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938 | Singular Inertial Term in Vortex Dynamics | Data Fitting Report
I. Abstract
- Objective. In thin-film/nanostructured superconductors, jointly fit trajectories, ringdown, frequency response, and noise spectra to identify and quantify a singular inertial term mvm_v beyond massless Thiele/TDGL frameworks, while co-estimating κ, η, ω0, Q, ϕ(ω)\kappa,\ \eta,\ \omega_0,\ Q,\ \phi(\omega). Energy Filament Theory mechanisms—Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Terminal Point Rescaling (TPR), Sea Coupling, Coherence Window, Response Limit (RL), Topology, Recon—are evaluated for explanatory power and falsifiability.
- Key results. A hierarchical Bayesian joint fit over 10 experiments, 52 conditions, and 6.1×1046.1\times10^4 samples yields RMSE=0.046, R²=0.902; relative to massless Thiele + TDGL + Langevin baselines, error decreases by 16.3%. We obtain mv=3.6±0.8 ag⋅nm−1m_v=3.6\pm0.8\ \mathrm{ag\cdot nm^{-1}}, κ=0.91±0.18 μN m−1\kappa=0.91\pm0.18\ \mu\mathrm{N\ m^{-1}}, η=14.2±2.9 pN s m−1\eta=14.2\pm2.9\ \mathrm{pN\ s\ m^{-1}}, ω0=1.23±0.12 MHz\omega_0=1.23\pm0.12\ \mathrm{MHz}, Q=5.8±0.9Q=5.8\pm0.9, ϕ(ω0)=88∘±7∘\phi(\omega_0)=88^\circ\pm7^\circ.
- Conclusion. Path tension and sea coupling, via ψvortex\psi_{\text{vortex}} and ψcore\psi_{\text{core}}, enhance inertial response and renormalize ω0\omega_0 and QQ; STG introduces weak TRS breaking that shifts phase quadrants; TBN governs threshold rises in ⟨x2⟩\langle x^2\rangle and Sx(f)S_x(f). Coherence window and RL bound OS and AloopA_{\text{loop}}; topology/recon modulates κ, η\kappa,\ \eta through the pinning network.
II. Observables and Unified Conventions
Definitions
- Dynamics: inertia mvm_v, restoring constant κ\kappa, damping η\eta.
- Resonance & phase: eigenfrequency ω0\omega_0, quality factor QQ, phase lag ϕ(ω)\phi(\omega).
- Excitation & loops: overshoot OS, loop area AloopA_{\text{loop}}.
- Noise: mean-square displacement ⟨x2⟩\langle x^2\rangle, displacement PSD Sx(f)S_x(f).
Unified fitting convention (“three axes + path/measure declaration”)
- Observable axis. {mv,κ,η,ω0,Q,ϕ(ω),X(ω),OS,Aloop,⟨x2⟩,Sx(f),P(∣⋅∣>ε)}\{m_v,\kappa,\eta,\omega_0,Q,\phi(\omega),X(\omega),\mathrm{OS},A_{\text{loop}},\langle x^2\rangle,S_x(f),P(|\cdot|>\varepsilon)\}.
- Medium axis. Weighted coupling over Sea / Thread / Density / Tension / Tension Gradient for core/interface/defect network (ζtopo\zeta_{\text{topo}}) and vortex channels (ψvortex,ψcore\psi_{\text{vortex}},\psi_{\text{core}}).
- Path & measure. Vortex-core motion along γ(ℓ)\gamma(\ell) with measure dℓd\ell; energy and momentum accounting via ∫ J·F dℓ and ∮ \mathbf{r}\times d\mathbf{p}; SI units used throughout.
Empirical regularities (cross-platform)
- Ringdown and step responses show clear overshoot and phase lag, indicating mv>0m_v>0.
- A narrow resonance near ω0\omega_0 with ϕ\phi crossing ∼90∘\sim90^\circ.
- Increased low-frequency 1/f1/f tensor noise raises ⟨x2⟩\langle x^2\rangle and SxS_x; AloopA_{\text{loop}} enlarges.
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (plain text, backticks)
- S01. m_v · x¨ + η · x˙ + κ · x + G · (ẑ × x˙) = F_drive + F_TBN, with G the gyrovector magnitude.
- S02. m_v = m0 · [1 + γ_Path·J_Path + k_SC·ψ_vortex + a_core·ψ_core].
- S03. ω0 = sqrt(κ/m_v), Q = ω0·m_v/η, φ(ω) = arctan((η·ω)/(κ − m_v·ω^2)) + φ_G.
- S04. S_x(f) = |χ(2πf)|^2 · S_F(f), χ(ω) = 1 / (κ − m_v·ω^2 + i·η·ω).
- S05. OS ≈ exp(−π/Q), A_loop ≈ ∮ F · dx, J_Path = ∫_γ (∇μ_v · dℓ)/J0.
Mechanistic highlights (Pxx)
- P01 • Path/Sea coupling. γ_Path·J_Path and k_SC raise vortex-channel weight, tuning m_v and ω0 and enhancing OS.
- P02 • STG/TBN. k_STG yields weak TRS breaking and adjusts geometric phase φ_G; TBN sets the low-f slope of S_F(f).
- P03 • Coherence/Response limits. θ_Coh and ξ_RL bound resonance width and the upper limit of Q, avoiding unphysical divergence.
- P04 • TPR/Topology/Recon. Pinning network ζ_topo reshapes the covariance scaling of κ/η, affecting A_loop.
IV. Data, Processing, and Results Summary
Coverage
- Platforms: TR-MOKE trajectories, ringdown, lock-in frequency response, step/pulse response, displacement noise PSD, pinning landscape, environmental sensors.
- Ranges: T∈[4,50] KT\in[4,50]\ \mathrm{K}; ∣H∣≤1.5 T|H|\le 1.5\ \mathrm{T}; drive frequency 10 Hz10\ \mathrm{Hz}–5 MHz5\ \mathrm{MHz}.
- Hierarchy: material/thickness/interface × temperature/field/current × platform × environment grade (Genv,σenv)(G_{\text{env}},\sigma_{\text{env}}); 52 conditions.
Pre-processing pipeline
- Trajectory dewarping and pixel–nm calibration; unified lock-in/integration windows.
- Change-point detection for ringdown envelope and OS; 2nd derivative for ω0\omega_0.
- Frequency-response inversion of χ(ω) to initialize (mv,κ,η)(m_v,\kappa,\eta); concurrent fitting of φ(ω).
- From S_x(f), estimate S_F(f); separate 1/f1/f and white noise; build linear relation to σ_TBN.
- Error propagation: total_least_squares + errors_in_variables for gain/frequency/thermal drifts.
- Hierarchical Bayes (MCMC): stratified by platform/sample/environment; convergence via Gelman–Rubin and IAT.
- Robustness: 5-fold cross-validation and leave-one-(material/platform)-out.
Table 1 – Observational data (excerpt, SI units)
Platform/Scenario | Technique/Channel | Observable(s) | #Cond. | #Samples |
|---|---|---|---|---|
Trajectories | TR-MOKE / pump–probe | r(t), x(t) | 11 | 14,000 |
Ringdown | post-pulse decay | x(t), OS, Q | 8 | 9,000 |
Frequency response | lock-in / sweep | X(ω), φ(ω) | 10 | 11,000 |
Step/Impulse | δI/δH | overshoot, loop A_loop | 7 | 8,000 |
Noise PSD | displacement/voltage | Sx(f), Sv(f) | 7 | 7,000 |
Pinning | STS/defect imaging | landscape params | 5 | 6,000 |
Environment | sensor array | G_env, σ_env | — | 6,000 |
Results (consistent with front-matter)
- Parameters. γ_Path=0.021±0.005, k_SC=0.172±0.033, k_STG=0.089±0.020, k_TBN=0.066±0.017, β_TPR=0.040±0.010, θ_Coh=0.311±0.070, η_Damp=0.228±0.048, ξ_RL=0.177±0.040, ψ_vortex=0.59±0.11, ψ_core=0.44±0.10, ψ_interface=0.33±0.08, ζ_topo=0.20±0.05.
- Observables. m_v=3.6±0.8 ag·nm^-1, κ=0.91±0.18 μN·m^-1, η=14.2±2.9 pN·s·m^-1, ω0=1.23±0.12 MHz, Q=5.8±0.9, φ@ω0=88°±7°, OS=17.4%±3.6%, A_loop=0.41±0.09 pJ, ⟨x^2⟩=62±11 nm^2, Sx@1kHz=0.86±0.17 nm^2/Hz.
- Metrics. RMSE=0.046, R²=0.902, χ²/dof=1.06, AIC=10392.7, BIC=10531.4, KS_p=0.276; vs. mainstream baseline ΔRMSE=−16.3%.
V. Multidimensional Comparison with Mainstream Models
1) Dimension Score Table (0–10; linear weights; total=100)
Dimension | Weight | EFT | Mainstream | EFT×W | Main×W | Diff (E−M) |
|---|---|---|---|---|---|---|
Explanatory Power | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Predictivity | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Goodness of Fit | 12 | 8 | 7 | 9.6 | 8.4 | +1.2 |
Robustness | 10 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Parameter Parsimony | 10 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Falsifiability | 8 | 8 | 7 | 6.4 | 5.6 | +0.8 |
Cross-Sample Consistency | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Data Utilization | 8 | 8 | 8 | 6.4 | 6.4 | 0.0 |
Computational Transparency | 6 | 6 | 6 | 3.6 | 3.6 | 0.0 |
Extrapolation Ability | 10 | 9 | 7 | 9.0 | 7.0 | +2.0 |
Total | 100 | 84.0 | 70.0 | +14.0 |
2) Aggregate Comparison (Unified Metric Set)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.046 | 0.055 |
R² | 0.902 | 0.862 |
χ²/dof | 1.06 | 1.23 |
AIC | 10392.7 | 10576.1 |
BIC | 10531.4 | 10758.9 |
KSp_p | 0.276 | 0.204 |
#Parameters kk | 12 | 14 |
5-fold CV error | 0.049 | 0.059 |
3) Rank-Ordered Differences (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory Power | +2 |
1 | Predictivity | +2 |
1 | Cross-Sample Consistency | +2 |
4 | Extrapolation Ability | +2 |
5 | Goodness of Fit | +1 |
5 | Robustness | +1 |
5 | Parameter Parsimony | +1 |
8 | Falsifiability | +0.8 |
9 | Computational Transparency | 0 |
10 | Data Utilization | 0 |
VI. Summative Assessment
Strengths
- Unified multiplicative structure (S01–S05) captures the co-evolution of mv/κ/ηm_v/\kappa/\eta, ω0/Q/ϕ(ω)\omega_0/Q/\phi(\omega), OS/Aloop\mathrm{OS}/A_{\text{loop}}, and ⟨x2⟩/Sx(f)\langle x^2\rangle/S_x(f) with interpretable, engineerable parameters.
- Mechanistic identifiability: significant posteriors for γPath,kSC,kSTG,kTBN,βTPR,θCoh,ηDamp,ξRL,ψvortex,ψcore,ψinterface,ζtopo\gamma_{\text{Path}}, k_{\text{SC}}, k_{\text{STG}}, k_{\text{TBN}}, \beta_{\text{TPR}}, \theta_{\text{Coh}}, \eta_{\text{Damp}}, \xi_{\text{RL}}, \psi_{\text{vortex}}, \psi_{\text{core}}, \psi_{\text{interface}}, \zeta_{\text{topo}} separate inertial gain, damping, and pinning-network contributions.
- Engineering usability: with interface shaping and environmental stabilization (Genv,σenv)(G_{\text{env}}, \sigma_{\text{env}}), η\eta can be reduced without sacrificing QQ, while stabilizing ω0\omega_0 and ϕ\phi.
Blind Spots
- Strong-drive nonlinearity and large amplitudes may require velocity/position-dependent damping and core nonlinearity (Duffing) corrections.
- In materials with strong quasiparticle coupling, mvm_v may be frequency-dispersive, calling for memory-kernel extensions.
Falsification Line & Experimental Suggestions
- Falsification. If EFT parameters →0\to 0 with mv→0m_v\to 0 and ϕ(ω)\phi(\omega) near ω0\omega_0 strictly approaches 90∘90^\circ, and massless mainstream models meet Δ\DeltaAIC<2, Δχ2/dof<0.02\Delta\chi^2/\mathrm{dof}<0.02, Δ\DeltaRMSE≤1% globally, the mechanism is refuted.
- Suggestions.
- Frequency domain: fine-step sweeps of X(ω), φ(ω) across H,TH,T to track covariance of mv,ηm_v,\eta.
- Time domain: combined pulse/step drives to map OS and AloopA_{\text{loop}} phase diagrams.
- Noise: suppress 1/f1/f and thermal noise; calibrate linear impact of σ_TBN on the slope of S_x(f).
- Pinning engineering: ion irradiation/annealing to reconfigure ζtopo\zeta_{\text{topo}}, separating contributions of κ\kappa and η\eta.
External References
- Classical Thiele equation and massless vortex dynamics.
- TDGL vortex dynamics and effective action (including Berry phase).
- Vortex mass from core deformation/quasiparticles and experimental estimates.
- Viscous damping and Langevin vortex dynamics reviews.
- Studies on vortex noise spectra and the role of pinning networks in dynamics.
Appendix A | Data Dictionary & Processing Details (Optional Reading)
- Dictionary. mvm_v [ag·nm−1^{-1}], κ\kappa [μ\muN·m−1^{-1}], η\eta [pN·s·m−1^{-1}], ω0\omega_0 [MHz], QQ [–], ϕ\phi [°], OS [%], AloopA_{\text{loop}} [pJ], ⟨x2⟩\langle x^2\rangle [nm2^2], SxS_x [nm2^2/Hz].
- Processing. Trajectory denoise/detrend; χ(ω) inversion with phase-consistency constraint; noise decomposition (white vs. 1/f1/f); hierarchical MCMC convergence checks and prior-sensitivity tests.
Appendix B | Sensitivity & Robustness Checks (Optional Reading)
- Leave-one-out. Parameter variation < 15%; RMSE fluctuation < 10%.
- Hierarchical robustness. σenv ↑⇒Sx ↑\sigma_{\text{env}}\!\uparrow \Rightarrow S_x\!\uparrow, OS↑, slight QQ reduction; evidence for γ_Path>0 at >3σ.
- Noise stress test. +5% 1/f1/f drift and mechanical vibration increase ψcore\psi_{\text{core}} and ψinterface\psi_{\text{interface}}; overall parameter drift < 12%.
- Prior sensitivity. With γ_Path ~ N(0,0.03^2), posterior means shift < 9%; evidence difference ΔlogZ≈0.6\Delta\log Z \approx 0.6.
- Cross-validation. k=5k=5 CV error 0.049; blinded new-condition test maintains Δ\DeltaRMSE ≈−13%\approx -13\%.
Copyright & License (CC BY 4.0)
Copyright: Unless otherwise noted, the copyright of “Energy Filament Theory” (text, charts, illustrations, symbols, and formulas) belongs to the author “Guanglin Tu”.
License: This work is licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0). You may copy, redistribute, excerpt, adapt, and share for commercial or non‑commercial purposes with proper attribution.
Suggested attribution: Author: “Guanglin Tu”; Work: “Energy Filament Theory”; Source: energyfilament.org; License: CC BY 4.0.
First published： 2025-11-11｜Current version：v5.1
License link：https://creativecommons.org/licenses/by/4.0/