Semi-Analytical Quantum-Hydrodynamic Model of Anomalous Water Diffusion in a Dissipative Medium

This work investigates a new microscopic semi-analytical model of anomalous self-diffusion of liquid water under high pressure, based on the Madelung quantum-hydrodynamic representation of the Schrödinger-Langevin (Costin) equation for open quantum systems. The quantum dynamics of the proton in the hydrogen bond (O-H···O) is modeled as an asymmetric vortex pair, where the ratio of oxygen and hydrogen quantum circulations (χ = Γ_O / Γ_H = 2) is deduced within the Quantum Theory of Atoms in Molecules (QTAIM) framework using the Poincaré index theorem for bounded domains. By linear expansion of the Biot-Savart integral in the localized perturbation limit (kd ≈ 1), we derive the exact instability eigenvalue Δ_OH(k, d) = ω₀² (1 - d_crit⁴ / d⁴), where the critical distance d_crit = 4 a_H = 2.11 Å is obtained from the Madelung velocity balance principle on the basis of the Bohr radius (a_H = 0.529 Å), showing strong agreement with high-resolution X-ray diffraction (HR-XRD) experimental data. The model incorporates environmental thermal decoherence via the introduction of Costin's friction term, which determines the damped instability growth rate σ_damped. The cooperative coupling factor (α ≈ 40) is independently derived from the spectral shifts in the cluster zero-point energy (ZPE) model and validated by inelastic neutron scattering (INS) and high-pressure Raman spectroscopy. The quantitative results obtained from the model show excellent agreement with the experimental data of Krynicki et al. (1978), precisely accounting for the 52% increase in the self-diffusion coefficient at a pressure of 100 MPa and explaining the anomalous isotopic shifts observed in heavy water (D₂O).