Justification of the damped-sinusoidal transverse velocity autocorrelation

Establish a theoretical justification for representing the molecular-dynamics transverse velocity autocorrelation function of lipid-bilayer fluctuations as an exponentially damped sinusoid, and explain the observed scaling of its oscillation frequency with wavevector.

Background

The continuum hydrodynamic model predicts exponential decay of transverse membrane velocity fluctuations, whereas the molecular-dynamics simulations exhibit damped oscillations. The authors empirically approximate the transverse velocity autocorrelation function by an exponentially damped sinusoid and observe a robust oscillation-frequency scaling proportional to q{3/2}. However, they do not derive or otherwise justify this functional form from the underlying molecular or hydrodynamic dynamics, leaving the origin of the oscillatory relaxation unresolved.

References

While we cannot justify this functional form at present, we notice a robust scaling of the oscillation frequency as $ \gammaqfit \sim q\bfrac{3}{2} $ in Fig.\ \ref{fig_gamma_q}.

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