content_image/329876.jpg content_image/329877.jpg content_image/329879.jpg
As discussed above, there exists a systematic procedure to obtain the Lagrange multipliers from the thermodynamically guided simulations [48]. However, their determination is a difficult task and for the calculation of the time friction matrix elements in section III E we will assume λ = 0 . 

## D. Poisson operator 

The general expression for the coarse-grained Poisson operator is given by 

<img src='content_image/329876.jpg'>

which for coarse graining from the Cahn-Hilliard to the Doi-Ohta level takes the form 

<img src='content_image/329877.jpg'>

where the mappings Π are given by (3.4), and the elements of the Poisson operator L (1) by (2.6). To understand the above expression, note that the integrations over r$_{3}$ and r$_{4}$ come from the contraction of the operator L (1) with δ Π /δ x , which includes both matrix multiplication and integration over the position label. The obtained quantity must then be averaged in order to obtain the expression at the Doi-Ohta level. Therefore, the integrations over r ′ 1 and r ′ 2 come from the spatial smoothing of this quantity, since the ensemble averaging also implies smoothing in space, as noted in the previous subsection. 

From the elements of L (1) in (2.6) and the mappings Π in (3.4), we conclude that the Poisson operator L (2) has the following form 

<img src='content_image/329879.jpg'>

(3.13) 

