Velocity-space diffusion due to resonant wave–wave scattering of electromagnetic and electrostatic waves in a plasma

1991 ◽  
Vol 45 (1) ◽  
pp. 103-113 ◽  
Author(s):  
Reiji Sugaya
Keyword(s):  
Distribution Function ◽  
Wave Scattering ◽  
Maxwell Equations ◽  
Velocity Distribution Function ◽  
Velocity Space ◽  
Electrostatic Waves ◽  
Resonant Wave ◽  
Unmagnetized Plasma ◽  
Energy And Momentum ◽  
Energy And Momentum Densities

The velocity-space diffusion equation describing distortion of the velocity distribution function due to resonant wave-wave scattering of electromagnetic and electrostatic waves in an unmagnetized plasma is derived from the Vlasov-Maxwell equations by perturbation theory. The conservation laws for total energy and momentum densities of waves and particles are verified, and the time evolutions of the energy and momentum densities of particles are given in terms of the nonlinear wave-wave coupling coefficient in the kinetic wave equation.

Momentum-space diffusion due to resonant wave–wave scattering of electromagnetic and electrostatic waves in a relativistic magnetized plasma

1996 ◽  
Vol 56 (2) ◽  
pp. 193-207 ◽  
Author(s):  
R. Sugaya
Keyword(s):  
Wave Equation ◽  
Diffusion Equation ◽  
Nonlinear Wave ◽  
Coupling Coefficient ◽  
Momentum Space ◽  
Wave Scattering ◽  
Magnetized Plasma ◽  
Electrostatic Waves ◽  
Resonant Wave ◽  
Energy And Momentum

The momentum-space diffusion equation and the kinetic wave equation for resonant wave–wave scattering of electromagnetic and electrostatic waves in a relativistic magnetized plasma are derived from the relativistic Vlasov–Maxwell equations by perturbation theory. The p-dependent diffusion coefficient and the nonlinear wave—wave coupling coefficient are given in terms of third-order tensors which are amenable to analysis. The transport equations describing energy and momentum transfer between waves and particles are obtained by momentum-space integration of the momentum-space diffusion equation, and are expressed in terms of the nonlinear wave—wave coupling coefficient in the kinetic wave equation. The conservation laws for the total energy and momentum densities of waves and particles are verified from the kinetic wave equation and the transport equations. These equations are very useful for the theoretical analysis of transport phenomena or the acceleration and generation of high-energy or relativistic particles caused by quasi-linear and resonant wave—wave scattering processes.

A study of the mass ratio dependence of the mixed species collision integral for isotropic velocity distribution functions

1982 ◽  
Vol 27 (1) ◽  
pp. 135-148 ◽  
Author(s):  
A. J. M. Garrett
Keyword(s):  
Distribution Function ◽  
Velocity Distribution ◽  
Particle Distribution ◽  
Velocity Distribution Function ◽  
Collision Integral ◽  
Distribution Functions ◽  
Velocity Space ◽  
Particle Distribution Function ◽  
Collision Term ◽  
Test Species

This paper is concerned with the Boltzmann collision integral for the one-particle distribution function of a test species of particle undergoing elastic collisions with particles of a second species which is in thermal equilibrium. This expression is studied as a function of the ratio of the masses of the test and host particles for the case when the test particle distribution function is isotropic in velocity space. The analysis can also be considered as referring to the zeroth-order spherical harmonic in velocity space of a general velocity distribution function. The resulting collision term, due originally to Davydov, is of Fokker–Planck form and effectively describes a diffusion in energy. The method of derivation employed here is more systematic than hitherto, and is used to calculate the first correction to the Davydov term. Differences between classical and quantum cross-sections are considered; the correction to the Davydov term is checked by means of a comparison with the exact solution of the associated eigenvalue problem for the special case of Maxwell interactions treated classically.

scholarly journals Kinetic analysis of propagating ion beam with leading and trailing edges as ICF energy driver

1989 ◽  
Vol 7 (2) ◽  
pp. 207-217 ◽  
Author(s):  
Takeshi Kaneda ◽  
Keishiro Niu
Keyword(s):  
Distribution Function ◽  
Electromagnetic Fields ◽  
Particle Velocity ◽  
Maxwell Equations ◽  
Ion Beam ◽  
Velocity Distribution Function ◽  
Radial Coordinate ◽  
Axial Coordinate ◽  
Beam Velocity ◽  
Trailing Edges

Analysis is given for nonstationary propagation of rotating ion beam which has a finite length on the basis of Vlasov–Maxwell equations. The beam velocity distribution function is assumed to have a form of product of a modification function g which depends on time and axial coordinate multiplied by a steady solution fb0 which is a known function of particle velocity and radial coordinate. Unknown function g is solved as the solution of Vlasov equation through electromagnetic fields induced by leading and trailing edges. These electromagnetic fields can be solved from the Maxwell equations by using beam distribution function and motion of electrons in background plasma.

Asymptotic solutions for the distribution function in non-equilibrium flows. Part 1. The weak shock

1968 ◽  
Vol 34 (1) ◽  
pp. 1-24 ◽  
Author(s):  
Roddam Narasimha
Keyword(s):  
Distribution Function ◽  
Weak Shock ◽  
Velocity Distribution Function ◽  
Velocity Space ◽  
Approach To Equilibrium ◽  
Cold Side ◽  
Interparticle Potential ◽  
Large Perturbation ◽  
Very High ◽  
Non Equilibrium

This paper is the first part of an investigation of the molecular velocity distribution function in non-equilibrium flows. In this part, the general features of the distribution are discussed and illustrated by a detailed study of its asymptotic expansions in different velocity domains for a weak shock, employing a simple relaxation model for the collisions. Using the strength of the shock as a small parameter, the Chapman–Enskog distribution is derived, in a less restrictive way than in previous analyses, as the first two terms in a suitable ‘inner’ asymptotic expansion of the distribution in velocity space, valid only for not very high velocities. It is shown that the consideration of two further limits, called the intermediate and outer, is necessary for a complete description of the distribution in velocity space. The uniformly valid composite expansion demonstrates the slow approach to equilibrium of fast molecules. The outer solution depends on integrals over the flow and is in general ‘global’, in contrast to the inner solution which is essentially local; this introduces certain asymmetries on a fine scale even in a weak shock. It is shown, for example, that fast molecules moving towards the hot side accumulate by collisionless streaming, whereas those moving towards the cold side attenuate like a molecular beam and represent essentially a ‘precursor’ of the hot side. A simple approximation for the distribution in the precursor is derived, and found to contain, in the outer limit, a large perturbation on the local Maxwellian; this results in an approach to equilibrium like $\exp (-|x|^{\frac{2}{3}})$ on the cold side.A heuristic extension of the argument to the true Boltzmann equation leads to the result that for molecules with an interparticle potential varying as the inverse m-power of the distance, the approach to equilibrium through the precursor is like $\exp (-|x|^l)$, where l = m/(m + 2).

scholarly journals On relaxation processes in a completely ionized plasma

2020 ◽  
Keyword(s):  
Distribution Function ◽  
Electric Field ◽  
External Electric Field ◽  
Electron Distribution ◽  
Electron Distribution Function ◽  
Relaxation Times ◽  
Relaxation Processes ◽  
Energy And Momentum ◽  
Energy And Momentum Densities ◽  
Relaxation Coefficients

Relaxation of the electron energy and momentum densities is investigated in spatially uniform states of completely ionized plasma in the presence of small constant and spatially homogeneous external electric field. The plasma is considered in a generalized Lorentz model which contrary to standard one assumes that ions form an equilibrium system. Following to Lorentz it is neglected by electron-electron and ion-ion interactions. The investigation is based on linear kinetic equation obtained by us early from the Landau kinetic equation. Therefore long-range electron-ion Coulomb interaction is consequentially described. The research of the model is based on spectral theory of the collision integral operator. This operator is symmetric and positively defined one. Its eigenvectors are chosen in the form of symmetric irreducible tensors which describe kinetic modes of the system. The corresponding eigenvalues are relaxation coefficients and define the relaxation times of the system. It is established that scalar and vector eigenfunctions describe evolution of electron energy and momentum densities (vector and scalar system modes). By this way in the present paper exact close set of equations for the densities valid for all times is obtained. Further, it is assumed that their relaxation times are much more than relaxation times of all other modes. In this case there exists a characteristic time such that at corresponding larger times the evolution of the system is reduced described by asymptotic values of the densities. At the reduced description electron distribution function depends on time only through asymptotic densities and they satisfy a closed set of equations. In our previous paper this result was proved in the absence of an external electric field and exact nonequilibrium distribution function was found. Here it is proved that this reduced description takes also place for small homogeneous external electric field. This can be considered as a justification of the Bogolyubov idea of the functional hypothesis for the relaxation processes in the plasma. The proof is done in the first approximation of the perturbation theory in the field. However, its idea is true in all orders in the field. Electron mobility in the plasma, its conductivity and phenomenon of equilibrium temperature difference of electrons and ions are discussed in exact theory and approximately analyzed. With this end in view, following our previous paper, approximate solution of the spectral problem is discussed by the method of truncated expansion of the eigenfunctions in series of the Sonine polynomials. In one-polynomial approximation it is shown that nonequilibrium electron distribution function at the end of relaxation processes can be approximated by the Maxwell distribution function. This result is a justification of Lorentz–Landau assumption in their theory of nonequilibrium processes in plasma. The temperature and velocity relaxation coefficients were calculated by us early in one- and two-polynomial approximation.

scholarly journals Velocity distribution function of hydrogen atoms in ion source discharges

2021 ◽  
Author(s):  
Tatsuhiro Tokai ◽  
Yuji Shimabukuro ◽  
Hidenori Takahashi ◽  
Keita Bito ◽  
Motoi Wada
Keyword(s):  
Distribution Function ◽  
Velocity Distribution ◽  
Velocity Distribution Function ◽  
Ion Source ◽  
Hydrogen Atoms

scholarly journals Velocity distribution function of spontaneously evaporating atoms

2020 ◽  
Vol 5 (10) ◽  
Author(s):  
Sergiu Busuioc ◽  
Livio Gibelli ◽  
Duncan A. Lockerby ◽  
James E. Sprittles
Keyword(s):  
Distribution Function ◽  
Velocity Distribution ◽  
Velocity Distribution Function

scholarly journals Linear Energy Density and the Flux of an Electric Field in Proca Tubes

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 640
Author(s):  
Vladimir Dzhunushaliev ◽  
Vladimir Folomeev ◽  
Abylaikhan Tlemisov
Keyword(s):  
Scalar Field ◽  
Electric Field ◽  
Higgs Field ◽  
Meissner Effect ◽  
Coupling Constant ◽  
Cylindrically Symmetric ◽  
Integral Characteristics ◽  
Energy And Momentum ◽  
Energy And Momentum Densities ◽  
Higgs Scalar

In this work, we study cylindrically symmetric solutions within SU(3) non-Abelian Proca theory coupled to a Higgs scalar field. The solutions describe tubes containing either the flux of a color electric field or the energy flux and momentum. It is shown that the existence of such tubes depends crucially on the presence of the Higgs field (there are no such solutions without this field). We examine the dependence of the integral characteristics (linear energy and momentum densities) on the values of the electromagnetic potentials at the center of the tube, as well as on the values of the coupling constant of the Higgs scalar field. The solutions obtained are topologically trivial and demonstrate the dual Meissner effect: the electric field is pushed out by the Higgs scalar field.

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