The Boltzmann Equation 1: Fundamentals

Boltzmann’s transport equation for a dilute gas with particles interacting with central, short range forces, and with bounding walls, is derived in detail, with emphasis on the use of the Stosszahlansatz. Boltzmann’s H–theorem is presented as a microscopic derivation of the law of entropy increase in non–equilibrium processes, and the Maxwell–Boltzmann equilibrium distribution is derived. Zermelo’s and Loschmidt’s arguments that the H–theorem is incompatible with the laws of mechanics are given and discussed. The Kac ring model is presented and used as a simple way to understand the application and the limitations of the Stosszahlansatz. It is concluded that the Boltzmann equation is statistical, rather than strictly mechanical, in nature, providing a description of the most probable non–equilibrium behavior of a dilute gas.