Lorentz-Boltzmann equation

The method discussed in Section 3 for the derivation of the generalized Boltzmann equation can be carried over, with some simple modifications, to derive an analogous kinetic equation for d, t). However, due to the special role of the tagged particle and the fact that the corresponding initial N-particle distribution is proportional to Ui, one obtains a linear equation for 4>d (vi, t) that for low densities reduces to the Lorentz-Boltzmann equation " ... 

Third, a further simplification of the Boltzmann equation is the use of the two-term spherical harmonic expansion [231 ] for the EEDF (also known as the Lorentz approximation), both in the calculations and in the analysis in the literature of experimental data. This two-term approximation has also been used by Kurachi and Nakamura [212] to determine the cross section for vibrational excitation of SiHj (see Table II). Due to the magnitude of the vibrational cross section at certain electron energies relative to the elastic cross sections and the steep dependence of the vibrational cross section, the use of this two-term approximation is of variable accuracy [240]. A Monte Carlo calculation is in principle more accurate, because in such a model the spatial and temporal behavior of the EEDF can be included. However, a Monte Carlo calculation has its own problems, such as the large computational effort needed to reduce statistical fluctuations. 

This equation can be interpreted as the drift term of a collisionless Boltzmann equation for the one-particle Wigner distribution p(q,p). To see that, let us explore the physical meaning of p(q,p) in this context. First note that p(q, p ) is in principle a Lorentz scalar. Thus an invariant solution of Eq. (59) is... 

This shows that the Lorentz force is not a thermodynamical force the magnetic field itself does not give rise to any net flow. As stated above, one assumes the field strengths and their spatial gradients to be first-order quantities while/ (k, r) and/ (k, r) in equation (9.27) will be assumed to be of zero and first order, respectively. Then in zero order the Boltzmann equation reduces to... 

This amplitude is found from the equation of motion of a harmonic oscillator affected by an a.c. field E(t). This approach yields the Lorentz and Van Vleck-Weisskopf lines, respectively, for a homogeneous and Boltzmann distributions of the initial a.c. displacements x(l0) established after instant to of a strong collision. The susceptibility corresponding to the Van Vleck-Weisskopf line in terms of our parameters is given by [66]... 

Here, k is the Boltzmann constant. The Lorentz-Bcrthelot rule has been adopted for the parameters working between different species of the mixture and the values are thus yilk= 9.% K and a j2=3.405 x 10 1 m, which correspond to the interaction parameters of neat argon. The critical temperature of the Lennard-Jones monoatomic fluids ) evaluated by the integral equation theory is 1.321 multiplied by ( f/k ), which is equal to 111.9 K and 223.8 K for the component 1 and 2, respectively. The temperature set in our present calculation is higher than any of these critical temperature. Thus the fluids discussed in this work are always at supercritical state. 

This kinetic description can be extended to the case where the tagged and fluid particles are assumed to interact via the same force law that holds for fluid particles. The mass m of the tagged particle is assumed to be large compared to that of a fluid particle mj ). Therefore, the mass ratio e = nijr/m is a small parameter in terms of which the Boltzmann- Lorentz collision operator may be expanded. If this expansion is carried out to the leading order, the Boltzmann-Lorentz operator reduces to a differential operator yielding a kinetic Fokker Planck equation for the tagged particle distribution F... 

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