The Maxwell equation

The evolution of the quantity $W(\mathbf{v})$ is obtained by multiplying both sides of the Boltzmann equation, Eq. (2.74), by $W$ and integrating over $\mathbf{v}$. Here we consider a molecular quantity that is function of the random velocity, i.e. $W(\mathbf{c})$, obtaining (after some algebra, mainly integrations by parts) the following equation [62,98]:

$$\begin{split}\frac{\partial(n \langle W \rangle)}{\partial t}... ...tial c_i} \right \rangle =\frac{\delta W}{\delta t} \end{split}$$ (2.115)

where $\mathbf{a}_i$ is the acceleration of particle $i$ due to external force fields and where $n \langle \cdot \rangle=\underset{\infty}{\iiint}d^3c \cdot P$ and the quantity $\delta W/\delta t$ represents the change of $W$ due to the collisions in the interval $\delta t$ and is expressed by the formula of Eq. (2.77) with the substitutions $\phi \to W$ and $\mathbf{v} \to \mathbf{c}$:

We recall the intuitive fact that for every summation invariant, that is quantities whose sum is conserved during collisions, the collisional term (i.e. right hand side of Eq. (2.121)) is zero.