For anisotropic collective expansion with a constant expansion rate, the distribution function would approach this quasi-equilibrium form, with an appropriately chosen set of anisotropy parameters.

Since the dissipative currents in our approach are different from those in Ref. [3], so are the inverse Reynolds numbers. In our case they describe how far the system is away from local “quasi-equilibrium.”

Note that no symmetry assumptions are made for the transition rate$W_{\mathit{k}{\mathit{k}}^{\prime }\to \mathit{p}{\mathit{p}}^{\prime }}$.

To reduce clutter we write${\tilde{\rho }}_r^{{\mu }_1\cdots {\mu }_{\ell }}(x)$ instead of ${\tilde{\rho }}_r^{⟨{\mu }_1\cdots {\mu }_{\ell }⟩}(x)$, remembering that the moments are traceless and have only spatial components in the LRF.

We point out that, although $\xi \sim {\pi }^{\mu \nu }\sim \mathcal{O}({\mathrm{Rn}}_{\pi }^{-1})$, it is $\mathcal{O}(1)$ in our power-counting scheme which is in terms of the Knudsen and modified inverse Reynolds numbers. It is therefore kept in Eq. (17).

Terms of second order in the inverse Reynolds number are absent in Eq. (21) because we only kept terms linear in the deviation ${\varphi }_{\mathit{k}}$ when expanding the collision integral around the leading-order distribution.

Equation (3) is just one such example; our results are generic and apply for any leading-order distribution with $x$-dependent anisotropy parameters, irrespective of its specific functional form.

Additionally, these terms include some of $\mathcal{O}({\mathrm{Kn}}^2)$ which in the isotropic formalism must be neglected for causality reasons [3, 23, 24]but which can be kept here because our formalism also includes the related comoving time derivatives of the anisotropy parameter(s).

Some terms in$({\mathcal{K}}^{(n)}{)}_{\mu \nu }$ would render the constitutive equations acausal and thus need to be discarded. This happens automatically in the linearized approximation studied here.