In the analytical derivation of the coarse-grained Poisson operator, several approximations that are related to the difference in length scales (3.2) are used. The crossover in length scale from the more microscopic Cahn-Hilliard level to the more macroscopic Doi-Ohta level is performed through the smoothing function χ ( r$_{1}$ − r$_{2}$ ). While the details of the derivation for some of the elements of L (2) are given in the Appendix A, here we give their final expressions L (2) ρ $_{g}$( r$_{1}$ , r$_{2}$ ) = ρ ( r$_{2}$ ) ∂χ ∂ r$_{2}$ . (3.14a) L (2) $_{gg}$( r$_{1}$ , r$_{2}$ ) = g ( r$_{2}$ ) ∂χ ∂ r$_{2}$ − ∂χ ∂ r$_{1}$ g ( r$_{1}$ ) , (3.14b) L (2) ϵg ( r$_{1}$ , r$_{2}$ ) = ϵ ( r$_{2}$ ) ∂χ ∂ r$_{2}$ + p ( r$_{1}$ ) ∂χ ∂ r$_{2}$ , (3.14c) L (2) P g $_{,β}$( r$_{1}$ , r$_{2}$ ) = ∂ ∂r$_{2}$$_{α}$ [( 2 p$_{αβ}$ ( r$_{2}$ ) − 1 3 P ( r$_{2}$ ) δ$_{αβ}$ ) χ ] + P ( r$_{2}$ ) ∂χ ∂r$_{2}$$_{β}$ , (3.14d) L (2) pg $_{,αβγ}$( r$_{1}$ , r$_{2}$ ) = ∂ ∂r$_{2}$$_{α}$ p$_{βγ}$ ( r$_{2}$ ) + 1 3 P ( r$_{2}$ ) δ$_{βγ}$ χ ] + ∂ ∂r$_{2}$$_{β}$ [( p$_{αγ}$ ( r$_{2}$ ) + 1 3 P ( r$_{2}$ ) δ$_{αγ}$ ) χ ] − 2 3 δ$_{αβ}$ ∂ ∂r$_{2}$$_{ν}$ [( p$_{νγ}$ ( r$_{2}$ ) + 1 3 P ( r$_{2}$ ) δ$_{νγ}$ ) χ ] − ( ∂ ∂r$_{2}$$_{γ}$ p$_{αβ}$ ( r$_{2}$ ) ) χ, (3.14e) L (2) l g $_{,β}$( r$_{1}$ , r$_{2}$ ) = − l r$_{1}$ ) P ( r$_{1}$ ) p$_{αβ}$ ( r$_{1}$ ) ∂χ ∂r$_{2}$$_{α}$ − 1 3 l ( r$_{1}$ ) ∂χ ∂r$_{2}$$_{β}$ − ∂l ( r$_{1}$ ) ∂r$_{1}$$_{β}$ χ. (3.14f) 17 where χ = χ ( r$_{1}$ − r$_{2}$ ). With this, we obtained all the elements of the coarse-grained Poisson operator L $^{(2)}$, since the elements L (2) g $_{P}$, L (2) gp , and L (2) g l are obtained from the antisymmetry condition for L $^{(2)}$. We note the occurence of the smoothing function χ ( r$_{1}$ − r$_{2}$ ) in the Poisson operator L $^{(2)}$. Indeed, when looking at the elements of L (2) which involve only the hydrodynamic fields ρ , g , and ϵ , we see that they differ from the appropriate elements of L (1) in (2.6) only in locality, i.e., instead of the Dirac delta function, we rather have a smoothing function χ ( r$_{1}$ − r$_{2}$ ). While the original Cahn-Hilliard model is local in space, the coarse-grained model, instead, takes into account the whole volume element v ( r$_{1}$ ) of the smoothing function χ ( r$_{1}$ − r$_{2}$ ). By assumption of the length scales comparison (3.2), this smoothing function behaves simply as a delta function on the Doi-Ohta level due to the difference in length scales. Alternatively, the expressions for the elements of the Poisson matrix (3.14d)-(3.14f) could be also obtained based only on the mathematical character of