Boltzmann collision integral

Now we will show that this expression can be transformed into the usual form of the Boltzmann collision integral with the quantum cross section. For this purpose we can use the following relations from the scattering theory ... 

With these relations it is easy to get the usual form of the quantum Boltzmann collision integral ... 

The general properties of the Boltzmann collision integral are well known and will not be discussed here. 

The first term in this expansion describes the binary collision and leads, as we have shown, to the Boltzmann collision integral. Among the terms of the order n, the first describes the three-particle collision and has, as will be shown, the same structure as the Boltzmann collision term. 

This collision integral has the shape of the usual Boltzmann collision integral for the scattering of a bound pair and a free particle and corresponds to the process... 

Boltzmann collision integral, now for the atom-atom collision. Of course, there are other four-particle processes but, according to our approximation, these were neglected. 

Taking into account that the Boltzmann collision integrals / and [/J, conserve the particle number, we get... 

TTius we have shown that if the correlations between the particles in the initial state are short ranged, if t t, and if we neglect the change in positions of two particles during a binary collision, then the binary collision term in Eq. (208) is equal to the Boltzmann collision integral. 

Finally, we note an important property of the conservation equations, Eqs. (3.30) and (3.32), in the context of the Boltzmann collision integral. Letting primes denote postcollisional values and unprimed denote the precollisional values, as in the Boltzmann collision integral, the conservation equations, when applied to a collision, lead to... 

Coefficient Equations.—To determine the coefficients of the expansion, the distribution function, Eq. (1-72), is used in the Boltzmann equation the equation is then multiplied by any one of the polynomials, and integrated over velocity. This gives rise to an infinite set of coupled equations for the coefficients. Only a few of the coefficients appear on the left of each equation in general, however, all coefficients (and products) appear on the right side due to the nonlinearity of the collision integral. Methods of solving these equations approximately will be discussed in later sections. 

Bather than carrying out the calculation for the general case, which yields rather unwieldy expressions, only equations sufficient to obtain certain approximations will be developed. If we multiply the Boltzmann equation, Eq. (1-39), by 1 = i%( 2)3r )) (0.9>)> the resulting equation is simply the equation of conservation of mass, since integrating unity over the collision integral gives zero ... 

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... 

In this equation, kg is the Boltzmann constant, T is the absolute temperature (Kelvin), mij = + mj), a,j is a length-scale in the interaction between the two molecules, and is a collision integral, which depends on the temperature and the in-... 

In terms of the collision integrals, the Boltzmann equation can be rewritten as... 

The two-particle Boltzmann collision term if and the three-particle contribution for k = 0 were considered in Section II. It was possible to express those collision integrals in terms of the two- and three-particle scattering matrices. It is also possible to introduce the T matrix in if for the channels k = 1, 2,3, that is, in those cases where three are asymptotically bound states. Here we use the multichannel scattering theory, as outlined in Refs. 9 and 26. 

The basis for the semiclassical description of kinetics is the existence of two well separated time scales, one of which describes a slow classical evolution of the system and the other describes fast quantum processes. For example, the collision integral in the Boltzmann equation may be written as local in time because quantum-mechanical scattering is assumed to be fast as compared to the evolution of the distribution function. 

The parameter the diffusion collision integral, is a function of k T/e, where is the Boltzmann constant and e is a molecular energy parameter. Values of tabulated as a function of k T/e, have been published (Hirschfelder et al., 1964 Bird et al., 1960). Neufeld et al., (1972) correlated using a simple eight parameter equation that is suitable for computer calculations (see, also, Danner and Daubert, 1983 Reid et al., 1987). Values of a and e/k (which has units of kelvin) can be found in the literature—for only a few species—or estimated from critical properties (Reid et al., 1987 Danner and Daubert, 1983). The mixture a is calculated as the arithmetic average of the pure component values. The mixture e is taken to be the geometric average of the pure component values. 

Qab = collision integral, which would be unity if the molecules were rigid spheres and is a function of k T/ ab for real gases (kg = Boltzmann s constant)... 

For elastic collisions, several different kinetic models have been proposed in order to close the Boltzmann hard-sphere collision term (Eq. 6.9). For inelastic collisions (e < 1), one must correctly account for the dependence of the dissipation of granular energy on the value of e. One method for accomplishing this task is to start from the exact (unclosed) collision integral in Eq. (6.68). From the definition of if given in Eq. (6.60), it can be... 

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). 

When H has reached its minimum value this is the well known Maxwell-Boltzmann distribution for a gas in thermal equilibrium with a uniform motion u. So, argues Boltzmann, solutions of his equation for an isolated system approach an equilibrium state, just as real gases seem to do. Up to a negative factor -k, in fact), differences in H are the same as differences in the thermodynamic entropy between initial and final equilibrium states. Boltzmann thought that his //-theorem gave a foundation of the increase in entropy as a result of the collision integral, whose derivation was based on the Stosszahlansatz. 

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