G() p\^i p\^j p\^k p\^l = (\^[ij]{}\^[kl]{}+ \^[ik]{}\^[jl]{}+\^[il]{}\^[jk]{}) \[idenG\] for any function of energy $G(\omega)$, we get \_1\^[ij]{} = \^[(1)ij]{} \[pi1aux\] which is traceless and transverse. The first order color current is (recall that $\zeta^{a(1)}$ must be independent of $p^\nu$) J\_1\^[a]{} = |[n]{} \^[a(1)]{} u\^\[j1a\]

The lowest order nontrivial contributions to the entropy current and the entropy production are S\^0\_1 = - \^[(1)ij]{}\^[(1)]{}\_[ij]{} [and]{} S\^i\_1 = 0 \[ecur\] in the rest frame, and S\^\_[;]{} = \^[(1)ij]{}\^[(1)]{}\_[ij]{} \[eprod\] respectively. The entropy flux (\[ecur\]) and the entropy production (\[eprod\]) computed from the first order solutions $(\zeta^{(1)},\zeta^{a(1)})$ to the variational equations are already quadratic in deviations from equilibrium, so we do not need to consider higher order contributions. Note that, because of the first order equation (\[var2\]), $\zeta^{a(1)}$ does not contribute to the entropy production at quadratic order.

### Second order solution

The variational equations at second order read

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

and R\[FR\[\^[b(2)]{}\]\] = g\_\^[b(1)]{} p\^\^[(1)]{} + \_\^[(1)]{} p\^p\^\^[b(1)]{}

The integrability conditions then read N \_\^[(2)]{}p\^p\^+ (\^[(1)]{})\^2 F + g\_0\^[a(1)]{}\^[a(1)]{} \^2 = 0 \[s1\] N \_[0j]{}\^[(2)]{}p\^i p\^j + g\_j\^[a(1)]{}\^[a(1)]{} p\^i p\^j = 0 \[s2\] corresponding to Eq. (\[int0\]) and \_[0]{}\^[b(1)]{}\^[(1)]{} = 0 \[s3\] corresponding to Eq. 