On the relationships between thermodynamic formalisms for complex fluids

In recent years, several seemingly distinct approaches have been developed which aim to describe the mechanics and thermodynamics of complex fluids under dynamical conditions. Two of these approaches of decidedly different flavor are the Matrix Model of thermodynamically driven systems developed by Jongschaap and the GENERIC formalism of Grmela and Ottinger. Herein, we examine the interrelationships between these two alternate approaches on an abstract level and work out direct connections for two specific examples of thermodynamically driven systems: a gas housed in a cylinder closed by a piston subject to external force and potential fields, and a dilute solution of Hookean dumbbells in a Bow field. It is demonstrated that the Matrix Model derives from the GENERIC formalism when the global thermodynamic system is split into two parts: a smaller open thermodynamic system with fewer internal variables, and its environment composed of external driving forces which are determined from the variables which are neglected in the smaller system. Thus the Matrix Model provides an explicit answer to the question of how to handle driven systems within the more global GENERIC formalism. It is also shown that the Matrix Model requires a prerequisite knowledge of the laws of motion for a given system, whereas these are explicitly embedded in the GENERIC structure.