How to formally write down the Boltzmann equation? Can someone write down the Boltzmann equation, not neglecting any of the variables of the involved functions and integrals? Specifically, how to concisely capture the "primed" variables in a sensible manner?
 A: 
Can someone write down the Boltzmann equation, not neglecting any of the variables of the involved functions and integrals?

The relativistic Boltzmann equation for a single particle species in the presence of an electromagnetic field can be written as:
$$
p^{\mu} \ \frac{ \partial f }{ \partial x^{\mu} } \partial_{\mu} f + \frac{ q }{ c } F^{\mu \nu} \ p_{\nu} \frac{ \partial f }{ \partial p^{\mu} } = \mathcal{Q} \tag{0}
$$
where $p^{\mu}$ is the contravariant momentum 4-vector, $f = f\left( \mathbf{x}, \mathbf{p}, t \right)$ is the momentum distribution function, $F^{\mu \nu}$ is the electromagnetic field tensor, $\mathcal{Q}$ is the collision operator of this species with all other particle species, $q$ is the charge of this particle species, and $c$ is the speed of light in vacuum.  Note that Equation 0 must be re-written for every particle species in the system.  I avoided using a subscript $s$ since it could be confused for a tensor index.  The functional form of $f$ and $\mathcal{Q}$ for each particle population are determined by the system and evolutionary time of the system, thus they are not explicitly defined here (nor should they be without further information).  The $p^{\mu}$ are random variables, not specific values or functions to be known.  The 3-vector form of $p^{\mu}$ is an argument of $f$ and is a random variable there as well.  The form of $F^{\mu \nu}$ can be written in terms of contravariant derivatives of the Lorentz covariant potentials.
Nothing is neglected here but things have been left in general form to avoid tying them to a specific case example.

Specifically, how to concisely capture the "primed" variables in a sensible manner?

I am not sure to what "primed" variables you are referring.  Perhaps you are asking about the nuances of the collision operator.  For binary Coulomb collisions (i.e., operators considering collsions between only two charged particles at a time), a good example with discussion can found in Hellinger & Trávnícek [2009].
