  A careful derivation of the generalized Langevin equation using "Zwanzig
flavor" projection operator formalism is presented. We provide arguments why
this formalism has better properties compared to alternative
projection-operator formalisms for deriving non-equilibrium,
non-thermodynamic-limit, equations. The two main ingredients in the derivation
are Liouville's theorem and optimal prediction theory.
  As a result we find that equations for non-equilibrium thermodynamics are
dictated by the formalism once the choice of coarse-grained variables is made.
This includes a microcanonical entropy definition dependent on the
coarse-grained variables. Based on this framework we provide a methodology for
succesive coarse-graining. As two special cases, the case of linear
coefficients and coarse-graining in the thermodynamic limit are treated in
detail. In the linear limit the formulas found are equivalent with those of
homogenization theory.
  In this framework there are no restrictions with respect to the
thermodynamic-limit or nearness to equilibrium. We believe the presented
approach is very suitable for the development of computational methods by means
of coarse-graining from a more detailed level of description.
