves

The properties of heavy-ion-acoustic (HIA) solitary structures associated with the nonlinear propagation of cylindrical and spherical electrostatic perturbations in an unmagnetized, collisionless dense plasma system has been investigated theoretically. Our considered model contains degenerate electron and inertial light ion fluids, and positively charged static heavy ions, which is valid for both of the non-relativistic and ultra-relativistic limits. The Korteweg-de Vries (K-dV) and modified K-dV (mK-dV) equations have been derived by employing the reductive perturbation method, and numerically examined in order. It has been found that the effect of degenerate pressure and number density of electron and inertial light ion fluids, and positively charged static heavy ions significantly modify the basic features of HIA solitary waves. It is also noted that the inertial light ion fluid is the source of dispersion for HIA waves and is responsible for the formation of solitary waves. The basic features and the underlying physics of HIA solitary waves, which are relevant to some astrophysical compact objects, are briefly discussed.

On the singularity of the Vlasov-Poisson system

The Vlasov-Poisson system can be viewed as the collisionless limit of the corresponding Fokker-Planck-Poisson system. It is reasonable to expect that the result of Landau damping can also be obtained from the Fokker-Planck-Poisson system when the collision frequency ν approaches zero. However, we show that the collisionless Vlasov-Poisson system is a singular limit of the collisional Fokker-Planck-Poisson system, and Landau's result can be recovered only as the ν approaches zero from the positive side.

Linear response theory for hydrodynamic and kinetic equations with long-range interactions

We apply the linear response theory to systems with long-range interactions described by hydrodynamic equations such as the Euler, Smoluchowski, and damped Euler equations. We analytically determine the response of the system submitted to a pulse or to a step function. We compare these results with those obtained for collisionless systems described by the Vlasov equation. We show that, in the linear regime, the evolution of a collisionless system (Vlasov) with the waterbag distribution is the same as the evolution of a collision-dominated gas without dissipation (Euler). In this analogy, the maximum velocity of the waterbag distribution plays the role of the velocity of sound in the corresponding barotropic gas. When submitted to a step function, these systems exhibit permanent oscillations. Other distributions exhibit Landau damping and relax towards a steady state. We illustrate this behaviour with the Cauchy distribution which can be studied analytically. We apply our results to the HMF model and obtain a generalized Curie-Weiss law for the magnetic susceptibility. Finally, we compare the linear response theory to the initial value problem for the linearized Vlasov equation and report a case of algebraic damping of the initial perturbation.

Gravothermal catastrophe: The dynamical stability of a fluid model

A re-investigation of the gravothermal catastrophe is presented. By means of a linear perturbation analysis, we study the dynamical stability of a spherical self-gravitating isothermal fluid of finite volume and find that the conditions for the onset of the gravothermal catastrophe, under different external conditions, coincide with those obtained from thermodynamical arguments. This suggests that the gravothermal catastrophe may reduce to Jeans instability, rediscovered in an inhomogeneous framework. We find normal modes and frequencies for the fluid system and show that instability develops on the dynamical time scale. We then discuss several related issues. In particular: (1) For perturbations at constant total energy and constant volume, we introduce a simple heuristic term in the energy budget to mimic the role of binaries. (2) We outline the analysis of the two-component case and show how linear perturbation analysis can be carried out also in this more complex context in a relatively straightforward way. (3) We compare the behavior of the fluid model with that of the collisionless sphere. In the collisionless case the instability seems to disappear, which is at variance with the linear Jeans stability analysis in the homogeneous case; we argue that a key ingredient to understand the difference (a spherical stellar system is expected to undergo the gravothermal catastrophe only in the presence of some collisionality, which suggests that the instability is dissipative and not dynamical) lies in the role of the detailed angular momentum in a collisionless system. Finally, we briefly comment on the meaning of the Boltzmann entropy and its applicability to the study of the dynamics of self- gravitating inhomogeneous gaseous systems.

Principal properties of the velocity distribution of dark matter particles on the outskirts of the Solar system

Abstract The velocity distribution of the dark matter particles on the outskirts of the Solar system remains unclear. We suggest to determine it using experimentally found properties of the oldest halo objects. Indeed, the oldest halo stars and globular clusters form a collisionless system, as well as dark matter particles do, and they evolved in the same gravitational field. If we accept this analogy, we can show that the velocity distribution of the dark matter particles should be highly anisotropic and have a sharp maximum near υ∼ 500 km s−1. The distribution is totally different from the Maxwell one. We analyse the influence of the distribution function on the results of dark matter detection experiments. It is found that the direct detection signal should differ noticeably from the one calculated from the Maxwell distribution with 〈υ〉≃ 220 km s−1, which is conventional for direct detection experiments (the ratio depends on the detector properties and typically falls within the range between 6 and 0.2). Moreover, the sharp distinction from the Maxwell distribution can be very essential to the observations of dark matter annihilation.

A new test for the Galactic formation and evolution - prediction for the orbital eccentricity distribution of the halo stars

We present theoretical calculations for the differential distribution of stellar orbital eccentricity in a galaxy halo, assuming that the stars constitute a spherical, collisionless system in dynamical equilibrium with a dark matter halo. In order to define the eccentricity e of a halo star for given energy E and angular momentum L, we adopt two types of gravitational potential, such as an isochrone potential and a Navarro–Frenk–White potential, which could form two ends covering in-between any realistic potential of dark matter halo. Based on a distribution function of the form f(E, L) that allows constant anisotropy in velocity dispersions characterized by a parameter β, we find that the eccentricity distribution is a monotonically increasing function of e for the case of highly radially anisotropic velocity dispersions (β≳ 0.6), while showing a hump-like shape for the cases from radial through tangential velocity anisotropy (β≲ 0.6). We also find that when the velocity anisotropy agrees with that observed for the Milky Way halo stars (β≃ 0.5–0.7), a nearly linear eccentricity distribution of N(e) ∝e results at e≲ 0.7, largely independent of the potential adopted. Our theoretical eccentricity distribution would be a vital tool of examining how far out in the halo the dynamical equilibrium has been achieved, through comparison with kinematics of halo stars sampled at greater distances. Given that large surveys of the SEGUE and Gaia projects would be in progress, we discuss how our results would serve as a new guide in exploring the formation and evolution of the Milky Way halo.

STATISTICAL MECHANICS OF COLLISIONLESS ORBITS. II. STRUCTURE OF HALOS

In this paper, we present the density, \rho, velocity dispersion, \sigma, and \rho/\sigma^3 profiles of isotropic systems which have the energy distribution, N(E)\propto[\exp(\phi_0-E)-1], derived in Paper I. This distribution, dubbed DARKexp, is the most probable final state of a collisionless self-gravitating system, which is relaxed in terms of particle energies, but not necessarily in terms of angular momentum. We compare the DARKexp predictions with the results obtained using the extended secondary infall model (ESIM). The ESIM numerical scheme is optimally suited for the purpose because (1) it relaxes only through energy redistribution, leaving shell/particle angular momenta unaltered, and (2) being a shell code with radially increasing shell thickness it has very good mass resolution in the inner halo, where the various theoretical treatments give different predictions. The ESIM halo properties, and especially their energy distributions, are very well fit by DARKexp, implying that the techniques of statistical mechanics can be used to explain the structure of relaxed self-gravitating systems.

Anomalous momentum transport and plasma heating in a collisionless return-current beam plasma system: Multifluid and kinetic approaches

The anomalous transport in a one-dimensional (1D), collisionless return-current beam plasma system is studied by means of electrostatic Vlasov and three-fluid simulations in a current-free counterstreaming electron beam situation. Despite of a lack of binary collisions in both approaches, the electron bulk drifts are found to slow down due to collisionless momentum transport and the electrons are heated. It is shown that in the multifluid plasma description, charge separation effects play a major role in this process. A 1D electrostatic Vlasov code simulation is performed with the same macroscopic plasma parameters to investigate the momentum transport mechanism if kinetic effects are taken into account. A comparison of the two different approaches shows that a stronger drift relaxation takes place in the multifluid plasma approach. Also, in both approaches the same drift relaxation level is reached within the same time Δt≈700 ωpe−1. This is due to the common mechanism of anomalous transport—charge separation—in multifluid and kinetic approach during the initial stage of the electron beam relaxation. By carrying out a wave-particle interaction analysis in velocity space, it is shown that, however, the essentially kinetic effects Landau damping and particle trapping stop further current relaxation when the bulk drifts velocity has reached the thermal velocity value. In conclusion, it is argued that in this situation, an additional anomalous transport term is required for a complete multifluid plasma description of the electron beam relaxation.

Self-organized relaxation in a collisionless gravitating system

We propose the self-organized relaxation process which drives a collisionless self-gravitating system to the equilibrium state satisfying local virial (LV) relation. During the violent relaxation process, particles can move widely within the time interval as short as a few free-fall times, because of the effective potential oscillations. Since such particle movement causes further potential oscillations, it is expected that the system approaches the critical state where such particle activities, which we call gravitational fugacity, is independent of the local position as much as possible. Here we demonstrate that gravitational fugacity can be described as the functional of the LV ratio, which means that the LV ratio is a key ingredient estimating the particle activities against gravitational potential. We also demonstrate that the LV relation is attained if the LV ratio exceeds the critical value b=1 everywhere in the bound region during the violent relaxation process. The local region which does not meet this criterion can be trapped into the presaturated state. However, small phase-space perturbation can bring the inactive part into the LV critical state.

Collisionless magnetic reconnection dynamics with electron inertia and parallel electron compressibility

AbstractCollisionless magnetic reconnection dynamics is considered by including the effects of electron inertia as well as parallel electron compressibility. A fluid treatment is adopted for both electrons and ions. Collisionless plasma dynamics properties near a two-dimensional X-type magnetic neutral line in the steady state are explored. The effects of electron inertia and parallel electron compressibility on the hyperbolicity (or lack thereof) of the magnetic field lines in the neutral layer are discussed. A unified linear tearing-mode formulation incorporating both electron inertia and parallel electron compressibility is given. The parallel-electron-compressibility branch is shown to couple in general to the electron-inertia branch in the presence of resistivity. A sufficient condition for linear stability in the Lyapunov sense for steady states of this collisionless plasma system signifying current confinement is deduced. Bounds on the equilibrium current gradient are shown to constitute sufficient conditions for nonlinear stability in the Lyapunov sense for steady states via nonlinear bounds for a suitable perturbation norm.

Gyro-kinetic simulation of global turbulent transport properties in tokamak experiments

A general geometry gyro-kinetic model for particle simulation of plasma turbulence in tokamak experiments is described. It incorporates the comprehensive influence of noncircular cross section, realistic plasma profiles, plasma rotation, neoclassical (equilibrium) electric fields, and Coulomb collisions. An interesting result of global turbulence development in a shaped tokamak plasma is presented with regard to nonlinear turbulence spreading into the linearly stable region. The mutual interaction between turbulence and zonal flows in collisionless plasmas is studied with a focus on identifying possible nonlinear saturation mechanisms for zonal flows. A bursting temporal behavior with a period longer than the geodesic acoustic oscillation period is observed even in a collisionless system. Our simulation results suggest that the zonal flows can drive turbulence. However, this process is too weak to be an effective zonal flow saturation mechanism.

Isolated and non-isolated dark matter haloes and the Navarro, Frenk and White profile

We compare the conclusions reached using the coarse-graining technique employed by Henriksen for a one degree of freedom (per particle) collisionless system to those presented in a paper by Binney based on an exact one degree of freedom model. We find agreement in detail, but in addition we show that the isolated 1D system is self-similar and therefore unrelaxed. Fine graining of this system recovers much less prominent wave-like structure than in a spherically symmetric isotropic 3D system. The rate of central flattening is also reduced in the 1D system. We take this to be evidence that relaxation of collisionless systems proceeds ultimately by way of short wavelength Landau damping. N-body systems, both real and simulated, can be trapped in an incompletely relaxed state because of a break in the cascade of energy to small scales. This may be due to the rapid dissipation of the small-scale oscillations in an isolated system to the existence of conserved quantities such as angular momentum, or to the failure in simulations to resolve sub-Jeans length scales. Such a partially relaxed state appears to be the Navarro, Frenk and White (NFW) state and is to be expected especially in young systems. The NFW core is shown to be isolated. In non-isolated systems, continuing coarse-grained relaxation should be towards a density core in solid body rotation.

Relaxation of a Collisionless System and the Transition to a New Equilibrium Velocity Distribution

In this paper, we present our conclusions from the numerical study of the collapse of a destabilized collisionless stellar system. We use both direct integration of the Vlasov-Poisson equations and an N-body tree code to obtain our results, which are mutually confirmed. We find that spherical and moderately nonspherical collapse configurations evolve to new equilibrium configurations in which the velocity distribution approaches a Gaussian form, at least in the central regions. The evolution to this state has long been an open question, and in this work we are able to clarify the process responsible and to support predictions made from statistical considerations (Lynden-Bell 1967; Nakamura 2000). The simulations of merging N-body systems show a transition to a Gaussian velocity distribution that is increasingly suppressed as the initial separation of centres is increased. Possible reasons for this are discussed.

Doping concentration evaluation using plasma propagation models in plasma immersion ion implantation (PIII) system

Plasma immersion ion implantation (PIII) is a high dose-rate implantation process technique in the area of semiconductor device fabrication used to fabricate various device structures like shallow junction, silicon on insulators and in the processing of flat panel display materials, trench doping, etc. The basic mechanism of ions source and their acceleration in PIII technique is different from that of the conventional ion-implantation. In this, the target is immersed in a plasma source and the implantation is done by accelerating the ions with a negative pulse bias voltage, applied to the target. The dynamics of ion transport and the implantation is different from line-of-sight implantation. In this paper, the doping of individual ions (Ar, He and N), in a collisionless PIII system is studied analytically when a negative pulse of 10 kV is applied to the target. The net ion doping concentration in one pulse duration has also been computed during the propagation of plasma sheaths.

Slow evolution of elliptical galaxies induced by dynamical friction

The main goal of this paper is to set up a numerical simulation for the study of the slow evolution of the density and of the pressure tensor profiles of an otherwise collisionless stellar system, as a result of the interactions with a minority component of heavier The effects that we would like to study are those attributed to slow collisional relaxation and generically called friction; in real cases, or in numerical simulations, the processes involved are complex, so that the relaxation associated with the granularity in phase space is generally mixed with and masked by evolution resulting from lack of equilibrium or from a variety of instabilities and collective processes. We start by revisiting the problem of the sinking of a satellite inside an initially isotropic, non-rotating, spherical galaxy, which we follow by means of N-body simulations using about one million particles. We then consider a quasi-spherical problem, in which the satellite is fragmented into a set of many smaller masses with a spherically symmetric initial density distribution. In a wide set of experiments, designed in order to bring out effects genuinely associated with dynamical friction, we are able to demonstrate the slow evolution of the density profile and the development of a tangentially biased pressure in the underlying stellar system, while we briefly address the issue of the circularization of orbits induced by dynamical friction on the population of fragments. The results of the simulations presented here and others planned for future investigations allow us to study the basic mechanisms of slow relaxation in stellar systems and thus may be of general interest for a variety of problems, especially in the cosmological context. Here our experiments are conceived with the specific goal to clarify some mechanisms that may play a role in the evolution of an elliptical galaxy as a result of the interaction between the stars and a significant population of globular clusters or of the merging of a large number of small satellites.

Collisionless collapse of stellar systems: Analytical vision

The present paper surveys some results on the collisionless system evolution that were obtained by the Gravitating System Dynamics Laboratory team from the Fessenkov Astrophysical Institute.

Shadowing-based Reliability Decay in Softened n -Body Simulations

A shadow of a numerical solution to a chaotic system is an exact solution to the equations of motion that remains close to the numerical solution for a long time. In a collisionless n-body system, we know that particle motion is governed by the global potential rather than by interparticle interactions. As a result, the trajectory of each individual particle in the system is independently shadowable. It is thus meaningful to measure the number of particles that have shadowable trajectories as a function of time. We find that the number of shadowable particles decays exponentially with time as e-μt and that for ∈ [~ 0.2, 1] (in units of the local mean interparticle separation ), there is an explicit relationship among the decay constant μ, the time step h of the leapfrog integrator, the softening , and the number of particles N in the simulation. Thus, given N and , it is possible to precompute the time step h necessary to achieve a desired fraction of shadowable particles after a given length of simulation time. We demonstrate that a large fraction of particles remain shadowable over ~100 crossing times even if particles travel up to about of the softening length per time step. However, a sharp decrease in the number of shadowable particles occurs if the time step increases to allow particles to travel farther than the softening length in 1 time step or if the softening is decreased below ~0.2.

Bunching of particles with nonzero longitudinal emittance

The dynamics of the bunching of particles with nonzero longitudinal emittance is studied in plane geometry. A self-consistent solution to the model time-dependent kinetic equation is obtained for a collisionless system of particles. The solution is shown to be nonunique. Two acceleration modes resulting in the creation of a virtual cathode and, therefore, in the bunching of particles are basically possible.