The Smoluchowski equation for colloidal suspensions developed and analyzed through the GENERIC formalism

The Smoluchowski equation (SE) and the mechanical stress tensor for the over-damped dynamics of colloidal particles is derived directly at the pair distribution level starting from a thermodynamic basis using the general equations for equilibrium non-equilibrium reversible and irreversible coupling (GENERIC) formalism. Within the GENERIC formalism, the effect of the non-trivial convection due to hydrodynamic interactions is incorporated for the first time. The method generates a thermodynamically valid set of transport equations for the colloidal dispersion, thus properly identifying the extra stress due to the presence of the colloids. The derivation connects a formal entropy expansion to the many-body terms that arise in both the transport equation and the stress tensor, thus unifying their origin and providing a systematic path forward for improvement in the theory. The analysis identifies the thermodynamically valid stress expression, thus clarifying a long-standing problem in the literature that arises when separate derivations are performed for the transport equations and the stress tensor. The results of previous investigators are analyzed within this framework. Comparison with alternate methods of deriving the many-body Smoluchowski equation provide new insight into the nature of the many-body terms.