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Thermohydrodynamics for a van der Waals fluid

Following a cell-method due to van Kampen for the calculation of a coarse-grained free energy functional for the van der Waals gas, we compute a corresponding entropy functional from microscopic principles. This entropy functional is one of the building blocks of the recently developed GENERIC framework. This framework allows to obtain in a thermodynamically consistent way the continuum hydrodynamic equations for a fluid able to display liquid-vapor coexistence. Surface tension appears naturally and the resulting model describes interfaces as diffuse regions, much in the same spirit as the gradient theory for equilibrium situations. We suggest that using interfacial forces in the integral form obtained in the microscopic derivation instead of third order derivatives of the density field might represent an advantage from a computational point of view.