Electron Kinetics in Time-Dependent Plasmas

According to the relevant power and momentum balance, Eqs. (38) and (39), the electron kinetics in steady-state plasmas is characterized by tbe conditions that at any instant the power and the momentum input from the electric field are dissipated by elastic and inelastic electron collisions into the translational and internal energy of the gas particles. This instantaneous complete compensation of the respective gain from the field and the loss in collisions usually does not occur in time-dependent plasmas, and often the collisional dissipation follows with a more or less large delay—for example, the temporally varying action of a time-dependent field. Thus, the temporal response of the electrons to certain disturbances in the initial value of their velocity distribution or to rapid changes of the electric field becomes more complicated, and the study of kinetic problems related to time-dependent plasmas naturally becomes more complex and sophisticated. Despite this extended interplay between the action of the binary electron collisions and the action of the electric field, the electron kinetics in time- 

When specifying the kinetic treatment to purely time-dependent plasmas with isotropic scattering in the conservative inelastic collision processes, from system (12) the simplified system (Wilhelm and Winkler, 1979 Winkler and Wuttke, 1992 Loffhagen and Winkler, 1994 Winkler, 1993 Winkler et al, 1995) 

To obtain a simpler struture of this mathematical problem, the system has often been reduced in the past by neglecting the first term in the second of Eqs. (44), i.e., the derivative of/i(U, t) with respect to time (Wilhelm and Winkler, 1979 Winkler and Wuttke, 1992 Loffhagen and Winkler, 1994). When this additional approximation is accepted, the anisotropic distribution can be eliminated by means of the second of Eqs. (44), and finally a partial differential equation of second order with additional terms of shifted energy arguments is obtained. Some remarks about the validity limits of this approximation will be made below in connection with the presentation of some results. 

However, in recent years, techniques for solving system (44) numerically without additional reductions or simplifications have been developed (Winkler et al., 1995 Winkler, 1993). This modem approach is used as the basis of the following explanations concerning the time-dependent two-term treatment. 

The system of partial differential equations of first order, Eqs. (44), usually has to be treated as an initial-boundary-value problem on an appropriate energy region 0 U U°° and for times t 0, where the time represents the evolution direction of the kinetic problem. Initial values for each of the distributions fo(U, i) and/,( /, t), suitable for the problem under consideration, have to be fixed, for example at t = 0. Appropriate boundary conditions for the system are given by the requirements /o([7 U°°, t) = 0 and /,(0, t) = 0. 

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