The Davis-Landau Sonic Onset Law: Universal Critical Velocity for Vortex Nucleation in Bose-Einstein Condensates

We report systematic validation of the Davis Law of Critical Velocity governing vortex nucleation in two-dimensional Gross-Pitaevskii dynamics:

$$ C = \frac{\tau}{K} \quad \Rightarrow \quad \beta = \frac{v_c}{c_s} = f(K_{\text{eff}}) \quad \text{(species-independent, geometry-indexed)} $$

Through GPU-accelerated Gross-Pitaevskii simulations across three atomic species (⁸⁷Rb, ²³Na, ⁷Li) spanning a 12:1 mass ratio, we demonstrate that the dimensionless critical velocity β = v_c/c_s is species-independent—a universal geometric invariant within each curvature regime, determined solely by obstacle geometry.

Key Results

1. Universal Critical Velocity:

   • Hard obstacles (tanh-wall): β_hard = 0.425 ± 0.000 across all three species
   • Semi-hard obstacles (Gaussian, V₀ = 50μ): β_soft = 0.486 ± 0.000 across all species
   • Excellent agreement with Frisch-Pomeau-Rica result (β ≈ 0.4)

2. Perfect Species Independence: The coefficient of variation is 0.0% for both obstacle types across a 12:1 mass ratio, definitively proving that β depends only on obstacle geometry, not atomic properties.

3. Davis Law Support: The relation C = τ/K from the Davis Field Equations correctly predicts that obstacle curvature controls the critical velocity. High-curvature (hard) obstacles nucleate vortices at lower β than low-curvature (soft) obstacles.

4. Quantum vs Classical Wake Dynamics: In the post-critical wake, coherent oscillations occur with phase φ ≈ 0° (symmetric breathing) rather than φ ≈ π (alternating von Kármán), implying suppression of the classical anti-phase shedding mode.

The Geometric Trichotomy

The functional relationship β = f(K_eff) is universal—independent of atomic mass, scattering length, and density—with distinct numerical values corresponding to distinct geometric classes.

Unifying Principle

The various measured thresholds are unified by a single principle:

• Geometry selects the regime (soft vs hard)
• Compressibility selects the prefactor (local c_s and amplification A)
• Barrier physics selects the delay (sonic onset vs nucleation threshold)

This validates the Davis Law prediction that coherence thresholds are determined by constraint geometry (C = τ/K), not microscopic physics.

Evidence Status

Included Files

Simulation Code

• multi_species_trichotomy.py — Main simulation testing β across Rb-87, Na-23, Li-7
• curvature_scaling_law.py — Tests C = τ/K scaling, K_eff operational definition

Figure Generation

• plot_davis_landau_law.py — Publication figures (universal β, v_c vs c_s)
• mach_summary_for_paper.py — M_rim vs v/c_s showing sonic onset mechanism
• phase_comparison_figure.py — VK vs GP symmetric modes diagnostic
• stability_islands_extended.py — Stability islands analysis

Data

• trichotomy_data.json — Pre-computed multi-species results
• superfluid_test_results_corrected.json — Validation data

Relation to Prior Work

This paper extends the Davis Framework for geometric physics:

• The Incompressibility of Topological Charge (Yang-Mills Mass Gap) — DOI: 10.5281/zenodo.17846521
• Holonomy-First Navier-Stokes Regularity — DOI: 10.5281/zenodo.18216597
• Field Equations of Semantic Coherence — DOI: 10.5281/zenodo.14784553

The critical velocity emerges as the threshold where topological defect creation (vortex nucleation) becomes energetically favorable—the "cost of distinguishability" in the information-geometric framework.

Keywords

Bose-Einstein condensate, critical velocity, superfluidity, Gross-Pitaevskii equation, vortex nucleation, Landau criterion, sonic onset, geometric trichotomy, Davis Law, quantum hydrodynamics, obstacle geometry, Mach number, effective curvature, species independence, universal constant

License

Creative Commons Attribution 4.0 International (CC BY 4.0)

Related Identifiers

• References: DOI 10.5281/zenodo.17846521 (Yang-Mills Mass Gap)
• References: DOI 10.5281/zenodo.18216597 (Navier-Stokes Regularity)
• References: DOI 10.5281/zenodo.14784553 (Field Equations of Semantic Coherence)