The Boltzmann Equation

Abstract

The phase coordinate of a particle is its position and momentum: $(\vec{r},\vec{p}\:)$. It is a six-dimensional variable, $\left (x,y,z,p_{x},p_{y},p_{z}\right )$, a point in a six-dimensional phase space. In this book, instead of momentum $\vec{p}$ we will mostly use two variables: particle energy E and a unit vector: $\vec{\Omega } =\vec{ p}/\vert \vec{p}\:\vert$, that defines the direction of particle travel. This increases the number of variables to seven: $\left (x,y,z,\Omega _{x},\Omega _{y},\Omega _{z},E\right )$. However, the dimensionality remains six, that is, only six variables are independent because $\vec{\Omega }$ is a unit vector: