In strong nonequilirium, the statistical operator describing the system depends on an infinite number of variables (BBGKY-hierarchy), contains information about all the previous states starting from an initial condition $\rho(t_0) = \rho_{rel}(t_0)$

$$ \rho(t) = \frac{1}{1-t_0}\int\limits_{t_0}^t \exp^{i(t_1-t)L}\rho_{rel}(t_1)dt_1 $$

and satisfies the inhomogenous Neumann equation

$$ \frac{\partial\rho(t)}{\partial t} + iL\rho(t) = -\epsilon(\rho(t)-)\rho_{rel}(t) $$

However, to describe the macroscopic state of a system at each time by appropriate observables

$$ \langle B_n(t) \rangle = Tr\{\rho_{rel}(t)B_n\} $$

it is often enough to use only the relevant (known) information contained in the relevant statistical operator, which can be obtained by maximizing the entropy and using in addition to the conserved quantities the mean values of additional variables as constraints

$$ \rho_{rel}(t) = \exp^{- \Phi(t)-\sum F_n(t)B_n} $$

where

$$ \Phi(t) = \ln Tr \left( \exp^{-\sum F_n(t)B_n} \right) $$

is the Messieux-Planck function.

After reading about some different applications of this MaxEnt-formalism, determining what are the appropriate relevant observables to determine the state of a nonequilibrium system looked often unsatisfactorally heuristic and handwaving to me.

So my question is:

Is there a general systematic method, at best motivated by some "first principles", to obtain the relevant variables needed to describe the relevant variables needed to describe the evolution of a nonequilibrium system?

A probably very stupid aside: the evolution of a system far away from equilibrium with many degrees of freedom needed to describe it towards its equilibrium state characterized by the conserved quantities (or their conjugate variables) only, remainds me of the coarse graining needed to describe a system at an effective scale and therefore renormalization comes to mind, not sure if there is a relationship between these two things or not ...