(Eq. 21)
Here M is the added mass, and β is the geometric factor with wall effect corrections, the integrands include the memory functions associated with the velocity autocorrelation functions in different domains (lubrication, bulk, and near-wall regimes in order of the first three terms on the right-hand side). The fourth term on the right-hand side is the force from other thermodynamic potentials, same as F(S) in (Eq. 7), and the fifth term is the random force term with colored noise to be consistent with the fluctuation-dissipation theorem for composite GLE [129, 130].
Effect of molecular forces is introduced as forcing functions in the GLE [129] and the effect of multiple particles including multiparticle HI can be introduced via density functional theory-based treatments [82, 83] to define F(S) from hydrodynamic and colloidal effects in addition to the specific contributions from molecular forces. If the memory functions are unknown, they can be obtained via deterministic approaches by solving the continuum hydrodynamic equations numerically [46, 128]. These disparate hydrodynamic fields and molecular forces can be integrated into a single GLE to realize a unified description of particle dynamics under the influence of molecular and hydrodynamic forces [85]. Another approach to integrating these forces is via the Fokker Planck approach using the sequential multiscale method paradigm [131].