Analytical determination of the Mori–Zwanzig memory kernel

Determine the analytical form of the Mori–Zwanzig memory kernel governing the transverse velocity autocorrelation function of a lipid bilayer coupled to surrounding water.

Background

The Mori–Zwanzig formalism yields an exact integro-differential equation for the transverse velocity autocorrelation function, with a wavevector-dependent memory kernel encoding the non-Markovian molecular dynamics. In the paper, the zero-frequency integral of this kernel is obtained indirectly from the time integral of the measured autocorrelation function, which permits calculation of the generalized membrane viscosity. The full time-dependent analytical kernel itself remains unresolved.

References

Equation eq_Cm is microscopically exact---though the analytical form of the memory kernel $ \Kq (t) $ is unknown.

— Molecular interfacial rheology: Lipid membrane shear viscosity  (2609.05373 - Xu et al., 4 Sep 2026) in Section “Bridging molecular and hydrodynamic descriptions”