Appendix A fluctuations, using the presented coarse-graining procedure. However, the phenomenological Doi-Ohta theory assumes low Reynolds number, i.e., neglects inertia. Deeper insight into the mixed viscous-inertial regime is therefore crucial in order to develop the connection between the two levels and to understand the role of inertia in this problem. We present a derivation of the elements L (2) P $_{g}$( r$_{1}$ , r$_{2}$ ) and L (2) l g ( r$_{1}$ , r$_{2}$ ) of the coarse-grained Poisson operator L (2) in detail. For the element L (2) P $_{g}$( r$_{1}$ , r$_{2}$ ) the general expression (3.12) gives which after substitution of the appropriate functional derivatives, L (1) c g element, and the integration over r$_{3}$ and r$_{4}$ becomes For further calculations, we use the difference between the Cahn-Hilliard and Doi-Ohta length scales (3.2), so that ∫ V d $^{3}$r ′ 1 χ ( r$_{1}$ − r ′ $_{1}$) χ ( r ′ 1 − r ′ $_{2}$) ≈ χ ( r$_{1}$ − r ′ $_{2}$) (see [45]). Then the last equation, after integration over r ′ $_{1}$, becomes After integration by parts, in which surface terms vanish, using approximation χ ( r$_{1}$ − r ′ $_{2}$) χ ( r$_{2}$ − r ′ $_{2}$) ≈ χ ( r$_{1}$ − r$_{2}$ ) χ ( r$_{2}$ − r ′ $_{2}$) valid due to the fact that the smoothing length is much smaller than the Doi-Ohta length scale, equation (3.2), and the identity ∂χ ( r − r ′ ) /∂ r ′ = − ∂χ ( r − r ′ ) /∂ r , we come to the expression 25 L (2) P g $_{,β}$( r$_{1}$ , r$_{2}$ ) = ∫ D c ρ$_{(}$$_{Q,}$$_{q}$$_{)}$ [ c ] × { ∂ ∂r$_{2}$$_{α}$ ∫ V d $^{3}$r ′ 2 2 ∂c ∂r ′ 2 α ∂c ∂r ′ 2 β χ ( r$_{1}$ − r$_{2}$ ) χ ( r$_{2}$ − r ′ $_{2}$) − ∂ ∂r$_{1}$$_{β}$ ∫ V d $^{3}$r ′ 2 |∇ c ( r ′ $_{2}$) | 2 χ ( r$_{1}$ − r$_{2}$ ) χ ( r$_{2}$ − r ′ $_{2}$) − ∂ ∂r$_{2}$$_{β}$ V d $^{3}$r ′ 2 |∇ c ( r ′ $_{2}$) | 2 χ ( r$_{1}$ − r$_{2}$ ) χ ( r$_{2}$ − r ′ $_{2}$) } . (A4) L (2) P $_{g}$( r$_{1}$ , r$_{2}$ ) = ∫ D c ρ$_{(}$$_{Q,}$$_{q}$$_{)}$ [ c ] ∫ V d $^{3}$r ′ 2 χ ( r$_{2}$ − r ′ $_{2}$) ∂c ∂ r ′ 2 ∂ ∂ r ′ 2 · ( 2 ∂c ∂ r ′ 2 χ ( r$_{1}$ − r ′ $_{2}$) ) . (A3) L (2) P $_{g}$( r$_{1}$ , r$_{2}$ ) = ∫ D c ρ$_{(}$$_{Q,}$$_{q}$$_{)}$ [ c ] ∫ V d $^{3}$r ′ 1 ∫ V d $^{3}$r ′ 2 χ ( r$_{1}$ − r ′ $_{1}$) χ ( r$_{2}$ − r ′ $_{2}$) × ∂c ∂ r ′ 2 ∂ ∂ r ′ 2 · ( 2 ∂c ∂ r ′ 2 χ ( r ′ 1 − r ′ $_{2}$) ) . (A2) L (2) P $_{g}$( r$_{1}$ , r$_{2}$ ) = ∫ D x ρ$_{y}$ [ x ] ∫ V d $^{3}$r ′ 1 ∫ V d $^{3}$r ′ 2 χ ( r$_{1}$ − r ′ $_{1}$) χ ( r$_{2}$ − r ′ $_{2}$) × ∫ V d $^{3}$r$_{3}$ ∫ V d $^{3}$r$_{4}$ δ Π$_{P}$ [ c ]( r ′ $_{1}$) δc ( r$_{3}$ ) L (1) c g ( r$_{3}$ , r$_{4}$ ) δ Π$_{g}$ [ g ]( r ′ $_{2}$) δ g ( r$_{4}$ ) , (A1)