f\_0 (…) (…)

From Eq. (\[10\]) we get the expression for the color currents \^= |[n]{}u\^+ p\^\^a \^a \[jdfull\] where we have defined $\bar{n}=g\left\langle \omega \right\rangle$.

For the entropy production, we have

S\^\_[;]{}= -Dp   (\_[col]{} )

The collision operator contains color independent and dependent parts, so, similarly to the decomposition used for $\mathbf{Z}$, we put \_[col]{} = I\^[(0)]{}\_[col]{} + I\^[a]{}\_[col]{} \^[a]{} We then have (\_[col]{} ) = I\^[(0)]{}\_[col]{}+ I\^[a]{}\_[col]{}\^a where we have used Eq. (\[01\]), so S\^\_[;]{}= - Dp  (I\^[(0)]{}\_[col]{}+ I\^[a]{}\_[col]{}\^a ) \[entaux\]

We shall write the collision operator as \_ = - R\[FR\[\]\] \[linea\] where we have put \_[col]{} f\_0 \_

In Eq. (\[linea\]), $\tau$ is the relaxation time, $F=F(\omega)$ is an arbitrary function of energy $\omega=-u_\mu p^\mu$, and $R$ is an operator enforcing the integrability conditions given in Eqs. (\[13\]). The projector $R$ explicitly reads R\[\] = - R\^[ab]{}(\_a ) \_b - R\_p\^() \[Rexplicit\] with R\_= u\_+ \^\_p\_and $R_{ab}=2\delta_{ab}$. Here, \^ = g\^+u\^u\^ is the spatial projector. Note that this form for $\mathbf{I}_{\beta}$ guarantees that the Second Law holds exactly. Moreover, it is flexible enough to include the important cases of Marle’s relativistic generalization of the BGK model [@liboff; @kremer], corresponding to $F(\omega)=T$, as well as the Anderson-Witting 