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for closed systems and it drives the reversible dynamics. Therefore, for the irreversible dynamics the entropy presents an important quantity — it does not decrease for closed systems. For the GENERIC formulation, it is then assumed that the irreversible contribution to the time evolution, ˙ x |$_{irrev}$ , is driven by the entropy gradient, i.e., that it is of the form ˙ x |$_{irrev}$ = M · ( δS/δ x ), with M a generalized friction matrix. This friction matrix contains transport coefficients and relaxation times associated to the corresponding dissipative effects. It is required for M to be (Onsager-Casimir) symmetric and the condition that it is positive semi-definite ensures that ˙ S ≥ 0 is fulfilled. 

From the above illustration, one concludes that the time evolution of x can be expressed in terms of four “building blocks” E , S , L and M as 

<img src='content_image/352909.jpg'>

The two different contributions to the time evolution, reversible and irreversible, are not independent. They are related by the two complementary degeneracy requirements 

<img src='content_image/352910.jpg'>

The first requirement expresses the reversible nature of the L contribution to the dynamics, demonstrating the fact that the reversible dynamics captured in L does not affect the entropy functional. The second one expresses the conservation of the total energy of an isolated system by the irreversible contribution to the system dynamics captured in M . 

The presented GENERIC form of the time evolution equations for the state variables x represents a mathematical structure which guarantees that the chosen model is thermody- namically consistent. Moreover, by concentrating on the thermodynamics building blocks E , S , L and M rather than on the time evolution equations in systematic coarse-graining pro- cedures, the obtained coarse-grained equations are guaranteed to be also thermodynamically admissible. 

## A. GENERIC formulation of the Cahn-Hilliard model 

In order to develop a systematic coarse-graining procedure which relates the more detailed Cahn-Hilliard level to the more macroscopic Doi-Ohta level, we must formulate the first one in the GENERIC structure. As a first step, we need to identify the list of independent 

