(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].