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part and a part arising from the diffusion. Since the hydrodynamic variables are simply taken over from the Cahn-Hilliard level, the appropriate hydrodynamic elements of the new friction matrix are the same as in (2.7). The diffusive part, on the other hand, is related to the configurational variable c , and the ensemble averaging must be performed in a similar way as for the Poisson operator in Section III D. For the analytical derivation of the coarse- grained friction matrix M (2) $^{′}$, $^{dif}$, in analogy to Eq. (3.11), one starts from 

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and then employs similar procedure and approximations as in the derivation of the elements of the Poisson matrix L (2) (see Appendix A). For example, the relaxation of the Doi-Ohta variable Q could be expressed as an irreversible part of the GENERIC time evolution equa- tion (3.18) as 

where we have used the approximation ( ∂Q ( r ) /∂t ) |$_{irrev}$ ≈ l ( r ) ( ∂P ( r ) /∂t ) |$_{irrev}$ , since the interfacial width l does not change much, compared to the other two configurational vari- ables. Using the entropy functional derivative (3.10) with λ = 0 , results in the matrix element M (2) $^{′}$, dif P ε being the only relevant term for the variable P which is left after the av- eraging (3.19). Then, putting the equation (3.20) in the Doi-Ohta equation (1.7a) for the relaxation of Q in a homogeneous case gives the following expression for the relaxation time 

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where V is the volume of the system. Details of the calculation are given in Appendix B. Similar expression could be obtained for the relaxation time τ$_{do}$$_{,}$$_{2}$ . The above formula, gives the expression for the Doi-Ohta relaxation time τ$_{do}$$_{,}$$_{1}$ calculated from the transformation of the element of the friction matrix M (1) , dif containing the relaxation parameters of the Cahn-Hilliard level at any time t . This is the result of the presented systematic coarse- graining procedure developed within the GENERIC formalism which shows in which way the Doi-Ohta relaxation times can be obtained from the more detailed level of description. However, the above relation is difficult to express analytically in form of the variables of the Doi-Ohta level, due to the diffusive terms coming from the Cahn-Hilliard level. On the other hand, the above expression could, in principle, be calculated numerically using short 

