Equilibrium Velocity Distribution Research Articles

Inhomogeneous hard rod systems: comparison of theoretical and numerical results

The exact dynamics of a system of N hard rods of different masses and the Boltzmann equation for an infinitely extended mixture of two kinds of hard rods are used to derive formulae for the equilibrium velocity distributions the collision frequency and the velocity autocorrelation function to first order in time. These formulae and a symmetry relation for the self-diffusion coefficients of a binary mixture are used to discuss some of the results of the molecular dynamics simulations performed by Marro and Masoliver (1985).

Disturbance of equilibrium velocity and rotational distributions of dissociating diatomic molecules

Disturbance of equilibrium velocity and rotational distributions of dissociating diatomic molecules

Monte-Carlo simulation on rarefied gas flow through two-dimensional slits (Cases of high pressure ratio)

The direct simulation Monte-Carlo method has been used to study the behaviour of the rarefied gas flow through the two-dimensional slits, where the pressure ratio between the upstream and the downstream of the slit is high enough to neglect the downstream pressure. In order to correct the error due to the limitation of the simulated physical-space, the equilibrium velocity distribution modified by the addition of the stream velocity normal to the boundary surface, instead of the equilibrium one, is assumed to the molecules which flow into the simulated region through the upstream boundary. About 16000 molecules of the hard sphere model and 800 cells are used in the calculation. The conductances of the slits calculated by the simulation are in good agreement with the experimental results.

The drift velocity of a hard-sphere Lorentz gas

Studies the Boltzmann equation for a Lorentz gas with scattering on stationary hard spheres in the presence of a constant field E. The exact initial and asymptotic time evolutions are given and compared with numerical calculations. Starting with an initial equilibrium velocity distribution the authors study the influence of the initial temperature T on the drift velocity of the Lorentz gas. The drift velocity quickly reaches a maximum and then decreases slowly towards zero. In particular an upper bound, close to 0.8 E1/2 lambda 1/2, exists for the drift velocity. Here lambda =( pi a2n)-1 is the mean free path, related to the density n and radius a of the scatterers. In an initially cool gas the drift velocity slows down as t-1/2 soon after the maximum is passed. In an initially hot gas, however, there are two asymptotic regimes. After a time of order lambda 1/2E-1/2 the drift velocity stays constant for a time interval whose length is proportional to T3/2, and eventually decays as t-1/2.

Comparison of Weibel Instability and Magnetic Field Generating Thermal Instability

Comparison of the mechanisms of the Weibel instability and the field generating thermal instability is made on the basis of the Boltzmann equation. The case k ⊥∇ T 0 //∇ n 0 is investigated, where k is the wave vector, T 0 and n 0 are the zeroth order temperature and the number density. In the collisional plasma ( k l m ≪1, where l m is the electron mean free path), the thermal instability is unstable, but the Weibel instability is stable. In the collisionless case ( k l m ≫1), the situation is vice versa. When the equilibrium velocity distribution function is expanded by the Legendre polynomials P l with the polar axis along ∇ T 0 , l =2 and l =1 components are essential, respectively, for the Weibel instability and the thermal instability.

Equilibrium distributions for relativistic free particles in thermal radiation within classical electrodynamics

The equilibrium velocity distribution of classical free particles interacting with random classical radiation is investigated using the model of Einstein and Hopf, and extended to allow relativistic particle velocities. The model considers a massive free particle which has an electric dipole oscillator mounted inside; the oscillator provides the interaction between the particle and the random radiation. In this paper we give the calculations leading to a Fokker-Planck equation for the equilibrium distribution of particle momenta, and evaluate the equation for a number of radiation spectra. We find the following results: (i) If the random classical radiation is the Rayleigh-Jeans law for thermal radiation, then the equilibrium particle distribution follows the Boltzmann distribution only at low temperatures where nonrelativistic particle velocities are involved. The Rayleigh-Jeans law does not lead to the Boltzmann distribution for relativistic free particles. (ii) If the random classical radiation is the zero-point radiation spectrum, then the equilibrium particle distribution is the Lorentz-invariant distribution. (iii) If the random classical radiation is the Planck spectrum with zero-point radiation, then the equilibrium particle distribution goes over asymptotically to the Lorentz-invariant distribution at velocities near the speed of light. Thus no finite number of free particles can form an equilibrium velocity distribution if zero-point radiation is present. The particles will diffuse to velocities ever closer to the speed of light. We conclude that equilibrium for a finite number of particles must involve some explicit mechanism for confinement if classical zero-point radiation is present.

Evaluation of cross sections of theLi6(d, α)αreaction

Measured cross sections for the $^{6}\mathrm{Li}(d, \ensuremath{\alpha})\ensuremath{\alpha}$ reaction have been evaluated critically, covering the range 0.06-10 MeV and extrapolation carried out to the low energy region using Gamow plots. The reaction rate parameter $〈\ensuremath{\sigma}\ensuremath{\nu}〉$ for thermonuclear reactions at equilibrium velocity distribution has been determined as a function of the plasma temperature $T$ of the reacting nuclei from the experimental and extrapolated cross section values ($T=2 \mathrm{keV} \mathrm{to} 1.0 \mathrm{MeV}$).NUCLEAR REACTIONS $^{6}\mathrm{Li}(d, \ensuremath{\alpha})$; ${E}_{d}=3 \mathrm{keV}\ensuremath{-}1 \mathrm{MeV}$ thermalized $\ensuremath{\sigma}({E}_{d})$ extrapolation, thermalized $〈\ensuremath{\sigma}v〉$ optimized integration.

Hot photo-carrier and hot electron effects in p- n junctions

When a p- n junction of semiconductor (with bandgap energy E g ) is illuminated by light beam (with photon energy hv) in the condition of E g > hv, such as in Ge p- n junction illuminated by CO 2 laser beam, an electromotive force (emf) was induced between the terminals of the p- n junction, which indicated the opposite polarity to the ordinary photovoltaic effect like a solar cell. Such an anomalous photovoltaic phenomenon was explained by an optically excited hot carrier effect, through the following experiment with electrical excitation. Using a rod of n- or p-type Ge with a p- n junction at the surface of its center and an ohmic contact at each terminal of the rod, the same kind of phenomena was observed when electric field is applied along the length of the rod. The perpendicularly induced voltage or current had the same polarity instead of the reverse change of the applied electric field, and increases with increasing the applied field strength. The perpendicularly induced emf was caused by warm or hot carriers crossing the potential barrier of the p- n junction, which is very sensitive to the departure from thermally equilibrium velocity distribution of carriers.

Lateral Momentum Flux in Boundary Currents

Abstract Some simple momentum advection effects are considered in a current aligned with the y axis on whichthere is superimposed a flow in the x-z plane. The cross flow coupled with horizontal shear in thecurrent tends to generate differences along the vertical in the longshore velocity, while vertical mixingtends to even out such differences. As in the scalar diffusion problem considered by Taylor, a balanceis possible between the two tendencies. The equilibrium velocity distribution may support considerablelateral momentum flux, which, in the case of zero rotation, is directed down the velocity gradient, allowingthe definition of an effective horizontal viscosity. When rotational effects are significant, both the senseand the magnitude of the momentum flux come to depend in a complex way on the total vorticity f + S,where f is Coriolis parameter and S the current shear. Some illustrative examples are calculated for cross flow produced by frictional effects in a boundary current. These show that...

An exact solution of the spin−spin autocorrelation function for a one−dimensional system of hard rods

It is shown exactly that, in a one−dimensional system of hard rods with spins, the autocorrelation function of any function of spin F(w) decays as t−1 at long times provided that 〈F〉eq exists and that g (0) ≠ 0, where g (v) is the linear velocity distribution function. As a consequence of this, when F (w) = w, the spin diffusion coefficient defined by the Kubo relation Ds = ℱ∞0 〈w (0) w (t) 〉 dt does not exist. The results are true for arbitrary initial equilibrium velocity and spin distributions, the only restriction being that they be symmetric.

Stabilization of Hydromagnetic Instabilities by Magnetization Currents?

THE importance of hydromagnetic instabilities in collisionless anisotropic plasmas has long been established, especially for cosmic plasmas and solar wind conditions1–4. These instabilities in their simplest version arise from the presence of the anisotropic plasma pressure or, in the picture of the kinetic theory, from the anisotropy of the particle equilibrium velocity distribution. The quasilinear kinetic stabilization of the firehose mode has been considered in connexion with the theory of the Earth's bow shock wave by Kennel and Sagdeev4. These papers, however, refer only to homogeneous plasmas in which magnetization currents are absent.

Linearized Variational Analysis of Single-Species, One-Dimensional Vlasov Plasmas

Analysis of linearized Vlasov plasmas by means of Hamilton's variational principle is illustrated by application to the initial-value problem for a neutralized, single-species plasma in one-spatial dimension with periodic boundary conditions. For any degree of approximation and for arbitrary equilibrium velocity distributions, the problem is reduced to the solution of a system of ordinary differential equations in time with constant coefficients and time-dependent driving terms; an exact particular solution has been found. The equations are discussed, including the relation of the equations to Eulerian treatments of the problem, and the use of the equations to obtain numerical results. Numerical examples have been worked out for a Maxwellian equilibrium distribution function, equilibrium distribution functions that are sums of two Maxwellian distributions, and for nonanalytic equilibrium distribution functions that are represented by cubic spline functions. Comparisons are made between numerical results from the variational method and precise solutions of the exact plasma dispersion relation.

Kinetic theory of subsonic and supersonic transport processes in weakly ionized gases

A statistical theory of the inter-component momentum and energy exchange in nonisothermal many-component systems is presented, in which the particles interact according to the elastic-sphere model. It is shown that the inter-component transport processes depend through transcendental functions on the drift velocities of the components, simple functional relations resulting for infra-sonic and ultra-sonic drift velocities. As an application, a theoretical description is given of: 1) the nonlinear electrical conduction of weakly ionized plasmas in strong electric fields, 2) the inter-component thermal nonequilibrium of a weakly ionized plasma in strong electric fields, and 3) the relaxation of energetic test particles by collisions with field particles having an equilibrium velocity distribution. By means of these concrete examples, the existence of subsonic and supersonic transport processes is discussed, respectively.

Electron Impact Excitation of the Dayglow

Calculations are described of the equilibrium velocity distributions of the photoelectrons produced in the F region by solar ionizing radiation. Detailed estimates are presented of the intensities and altitude profiles of emission features of atomic oxygen, molecular nitrogen, and molecular oxygen, appearing in the dayglow as a result of photoelectron impacts.

Transverse instabilities in a collisionless plasma

Transverse instabilities in an electron gas with a neutralizing background, in the absence of a permanent magnetic field are discussed, mostly when the equilibrium velocity distribution is slightly spheroidal, i.e. of the formwith α ˜ γ and with the functional form of f0 not given explicitly. It is found that two types of unstable waves exist; one grows at a rate proportional to ε½‖ cos θ‖3 whenever ε > 0, the other at a rate proportional to(ε cos2θ)½ whenever ε cos 2θ > 0, where ε = (γ2/α2) — 1, and θ is the angle the wave vector makes with the axis of symmetry of the velocity spheroid. It is shown that within the approximation we are working the complex phase velocity is pure imaginary, i.e. the unstable waves are standing waves. The effect of a small permanent magnetic field is considered. The phase velocity becomes complex, at least for propagation parallel to the magnetic field and the instability rate decreases by a factor where H0 is the permanent magnetic field and 〈H2〉½ the r.m.s. maximum field that grows before non-linear effects take over. The maximum instability rate is compared with the collision rate and is found to be higher in low density, high temperature plasmas such as the solar wind. An application to collisionless shock theory is suggested.

Linear Response Theory of Longitudinal Plasma Excitations

AbstractThe plasma convolution equation equivalent to the initial problem of linearized BOLTZMANN‐VLASOV and POISSON'S equations of an isotropic, one‐dimensional and nonrelativistic plasma is derived. The integral equation obtained involves both the time t and space coordinate x.The solution of this equation is exhibited in terms of a forcing function and a resolvent kernel. The forcing function is an exciting electric field caused by the initial disturbance of plasma equilibrium. The resolvent kernel obeys an integral kernel equation which also involves the kernel of the convolution plasma equation and the resolvent is interpreted as a plasma response to the unit impulse disturbance. The plasma kernel is directly expressed by the equilibrium velocity distribution. The plasma response and the equilibrium distribution are related in the same way as the response of a transmission‐line is related to its transfer function and the integral‐kernel equation plays a key role in the formulation of problems of plasma analysis and synthesis. To analyse plasma, is to determine its electric response for a given equilibrium velocity distribution of plasma components and to synthesize plasma is to design an equilibrium distribution of plasma components for a required plasma response.In this paper the plasma analysis is carried out and the example of Lorentzian and Maxwellian plasma equilibrium distributions are considered. The plasma convolution equation equivalent to the one‐point boundary problem, (x = 0), of linearized BOLTZMANN‐VLASOV and POISSON'S equations is presented. The absence of discrete spectrum solutions is evident from the plasma convolution equations but the oscillation properties of plasma are preserved for the examples considered. The complete analytical results are obtained for a Lorentzian plasma. The electric response of the plasma is shown to be a temporal oscillation travelling with the velocity U = x/t and the amplitude of the travelling oscillation depending on U and decaying as t−1. For a Maxwellian plasma the asymptotic expansion of the electric response is obtained for short and long times and a fixed x. The result for the short‐time limit is the same as in the Lorentzian plasma case and for the long time approximation the plasma response is shown to be an oscillation decaying as t−1.

On non-Landau damped solutions to the linearized vlasov equation

Because of the existence of perturbations on the equilibrium velocity distribution of an electron plasma that are entire functions and yet do not give the damping predicted by Landau, the Landau calculation is re-examined critically, for the source of these solutions. It is shown that certain tacit assumptions go into the Landau calculation which were not proved there, but are demonstrated to be true. A broad class of initial perturbations is constructed that satisfies all the Landau criteria but does not yield Landau damping. These results derived for a Maxwell equilibrium distribution are generalized to a broader class of equilibrium distributions. Newcomb’s result that an increasing electric potential cannot be monotonic is quite simply proved as a by-product of the integral equation for the plasma.

Dispersion Function for a Plasma with a Cauchy Equilibrium Distribution

The dispersive properties of a plasma with a generalized Cauchy equilibrium velocity distribution of the form (v2 + a2)−3 are derived for certain modes of propagation. The plasma dispersion function is found in a simple algebraic form which makes the solution of the dispersion equations readily accessible. A quartic dispersion relation which is obtained for right-hand circularly polarized electromagnetic waves propagating along a uniform static magnetic field is solved for all frequencies including cyclotron resonance. The complex index of refraction for this mode is plotted for several values of the temperature and the plasma-cyclotron frequency ratio. In general, thermal effects reduce the infinite discontinuity in phase at the cyclotron resonance to a finite value and cause thermal damping of the propagating wave below resonance. The results are compared with previous numerical analyses using the Maxwell distribution.

Two Models of Brownian Motion

Two simple models of Brownian motion are presented. They are used to examine the circumstances under which the detailed velocity dependence of collisions may be adequately represented by a continuous frictional damping. The differences between the equilibrium velocity distributions predicted by a damping approximation and an exact treatment are discussed.

Oscillations of a Nonuniform Plasma

The oscillations of a plasma which is confined by an rf field are investigated. The confining potential is approximated as ψ(x) = ½mω02x2. Longitudinal plasma oscillations in the x direction are determined from the self-consistent Boltzmann transport equation without the collision term (Vlasov equation). This equation is linearized about equilibrium velocity distribution f0 = exp [−(ψ + ½mv2)/κT]. By expanding the electric field in Hermite polynomials, it is possible to reduce exactly the resulting integro-differential equation to an infinite system of linear equations for the expansion coefficients. The resonant frequencies are the roots of the determinant of the system. The frequency spectrum so obtained is quite unlike those obtained for Sturm-Liouville problems. This spectrum contains the integral multiples of ω0 as limit points. As e2n/mω02 → 0, the resonant frequencies coalesce into these limit points, each of these frequencies μω0 (μ = integer) being infinitely degenerate. Since all frequencies are real, the oscillations are not damped. The resonant frequencies are determined approximately as functions of e2n/mω02 as the roots of principal sub-determinants of finite order N. This procedure converges rapidly with increasing N.