II. GENERIC FORMALISM The goal is to establish a systematic, thermodynamically guided method which determines the coarse-grained Doi-Ohta model of a phase separating binary fluid from the more detailed Cahn-Hilliard model, in the region where the considered system can be described with both approaches. To connect these two different levels of description, the General Equation for the Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) framework is used [41–43]. This approach allows to specify the mathematical structure which should be left invariant in the coarse-graining procedure and hence assures thermodynamically consistent results. In the following, we first describe the GENERIC structure and formulate the Cahn- Hilliard model within this framework. The coarse-graining procedure which derives the more macroscopic Doi-Ohta model is then presented. In particular, we present some mathematical challenges which occur within the coarse-graining procedure, as well as the efficient way to extract the interface relaxation times of the Doi-Ohta level from the underlying morphology. The main points of the GENERIC framework of nonequilibrium thermodynamics [41–43] can be summarized in the following way. In analogy to equilibrium thermodynamics, one needs to choose a complete set of variables x , which describes the situation of interest to the desired detail. For a system out of equilibrium the time evolution of all the relevant state variables can be divided into the reversible and the irreversible contributions These two contributions are obtained by considering separately the energy E and the entropy S — the two generators of reversible and irreversible dynamics, respectively. This can be explained by using the analogy with the classical Hamiltonian mechanics as following. Using the classical Hamiltonian mechanics, the reversible contribution of the time evolution, ˙ x |$_{rev}$ , is related to the energy gradient by way of a Poisson operator L , so that it is ˙ x |$_{rev}$ = L · ( δE/δ x ). The Poisson bracket associated to L , given by { A, B } = ( δA/δ x , L · δB/δ x ) with appropriate scalar product ( ., . ), must be antisymmetric and satisfy Lebniz’ rule and the Jacobi identity, which are both features that capture the nature of reversibility. The motivation for formulating the irreversible part of the dynamics comes from the reversible part. For the reversible dynamics, the energy plays a distinct role — it is a conserved quantity 6 ˙ x = ˙ x |$_{rev}$ + ˙ x |$_{irrev}$ . (2.1)