Maxwellian density

For thv> 1, the expectation and the variance approach, respectively, zero and kBT/m, the transition density function becomes independent of the initial condition v0 and equals the Maxwellian density... 

Using the Maxwellian density distribution function (4.345), the kinetic granular heat flux becomes equal to (2.254) ... 

An important specific feature of the present experiment is worth noting. The X-ray photons have energies that are several orders of magnitude larger than those of optical photons. The pump and probe processes thus evolve on different time scales and can be treated separately. It is convenient to start with the X-ray probing processes, and treat them by Maxwellian electrodynamics. The pumping processes are studied next using statistical mechanics of nonlinear optical processes. The electron number density n(r,t), supposed to be known in the first step, is actually calculated in this second step. 

In the above Maxwellian description of X-ray diffraction, the electron number density n r, t) was considered to be a known function of r,f. In reality, this density is modulated by the laser excitation and is not known a priori. However, it can be determined using methods of statistical mechanics of nonlinear optical processes, similar to those used in time-resolved optical spectroscopy [4]. The laser-generated electric field can be expressed as E(r, f) = Eoo(f) exp(/(qor— flot)), where flo is the optical frequency and qo the corresponding wavevector. The calculation can be sketched as follows. 

To solve eqn. (294) for the doublet density, the hierarchy of the equation must be broken in a manner analogous to the super-position approximation of Kirkwood or that of Felderhof and Deutch [25], which was presented in Chap. 9, Sect. 5. Furthermore, it is not unreasonable to assume that the system is quite near to thermal equilibrium. Were the system at thermal equilibrium, then collisions would not change the velocity distribution of the particles and the equilibrium distribution would be of the usual Maxwellian form, 0 (v,), etc. These are the solutions of the psuedo-Liouville equation... 

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... 

Here ey is the cross section for a collisionally induced transition and v is the thermal velocity of the colliding particle, < av > is the average value of ov for a Maxwellian velocity distribution. Assuming a typical dipole moment of 1 Debye, a = 1CT15 cm"2, v = 5 x 104 cm/s, one obtains the density n % 103/X3. Thus for the detection of an emission line in the centimeter-wave region (X = 1 cm) the density within the cloud is expected to be of the order of 103 cm-3. On the other hand, a detection of a millimeter-wave transition in emission at X = 1 mm requires densities of the order of 10s to 106 particles/cm3. 

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]. 

From these time-scales, it may be assumed in most circumstances that the free electrons have a Maxwellian distribution and that the dominant populations of impurities in the plasma are those of the ground and metastable states of the various ions. The dominant populations evolve on time-scales of the order of plasma diffusion time-scales and so should be modeled dynamically, that is in the particle number continuity equations, along with the momentum and energy equations of plasma transport theory. The excited populations of impurities on the other hand may be assumed relaxed with respect to the instantaneous dominant populations, that is they are in a quasi-equilibrium. The quasi-equilibrium is determined by local conditions of electron temperature and electron density. So, the atomic modeling may be partially de-coupled from the impurity transport problem into local calculations which provide quasi-equilibrium excited ion populations and effective emission coefficients (PEC coefficients) and then effective source coefficients (GCR coefficients) for dominant populations which must be entered into the transport equations. The solution of the transport equations establishes the spatial and temporal behaviour of the dominant populations which may then be re-associated with the local emissivity calculations, for matching to and analysis of observations. 

We have made the additional approximation of assuming that the number of collisions Z at any point is independent of d, the distance between plates. This is justifiable if the mean speed 5 F L/d, where F L/d is the difference in velocity between two layers of gas separated by a mean free path. Und( r such conditions the molecular density in each layer is constant and most collisions then take place between molecules that have essentially the same relative Maxwellian distribution. When this condition is not satisfied, there will be important density gradients and thermal gradients, so that the entire analysis does not apply. This condition is the equivalent of saying that the velocity of the moving plate is small compared to the velocity of sound. 

If a is used to represent the accommodation coefficient of energy transfer between molecules and plates, then the gas between the two plates may be assumed to consist of two independent Maxwellian distributions of densities Ni and N2 and temperatures Ti and T y where the plate temperatures are and T2, respectively, and the total gas density is Nt = Ni + AT2. 

Notice that the above equations also follow from the stationary solution of the Vlasov equation with the appropriate boundary conditions (Maxwellian distributions at the infinity and zero value of distribution functions with positive radial velocity at the grain surface). It should be noted, that in the derivation of the density for bound ions, we also start from the Maxwellian distribution, though the finite trajectories do not reach the infinity and, therefore, cannot be... 

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