Boltzmann density operator function

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]. 

Between the limits of small and large r, the pair distribution function g r) of a monatomic fluid is determined by the direct interaction between the two particles, and by the indireet interaction between the same two partieles through other particles. At low densities, it is only the direct interaction that operates through the Boltzmann distribution and... 

Z is the partition function, a is the scalar quantity conjugate to A and ks is the Boltzmann constant a could be, e.g., an electric field and A the electric dipolar moment operator of the system. We assume that at the initial time the system is not far from equilibrium and expand the density to first order in a (see Ref. [62]). The expected value of an observable B is expressed as the Mori scalar product that refers to the equilibrium density [62, 63]... 

A more sophisticated method to solve the Boltzmann equation is the variational method (for a review see, e.g., Ziman (I960)). If the function 0y is known (see eq. (11)), the expressions for the electrical current density, and the heat current density, y, are given (for the determination of 0y from eq. (11) we use the inverse collision operator Q ) by ... 

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