Maxwellian distribution function

This result is purely statistical. Replacing the distribution function by particular expressions, depending on the temperature, is the last operation When a dynamical process occurs the equilibrium distribution function (maxwellian) should be modified, and the greater the reaction rate compared to the relaxation rates of both the velocities and the intramolecular states, the greater the modification Thus it is only for low reaction rates that equilibrium distribution functions can be inserted in the formulas above, and that the reaction rate depends on the temperature, but neither on the time nor on the concentrations. 

The equation (1) with the collision term described for binary collisions, it isn t lineal, is for that reason that the solution is very difficult. Nevertheless, exists a solution for Boltzmann equation, it isn t trivial and is very important and is known like distribution function Maxwellian. For this case the Boltzmann equation presents a non reversible behavior and distribution function lays to distribution Maxwellian, this represent the situation of an uniform gas in stationary state. 

The velocity distribution of the electrons in a plasma is generally a complicated function whose exact shape is detennined by many factors. It is often assumed for reasons of convenience in calculations tliat such velocity distributions are Maxwellian and tliat tlie electrons are in tliennodynamical equilibrium. The Maxwell distribution is given by... 

The Burnett Expansion.—The Chapman-Enskog solution of the Boltzmann equation can be most easily developed through an expansion procedure due to Burnett.15 For the distribution function of a system that is close to equilibrium, we may use as a zeroth approximation a local equilibrium distribution function given by the maxwellian form ... 

It was shown, in Eqs. (1-73), (1-74), (1-75), that a = 1, afy r> => 0, a = 0. As the zeroth approximation we shall assume that A mid /a are zero (their effects are negligibly small) if Eqs. (1-86) and (1-87) are multiplied by /a and A, respectively, we obtain the condition that og0 and -oSi are zero higher order equations would show that all the coefficients are zero. Thus, the coefficients are proportional to some power of /a (or A). The zero-order approximation to the distribution function is just the local maxwellian distribution... 

Consider electrons of mass m and velocity v, and atoms of mass M and velocity V we have mjM 1. The distribution function for the electrons will be denoted by /(v,<) (we assume no space dependence) that for the atoms, F( V), assumed Maxwellian as usual, in the collision integral, unprimed quantities refer to values before collision, while primed quantities are the values after collision. In general, we would have three Boltzmann equations (one each for the electrons, ions, and neutrals), each containing three collision terms (one for self-collisions, and one each for collisions with the other two species). We are interested only in the equation for the electron distribution function by the assumption of slight ionization, we neglect the electron-electron... 

Electron energy distribution function The distribution function of electrons in a plasma. That of a low-pressure radiofrequency plasma generally consists of two Maxwellian distributions, that is, fast and slow electrons. 

For low-pressure plasmas containing mainly inert gases the electrons can be characterized by a Maxwellian electron energy distribution function (EEDF). How-... 

Two other attempts, without the use of a distribution function, are worth mentioning, as these are operationally related to experiments and serve to give a rough estimate of the thermalization time. Christophorou et al. (1975) note that in the presence of a relatively weak external field E, the rate of energy input to an electron by that field is (0 = eEvd, where vd is the drift velocity in the stationary state. Under equilibrium, it must be equal to the difference between the energy loss and gain rates by an electron s interaction with the medium. The mean electron energy is now approximated as (E) = (3eD )/(2p), where fl = vd /E is the drift mobility and D is the perpendicular diffusion coefficient (this approximation is actually valid for a Maxwellian distribution). Thus, from measurements of fl and D the thermalization time is estimated to be... 

However, because of the Maxwellian distribution function for molecular speeds, not all H2 molecules will be faster than all 02 molecules and some H2 molecules will have velocities near 0 m/s. Some 02 molecules will be moving faster than the average speed of H2 molecules. 

Prove the assertion in the text that the relative velocity of two sets of particles having individual Maxwellian velocity distribution functions also has a Maxwellian distribution with the masses replaced by the reduced mass. 

From Eq. (2.107), the distribution function of the centre-of-mass velocity will be a Maxwellian with a mass of mi + m2. 

The essential characteristic of the equilibrium correlations is that they originate in a system starting from non-correlated states. We recall also that the correct form of the equilibrium correlations can be obtained if one admits that for long times the velocity distribution function takes a Maxwellian form. 

Here, is the mean streaming velocity of particles approaching the wall and (1 — a) is the fraction of fluid particles reflected at the wall, so the first term represents the distribution of particles adsorbed. The velocity distribution functions, /(v), are assumed to be Maxwellian,... 

It should be noted that a Maxwellian form of fie) is a reasonable approximation to the actual distribution at low electron energies. This observation is indicated in Figure 5 i24). However, the first ionization potential of most atoms and molecules is above eV. Thus, many of the important homogeneous processes that occur in glow discharges, such as ionization, take place as a result of high energy electrons in the "tail" of the distribution. These electrons are precisely the ones that are not adequately described by a Maxwellian distribution function. 

A sufficiently rarefied gas, or a mixture of gases, consists of a number of neutral molecules of species 1 and 2 (which may or may not be the same). We may assume a distribution of velocities (measured in the laboratory frame), fi ( ) d3u, that may be modeled by a Maxwellian distribution function, with i = 1 or 2, as long as the duration of the average collision is short compared to the time between collisions. For binary collisions, one usually transforms from laboratory coordinates, Vj, to relative ( >12) and center-of-mass (1>cm) velocities,... 

The random velocities of atoms and molecules are described by velocity distribution functions which can often be approximated by a Maxwellian distribution (as in Eq. 2.10). If radiating atoms have such a distribution, the resulting line profile is a Gaussian,... 

In the Lorentz gas approximation, this term is proportional to the number densities of atoms of type A and B, nA and nB, because the probability of finding an atom of the light species with a speed between vA and vA = dvA is given by the Maxwellian distribution function,... 

Condition (c) requires that the stationary solution of the Fokker-Planck equation should be the Maxwellian distribution function. Substitution leads to... 

In order to evaluate the collisional integrals Pc, qc, and y, explicitly, it is important to know the specific form of the pair distribution function /(2)(vi, ri V2, ry. t). The pair distribution function /(2) may be related to the single-particle velocity distribution function f by introducing a configurational pair-correlation function g(ri, r2). In the following, we first introduce the distribution functions and then derive the expression of /(2) in terms of f by assuming /(1) is Maxwellian and particles are nearly elastic (i.e., 1 — e 1). 

Plasmas typical of C02 laser discharges operate over a pressure range from 1 Torr to several atmospheres with degrees of ionization, that is, nJN (the ratio of electron density to neutral density) in the range from 10-8 to 10-8. Under these conditions the electron energy distribution function is highly non-Maxwellian. As a consequence it is necessary to solve the Boltzmann transport equation based on a detailed knowledge of the electron collisional channels in order to establish the electron distribution function as a function of the ratio of the electric field to the neutral gas density, E/N, and species concentration. Development of the fundamental techniques for solution of the Boltzmann equation are presented in detail by Shkarofsky, Johnston, and Bachynski [44] and Holstein [45]. 

The distribution function/(v) is Maxwellian at local equilibrium, and is defined by... 

Fig. 18.7. The integral over the bound-free Gaunt factor which enters the expression for the radiative recombination coefficient. Results are for the k distribution family for capture into the ground n = 1 shell of hydrogen. The curves are generated by numerical quadrature over the distribution function. The limiting Maxwellian curve is the analytic expression elH kT Ei(Ib /ET)/T 2 and corresponds to k —> oo. The x-axis coordinate is Tefr = 2E/Z...

If the gas phase is homogeneous the temperature is introduced through maxwellian distribution functions. The result is ... 

E. Collision Frequency between Maxwellian Molecules. Finally, we can calculate the average number of collisions made by a molecule going through a Maxwellian gas if the molecule does not have a fixed velocity V, but has instead a velocity distribution which is itself Maxwellian. This may be done by multiplying Zc [Eq. (VII.8D.4)] by the Maxwellian distribution function and averaging over all values of Vx ... 

A still more rigorous calculation, made by Chapman and Enskog, in which the distribution function is no longer assumed Maxwellian gives... 

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