Maxwellian distribution

The rate coefficient for elastic scattering between two species with non-isothennal Maxwellian distributions is then... 

The amount of energy lost in unit time, the energy-loss frequency, is Vgg = kpN (t). The energy-loss rate coefficient for two-temperature Maxwellian distributions is... 

Maxwell found that he could represent the distribution of velocities statistically by a function, known as the Maxwellian distribution. The collisions of the molecules with their container gives rise to the pressure of the gas. By considering the average force exerted by the molecular collisions on the wall, Boltzmann was able to show that the average kinetic energy of the molecules was... 

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... 

It is calculated in the S-matrix formalism and averaged over impact distances b and velocities v with Maxwellian distribution f(v)... 

Maxwellian distribution 129 infinite-order sudden (IOS) approximation 155-6 semi-classical calculation 136-7 Sack s model rotational relaxation 19 strong collision model 219 scattering see isotropic scattering spectra ... 

Figure 13a-e shows the EEDFs of a CH4/H2 plasma as a function of pressure. The shape of the EEDF at 50 mTorr corresponding to Figure 13a is different from that of an Ar plasma at the same pressure. There is a hump at " 6 eV. The hump gradually disappears with decreasing pressure. The shape of the EEDF at 30 mTorr in Figure 13c is almost a straight line, which means a Maxwellian distribution. The shape of the EEDF at 20 mTorr in Figure 13d deviates from a straight line and comes close to a Druyveysten distribution. It still keeps a Druyveysten distribution at 10 mTorr in Figure 13e. Accordingly, the transition from a Maxwellian distribution to a Druyveysten distribution occurs at 20 mTorr in a CH4/H2 plasma.

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. 

The determination of electron concentration by the frequency shift method is limited to time resolution greater than a few hundred nanoseconds and is therefore not applicable to liquids. The microwave absorption method can be used virtually down to the pulse width resolution. Under conditions of low dose and no electron loss, and assuming Maxwellian distribution at all times, Warman and deHaas (1975) show that the fractional power loss is related to the mean electron energy (E) by... 

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. 

Additional information concerning the fast electron distribution can be obtained in such experiments by direct measurement of forward escaping electrons using a calibrated stack of radiochromic films [52] that can provide information of the angular and energy distribution of fast electrons. These measurements were performed in the same experiment described above [31] and the summary of those measurements is reported in Fig. 7.10 and reveal a distribution that is consistent with a relativistic Maxwellian distribution with a characteristic temperature of 160keV. 

We generate the start-up configuration - all particles in the box are assigned positions r, and velocities v . Velocities are randomly distributed according to a Maxwellian distribution for some given temperature. 

Moreover, we have seen that if one waits long enough, the velocity distribution p(00)(t) tends toward the Maxwellian distribution po1 (see Eq. (71)). We thus have ... 

This equation is readily transformed to an integral equation for different from i and in <— k,- Y(z] — k )) never appear in two successive collision operators because otherwise we would get a negligible contribution in the limit of an infinite system moreover as these dummy particles have zero wave vectors in the initial state, they have a Maxwellian distribution of velocities (see Eq. (418)). This allows us to write Eq. (A.74) in the compact form ... 

One can show (see, for example, ref. 24) that the Maxwellian distribution of velocities... 

If one follows the approach of Landau and Teller [11], who in dealing with vibrational relaxation developed an expression by averaging a transition probability based on the relative molecular velocity over the Maxwellian distribution, one can obtain the following expression for the recombination rate constant [6] ... 

The addition of constraints to the equations of motion have also been used to produce thermostats at surfaces which control the flux of heat in and out of the substrate. For example, Riley et al. have proposed a velocity reset procedure which regulates atomic motion by coupling the current velocity of each atom with a velocity chosen from a Maxwellian distribution . In a similar scheme, Agrawal et al. have added a friction term to atomic velocities which depends in part on the difference between the current temperature of the surface region and that desired for the substrate . This approach was... 

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,... 

Central to the categorization of plasmas are electron temperature and electron density. Electrons have a distribution of energies, so it is useful to assume a Maxwellian distribution, in terms of electron energy, E, such that... 

The mechanism by which equilibrium is attained can only be visualized in terms of microscopic theories. In the kinetic sense, equilibrium is reached in a gas when collisions among molecules redistribute the velocilies lor kinetic energies) of each molecule until a Maxwellian distribution is reached for the whole bulk. In the case of the trend toward equilibrium for two solid bodies brought into physical contact, we visualize the transfer of energy by means of free electrons and phonons (lattice vibrations). 

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