Boltzmann equation, collision term generalized

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... 

The most recent effort in this direction is the work of Cohen,8 who established a systematic generalization of the Boltzmann equation. This author obtained the explicit forms of the two-, three-, and four-particle collision terms. His approach is formally very similar to the cluster expansion of Mayer in the equilibrium case. 

Choh and Uhlenbeck6 developed Bogolubov s ideas and extended his formal results. They established a generalized Boltzmann equation which takes account of three-particle collisions. The extension of their results to higher orders in the concentration poses no problem in principle, but it appears difficult, in this formalism, to write a priori the collision term with an arbitrary number of particles. 

This relation enable us to simplify the formulation of the general equation of change considerably. Fortunately, the fundamental fluid dynamic conservation equations of continuity, momentum, and energy are thus derived from the Boltzmann equation without actually determining the form of either the collision term or the distribution function /. 

T source term in generalized Boltzmann type of equation representing the effects of particle coalescence, breakage and collisions J c)) collision term in the Boltzmann equation... 

The function ffjl is derived analytically from the hard-sphere-collision integral, and readers interested in the exact forms are referred to Tables 6.1-6.3 of Chapter 6. One crucial issue is the description of the equilibrium distribution with QBMM. In fact, since the nonlinear collision source terms that drive the NDF and its moments to the Maxwellian equilibrium are approximated, the equilibrium is generally not perfectly described. The error involved is generally very small, and is reduced when the number of nodes is increased, but can be easily overcome by using some simple corrections. Details on these corrections for the isotropic Boltzmann equation test case are reported in Icardi et al. (2012). 

In the analysis of the higher-order collision terms, in the generalized Boltzmann equation, it is important to determine ... 

Since the leading divergences in the generalized Boltzmann equation are associated with sequences of binary collisions, the resummations are usually carried out by first expressing the 5-particle streaming operators 5, (xi,..., x ) in Eq. (208) in terms of sequences of binary collisions that take place between the s particles. This is accomplished by means of the binary collision expansion, which has proven to be one of the most useful tools in the kinetic theory of gases. " ... 

In the classical Boltzmann equation for electron transport, scattering is included via a dissipation term using x, the average collision time. A master equation approach is basically a generalization of the Boltzmann equation to a fully quantum mechanical system. The master equation is based on the... 

The relationship between the general correlation function approach and the dilute gas should, in principle, be clear-cut. In fact, Equation (3) can be derived from the corresponding equation (2). Many authors have attempted to generalize the Boltzmann equation for all densities alternatively, many authors have attempted to write the fluxes of Equation (1) in terms of the dynamics of molecular collisions. But these are difficult problems. 

The set of q coupled equations on the form (2.289), with the generalized collision term defined by (2.290), comprises the generalization of the Boltzmann equation for mono-atomic molecules to q species. In each of these equations, the distribution functions for all the species appear on the RHS of the equation under the integration sign. Moreover, the sum of integrals describes the entry and exit of particles of s in or out of the phase element. 

In contrast to the formally analogous van t Hoff equation [10] for the temperature dependence of equilibrium constants, the Arrhenius equation 1.3 is empirical and not exact The pre-exponential factor A is not entirely independent of temperature. Slight deviations from straight-line behavior must therefore be expected. In terms of collision theory, the exponential factor stems from Boltzmann s law and reflects the fact that a collision will only be successful if the energy of the molecules exceeds a critical value. In addition, however, the frequency of collisions, reflected by the pre-exponential factor A, increases in proportion to the square root of temperature (at least in gases). This relatively small contribution to the temperature dependence is not correctly accounted for in eqns 2.2 and 2.3. [For more detail, see general references at end of chapter.]... 

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