$\rho(T)$ is the energy density as obtained from the equation of state. The prescription amounts to matching the local nonequilibrium state of the flowing real matter to a fiducial perfect fluid. We will show later that the stress tensor $\Pi^{\mu\nu}$ is transverse, i.e. $u_\mu\Pi^{\mu\nu}=0$, so the prescription is consistent. With $u^\mu$ and $T$, we can define $\beta^\mu=u^\mu/T$ and then construct f\_0(x\^,p\^) = e\^[-\_p\^]{} We write the distribution function as = f\_0 \[+(1+f\_0)\] f\_0 \[+\] \[fandchi\] The entropy production S\^\_[;]{}=-Dp{\_[col]{}} then reads S\^\_[;]{}=-Dp{\_[col]{}(+)} \[smumubis\] where we have used that Dp{\_[col]{}(f\_0)} = 0

Given the expression for the entropy production given by Eq. (\[smumubis\]) and taking into account that we will consider a linear collision operator, in order to satisfy the H-theorem we introduce a new variable $\mathbf{Z}$ such that e\^ = + The solution to the variational problem entails finding $\mathbf{Z}$ as a function of the Lagrange multipliers to be introduced shortly. The outcome is a closure for the distribution function $\mathbf{f}$, i.e., an expression for $\mathbf{f}$ in terms of the variables of the effective theory, which are the usual hydrodynamic variables and the Lagrange multipliers. The latter encode the back reaction of the distribution function on the hydrodynamic modes.

The 