content_image/369839.jpg content_image/369838.jpg content_image/369837.jpg content_image/369836.jpg
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. 

## Appendix A 

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 

<img src='content_image/369839.jpg'>

which after substitution of the appropriate functional derivatives, L (1) c g element, and the integration over r$_{3}$ and r$_{4}$ becomes 

<img src='content_image/369838.jpg'>

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 

<img src='content_image/369837.jpg'>

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 

<img src='content_image/369836.jpg'>

