closed dynamics based on hydrodynamic variables alone.

We obtain a closed theory by identifying a tensor $\lambda^{\mu\nu}$ (which is introduced in Section \[eff\]) through which the system couples to the hydrodynamic variables. One can think of the introduction of the nonhydrodynamic variable $\lambda^{\mu\nu}$ as a simple way of modeling the back reaction of the distribution function (which could correspond to a highly nonequilibrium situation) on the hydrodynamic modes (i.e. those modes associated to conservation laws and hence relaxing much more slowly). A somewhat similar situation is discussed in [@tanos; @tanos2; @tanos3; @anile; @muscato] in the context of the Entropy Maximum Principle (EMP), in [@GKpaper] (see also [@GKbook]) where the moments of the collision term of the classical Boltzmann equation are interpreted as independent variables rather than as infinite moment series, in [@mauricio] and [@aniso] in the context of “anisotropic hydrodynamics”, in [@nagy] in the framework of divergence-type theories [@geroch] (see also [@dev; @app; @linking]), in [@denicol] where the closure is obtained from an expansion of the distribution function in moments [*at all orders*]{}, and in [@christen; @christen2] in relation to the Entropy Production Principle (EPP).

Here, the tensor $\lambda^{\mu\nu}$ is identified with a Lagrange multiplier in a well-defined variational problem 