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 . The general expression for the coarse-grained Poisson operator is given by which for coarse graining from the Cahn-Hilliard to the Doi-Ohta level takes the form 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 L $^{(2)}$( y ) = 〈 ( δ Π ( x ) δx ) T · L $^{(1)}$( x ) · ( δ Π ( x ) δx ) 〉$_{y}$ . (3.11) L (2) ij ( 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$_{i}$[ x ]( r ′ $_{1}$) δx$_{k}$ ( r$_{3}$ ) L (1) kl ( r$_{3}$ , r$_{4}$ ) δ Π$_{j}$[ x ]( r ′ $_{2}$) δx$_{l}$ ( r$_{4}$ ) , (3.12) D. Poisson operator L $^{(2)}$( r$_{1}$ , r$_{2}$ ) =      0 L (2) ρ g 0 0 0 0 L (2) g ρ L (2) gg L (2) g ϵ L (2) g P L (2) gp L (2) g l 0 L (2) ϵg 0 0 0 0 0 L (2) P g 0 0 0 0 0 L (2) pg 0 0 0 0 0 L (2) l g 0 0 0 0     $^{.}$ (3.13) 16