and expand the nonequilibrium correction $\mathbf{Z}$ and the Lagrange multipliers as follows

&= \^[(1)]{} + \^[(2)]{}\
\^[a]{} &= \^[a(1)]{} + \^[a(2)]{}\
\_ &= \_\^[(1)]{} + \_\^[(2)]{}\
\^a\_ &= \_\^[a(1)]{} + \_\^[a(2)]{}

\[zetas2nd\]

We now go over to solve the equations at first and second order in $\tau$. We note that the solution given here closely follows the one given in [@linking]. For the reader’s convenience, a brief summary of the logical steps carried out to obtain the closure for the distribution function can be found in Section (\[sumclos\]).

### First order solution

At first order, the variational equations are R\[FR\[\^[(1)]{}\]\] = \^[(1)]{}\_ p\^p\^ \[var1\] and R\[FR\[\^[b(1)]{}\]\] = 0 \[var2\] Eqs. (\[ra\]) and (\[var2\]) imply that $\zeta^{b(1)}$ must be independent of $p^\nu$, as it must be given that $\zeta^{b(1)}$ are color fugacities.

From Eq. (\[var1\]) we get \^[(1)]{}\_ p\^p\^p\^= 0 \[l0var\] Setting $\rho=k$ we obtain $\lambda^{(1)}_{0k}=0$. Without loss of generality we can take $\lambda^{(1)}_{00}=0$. Since we are dealing with a conformal theory $\left\langle p^i p^j \right\rangle = \delta^{ij} \left\langle \omega^2 \right\rangle/3$, and we get $\lambda^{(1)i}_i = 0$. We thus obtain \^[(1)]{}= \^[(1)]{}\_[ij]{} p\^i p\^j \[zeta10\]

The first order shear tensor is then \_1\^ = p\^p\^\^[(1)]{} We find that $\Pi^{00}=\Pi^{k0}=0$. Using that 