Kinetic Theory of Long-Range Systems: Klimontovich, Vlasov and Lenard–Balescu Equations

Abstract

AbstractThis chapter describes the derivation of kinetic equations for long-range systems. This is done on the basis of the Klimontovich equation, following an approach which is alternative but equivalent to the BBGKY hierarchy. After the derivation of the Klimontovich equation, that mathematically is equivalent to the full equations of motion, the Vlasov and the Lenard-Balescu equations are obtained. The former approximates the interaction in the system with a mean-field force, and for this it is considered a collisionless approximation; the latter takes into account the collisional corrections to the Vlasov equation. In the last section, with the background of the general theory presented in chapter 2 and in this chapter, there is a description of the physical meaning of the Kac’s prescription, often used as a convenient tool for the study of long-range systems.