BBGKY Hierarchy, Kinetic Theories and the Boltzmann Equation

Abstract This chapter introduces to the study of the dynamics of many-body systems. The reduced distribution functions and their dynamics, determined by the equations of the BBGKY hierarchy, are presented. It is explained how kinetic theories are based on plausible approximations of the first equations of the hierarchy; this leads to a truncation of the hierarchy and to a closed equation for the one-particle distribution function. The derivation of the Boltzmann equation, suitable for diluted short-range systems, is then offered in detail. Through the study of the important H-theorem, it is shown that the macroscopic irreversibility is built upon in the kinetic equations. The apparent contradiction with the reversibility of the microscopic equations of motion is resolved by the appropriate interpretation of the physical meaning of the collisional term of the kinetic equations.