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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 

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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 

