For clarity, we work at finite $D$, which in kinetic theory amounts to truncating the moment expansion at some very large number (e.g., ${10}^{100}$ moments). This is physically acceptable: the distribution function is already a coarse-grained object, and kinetic theory ceases to hold at too fine momentum resolution (Ref. [27], Sec. 3.1).

In kinetic theory, one usually finds that, for each $k>0$, there is a continuum of “improper” solutions of (4), with $|\mathrm{Re}\omega |$ extending up to $k$, but never beyond [5].

For any invertible $D\times D$ matrix $\mathbb{S}$, we can write ${\mathbb{E}}^{\mu }={\mathbb{S}}^{†}{\tilde{\mathbb{E}}}^{\mu }\mathbb{S}$ and $\mathbb{\sigma }={\mathbb{S}}^{†}\tilde{\mathbb{\sigma }}\mathbb{S}$, where ${\tilde{\mathbb{E}}}^{\mu }$ and $\tilde{\mathbb{\sigma }}$ are Hermitian and inherit the positivity properties of the originals. Plugging these decompositions into (3), we obtain an equivalent theory ${\tilde{\mathbb{E}}}^{\mu }{\partial }_{\mu }\tilde{\mathrm{\Psi }}=-\tilde{\mathbb{\sigma }}\tilde{\mathrm{\Psi }}$, with $\tilde{\mathrm{\Psi }}=\mathbb{S}\mathrm{\Psi }$.

Density-frame hydrodynamics posits that, in any inertial frame (not necessarily comoving), the conserved fluxes $J_I^k$ admit an expansion in powers of spatial derivatives of the conserved densities $J_I^0$. Equivalently, that the hydrodynamic eigenfrequencies $\omega$ can be expanded in powers of the spatial wave vector $k^j$.