to this approximation in what follows.

Effective theory {#eff}
================

Obtaining a closure for $\mathbf{f}$ {#closure}
------------------------------------

We will now obtain an expression for the one-particle distribution matrix in terms of variables of the effective theory. To do so, we will rely on the EPP [@epvm] discussed in the Introduction.

This method allows one to find the distribution function which extremizes the entropy production $S^\mu_{;\mu}$, subject to the constraints that the conserved currents take on known values. It provides a prescription to associate a distribution function to given macroscopic currents, yielding a nonlinear closure that generalizes the well-known Grad’s quadratic ansatz [@deGroot; @is1; @ferz; @liboff; @kremer] in a nontrivial way [@linking; @progress].

The EPP does not rely on a gradient expansion, which is usually invoked when deriving hydrodynamics from kinetic theory. This results in effective theories capable of describing highly nonequilibrium and nonlinear situations quite reliably as compared to microscopic approaches [@epvm; @tanos; @anile; @tanos2; @tanos3; @muscato; @christen; @christen2].

### Deviations from the unperturbed state

For simplicity, in what follows we will neglect quantum statistics. Given $\rm{tr}(\mathbf{T}^{\mu\nu})=T^{\mu\nu}$, we can define a flow velocity $u^\mu$ and a temperature $T$ by using the Landau-Lifshitz prescription. We have u\_T\^ = (T) u\^ where 