Time asymptotic behavior for Boltzmann equations and their derivatives is an interesting and important problem for both physicists and mathematicians. The dynamics of dilute electrons and plasma can be modeled by Vlasov-Poisson- Boltzmann equation, for which, the equilibrium state can be a global Maxwellian. In Chapter 2 of this thesis, we show that the rate of convergence to equilibrium is of polynomial decay for an arbitrarily large degree, by using a method developed for the Boltzmann equation without external force in [14]. In particular, the idea of this method is to show that the solution f cannot stay near any local Maxwellians for long. The improvement in this thesis is to handle the effect from the external force governed by the Poisson equation. Moreover, by using the macro-micro decomposition, we simplify the estimation on the time derivatives of the deviation of the solution from the local Maxwellian with same macroscopic components. In Chapter 3, we demonstrate three interesting problems in the application of the heliostats with spinning-elevation tracking mode. The spinning-elevation tracking method was first invented by Chen, et al. Different with the integral form, the derivative form of the surface expression described in the first section of Chapter 3 is derived from differential equations and based on the theory of non-imaging focusing heliostat proposed by Chen, et al. in 2001. By using the derivative method, we obtained the same result with that of the previous work of Chen, et al. in 2006. The comparison of the derivative form of fixed aberration correction surface has been made with that of integral form surface as well as that of spherical surface in concentrating the solar ray. The second section of Chapter 3 discusses the tracking speed for different types of sun tracking algorithms, particularly the latitude-oriented spinningelevation tracking mode in the solar energy application. It is found that the tracking speed of spinning axis of a latitude-orientated type of heliostat varies drastically during solar noon around days of equinox. We analyze the time intervals during which the heliostat requires faster tracking speed, and discuss the causes and possible solutions. The last section of Chapter 3 discusses the flux distribution of a quasi-single stage solar furnace which consists of a non-imaging focusing heliostat as the primary stage and a much smaller spherical concentrator as a secondary. As the optics of the primary stage heliostat is of non-imaging nature, the analytical method for studying the flux distribution of the hot spot of this type of solar furnace would be complicated. Therefore, a digital simulation approach has been employed. Flux distributions of the hot spot for several different incident angles, which have covered all the extreme cases of operating conditions have been simulated. Simulation result shows that a solar furnace using an 8m×8m non-imaging focusing heliostat with 289 mirrors coupled with a spherical concentrator with 0.7m aperture and 27cm focal length is capable of achieving flux concentration of 25,000 suns. Concentration contours of flux distribution for several interesting cases are presented, the different working areas of high flux footage from 5,000 suns to 15,000 suns have been compiled and photos showing the ”powerful etching” of a steel plate by the hot spot of the solar furnace are also included. Keywords: VPB system, equilibrium, asymptotic, macro-micro decomposition, spinning-elevation, solar furnace