symmetry and the degeneracy requirement for the matrix M (1 ) , dif are satisfied by construc- tion. Finally, by inserting the above building blocks E , S , L and M into the GENERIC equation (2.2), the full set of time evolution equations takes the form where Π = ( p − 1 $_{2}$κ$_{E}$ |∇ c | $^{2}$) 1 + κ$_{E}$ ( ∇ c ) ( ∇ c ) is the pressure tensor, τ = η ˙ γ − ˆ κ ( ∇ · v ) 1 is the viscous stress tensor, and j q = − λ $^{q}$∇ T represents the conductive flow of internal energy. For an isothermal, incompressible flow, these time evolution equations take the form of the standard Cahn-Hilliard model. Similar set of hydrodynamic equations for phase separating fluid mixtures has been derived from an underlying microscopic dynamics [46]. A specific feature of the GENERIC formalism is that it is applicable at different levels of description associated with different length and time scales. Each level of description is described by an appropriate set of state variables and building blocks. The mathematical structure of GENERIC offers a possibility to perform coarse graining by focusing not on the time evolution equations, but on the building blocks E , S , L and M , see [47]. The transition between a more detailed (Cahn-Hilliard) level 1 to a less detailed (Doi-Ohta) level 2, which involve the derivation of the building blocks from level 1 to level 2, can then be performed by systematic procedures [48]. The possibility to perform coarse-graining by focusing on the basic building blocks guarantees that the thermodynamic structure of the problem will be preserved. A systematic coarse-graining procedure is based on the idea of the separation of time scales. In this way, it is assumed that there exists a division of the “fast” and “slow” 11 III. COARSE GRAINING FROM CAHN-HILLIARD TO DOI-OHTA LEVEL ∂ρ ∂t = −∇ · ( ρ v ) , (2.9a) ∂ g ∂t = −∇ · ( vg ) − ∇ · ( Π + τ ) , (2.9b) ∂ϵ ∂t = −∇ · ( ϵ v ) − p ∇ · v − τ : ( ∇ v ) T − ∇ · j q ∂c ∂t = − v · ∇ c ) + ∇ · [ M T ∇ − ∂s ( ρ, ϵ, c ) ∂c − κ$_{E}$ T ∇ $^{2}$c , (2.9d) + κ$_{E}$ ( ∇ $^{2}$c ) ∇ · [ M T ∇ ( − ∂s ( ρ, ϵ, c ) ∂c − κ$_{E}$ T ∇ $^{2}$c )] , (2.9c)