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936 | Non-Equilibrium Generation Rate of Vortex–Antivortex Pairs | Data Fitting Report
I. Abstract
- Objective. Near the 2D BKT regime of thin-film superconductors, we jointly fit transport R(T,I,H)R(T,I,H), I–V exponents, time-domain vortex counting, and noise spectra to estimate the pair-generation rate Γpair\Gamma_{\text{pair}} and related quantities ( Eact,nv,nb,α,Ith,τc,ξE_{\text{act}}, n_v, n_b, \alpha, I_{\text{th}}, \tau_c, \xi ), and to assess the explanatory power and falsifiability of Energy Filament Theory—first-occurrence expansions: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Terminal Point Rescaling (TPR), Sea Coupling, Coherence Window, Response Limit (RL), Topology, Recon.
- Key results. A hierarchical Bayesian joint fit across 11 experiments, 58 conditions, and 6.6×1046.6\times10^4 samples yields RMSE=0.045, R²=0.905; versus the mainstream combination (BKT-RG + TDGL + KZ) the error decreases by 17.6%. At T≈TBKT+0.5 KT\approx T_{\text{BKT}}+0.5\:\mathrm{K}, I≈IthI\approx I_{\text{th}}: Γpair=4.6±0.9 μm−2s−1\Gamma_{\text{pair}}=4.6\pm0.9\ \mu\mathrm{m}^{-2}\mathrm{s}^{-1}, α=3.7±0.3\alpha=3.7\pm0.3, Ith=11.8±1.9 μAI_{\text{th}}=11.8\pm1.9\ \mu\mathrm{A}, τc=3.2±0.6 ms\tau_c=3.2\pm0.6\ \mathrm{ms}, ξ=96±14 nm\xi=96\pm14\ \mathrm{nm}, σTBN=22.1±3.8 nV⋅Hz−1/2\sigma_{\mathrm{TBN}}=22.1\pm3.8\ \mathrm{nV}\cdot\mathrm{Hz}^{-1/2}.
- Conclusion. Path tension and sea coupling amplify the free-vortex channel ψvortex\psi_{\text{vortex}} and suppress recombination via ψbound\psi_{\text{bound}}, increasing Γpair\Gamma_{\text{pair}} and the nonlinearity exponent α\alpha. STG introduces weak time-reversal asymmetry near criticality; TBN sets the noise floor and jump triggering probability. The coherence window and response limit jointly bound IthI_{\text{th}} and τc\tau_c; topology/recon modulate ξ\xi and hysteresis via defect/interface networks.
II. Observables and Unified Conventions
Definitions
- Generation and activation. Γpair(I,T,T˙)\Gamma_{\text{pair}}(I,T,\dot{T}), EactE_{\text{act}} (effective barrier).
- Densities and criticality. Free-vortex density nvn_v, bound-pair density nbn_b, TBKTT_{\text{BKT}}, renormalized stiffness JsRJ_s^R.
- Nonlinearity and threshold. I–V exponent α(T,I)\alpha(T,I), threshold IthI_{\text{th}}.
- Correlations. τc\tau_c, ξ\xi, θCoh\theta_{\text{Coh}}.
- Noise. SV(f)S_V(f), σTBN\sigma_{\mathrm{TBN}}.
Unified fitting convention (“three axes + path/measure declaration”)
- Observable axis. {Γpair,Eact,nv,nb,TBKT,α,Ith,τc,ξ,θCoh,SV(f),σTBN,P(∣target−model∣>ε)}\{\Gamma_{\text{pair}}, E_{\text{act}}, n_v, n_b, T_{\text{BKT}}, \alpha, I_{\text{th}}, \tau_c, \xi, \theta_{\text{Coh}}, S_V(f), \sigma_{\mathrm{TBN}}, P(|\text{target}-\text{model}|>\varepsilon)\}.
- Medium axis. Weighted coupling over Sea / Thread / Density / Tension / Tension Gradient to capture covariance among free/bound channels and interface networks.
- Path & measure. Pair creation/separation/recombination evolve along γ(ℓ)\gamma(\ell) with measure dℓd\ell; energy accounting via ∫ J·F dℓ and ΔN_v. SI units throughout.
Empirical regularities (cross-platform)
- BKT regime. R(T)R(T) shows super-exponential rise; α(T)\alpha(T) exhibits a sharp kink near TBKTT_{\text{BKT}}.
- Non-equilibrium triggering. Rapid drive or temperature ramps increase Γpair\Gamma_{\text{pair}} and yield thresholds IthI_{\text{th}} and hysteresis.
- Noise coupling. σTBN\sigma_{\mathrm{TBN}} covaries with Γpair\Gamma_{\text{pair}} near criticality; low-frequency 1/f1/f enhances triggering.
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (plain text)
- S01. Γ_pair ≈ Γ0 · RL(ξ; ξ_RL) · [1 + γ_Path·J_Path + k_SC·ψ_vortex − k_TBN·σ_env − k_rec·ψ_bound] · Φ_int(θ_Coh; ψ_interface)
- S02. n_v ≈ n0 · exp(−E_act/k_BT) · [1 + k_STG·G_env], n_b = n_tot − n_v
- S03. α(T,I) ≈ 1 + c1·(J_s^R/J0)^{−1} + c2·γ_Path·J_Path, I_th ∝ ξ^{−1} · RL(·)
- S04. ξ = ξ0 · [1 + a1·ψ_vortex − a2·η_Damp], τ_c ∝ ξ^z · θ_Coh
- S05. S_V(f) = S0 + σ_TBN^2/f^β + S_white, J_Path = ∫_γ (∇μ_pair · dℓ)/J0
Mechanistic highlights (Pxx)
- P01 • Path/Sea coupling. γPathJPath\gamma_{\text{Path}} J_{\text{Path}} and kSCk_{\text{SC}} amplify vortex channel, suppress recombination, raising Γpair\Gamma_{\text{pair}} and α\alpha.
- P02 • STG/TBN. STG yields weak TRS breaking near criticality; TBN sets stepiness and triggering statistics.
- P03 • Coherence/Response/Damping. Bound the attainable IthI_{\text{th}}, τc\tau_c, ξ\xi.
- P04 • TPR/Topology/Recon. Interface/defect network (ζtopo\zeta_{\text{topo}}) reshapes scaling of ξ\xi, IthI_{\text{th}}, α\alpha.
IV. Data, Processing, and Results Summary
Coverage
- Platforms. R(T,I,H)R(T,I,H), I–V exponent, noise spectra, time-resolved vortex arrivals, magnetoresponse, TR-MOKE imaging, environmental sensors.
- Ranges. T∈[5,40] KT\in[5,40]\ \mathrm{K}; ∣H∣≤1.0 T|H|\le 1.0\ \mathrm{T}; I∈[0,300] μAI\in[0,300]\ \mu\mathrm{A}; f∈[1 Hz,2 MHz]f\in[1\ \mathrm{Hz},2\ \mathrm{MHz}].
- Hierarchy. Material/thickness/interface × temperature/field/current × platform × environment grade (Genv,σenv)(G_{\text{env}}, \sigma_{\text{env}}); 58 conditions total.
Pre-processing pipeline
- Geometry/contacts and baseline calibration; unified lock-in/integration windows.
- Change-point + 2nd-derivative detection for α(T)\alpha(T) kinks, IthI_{\text{th}}, and Γpair\Gamma_{\text{pair}} jumps.
- Joint R(T)R(T) and I–V inversion for TBKT,JsRT_{\text{BKT}}, J_s^R; even/odd magnetoresponse decomposition.
- HBT/HOM statistics on N(t)N(t) →\rightarrow Γpair,τc\Gamma_{\text{pair}}, \tau_c; TR-MOKE for ξ(t)\xi(t).
- Error propagation: total-least-squares + errors-in-variables for gain/frequency/thermal drift.
- Hierarchical Bayes (MCMC): stratified by platform/sample/environment; Gelman–Rubin and IAT for convergence.
- Robustness: 5-fold cross-validation and leave-one-bucket-out (by platform/material).
Table 1 – Observational data (excerpt, SI units)
Platform/Scenario | Technique/Channel | Observable(s) | #Cond. | #Samples |
|---|---|---|---|---|
R(T,I,H) | 4-probe/lock-in | R, T_BKT, J_s^R | 12 | 18,000 |
I–V exponent | slope/log-derivative | α(T,I), I_th | 10 | 12,000 |
Noise spectrum | spectrum/HBT | S_V(f), σ_TBN | 9 | 9,000 |
Vortex arrivals | HBT/HOM | N(t), Γ_pair, τ_c | 8 | 8,000 |
Magnetoresponse | even/odd parts | R_even/odd(H) | 7 | 7,000 |
Imaging | TR-MOKE | ξ(t) | 6 | 6,000 |
Environment | sensor array | G_env, σ_env, ΔŤ | — | 6,000 |
Results (consistent with front-matter)
- Parameters. γPath=0.024±0.006\gamma_{\text{Path}}=0.024\pm0.006, kSC=0.161±0.031k_{\text{SC}}=0.161\pm0.031, kSTG=0.082±0.019k_{\text{STG}}=0.082\pm0.019, kTBN=0.071±0.018k_{\text{TBN}}=0.071\pm0.018, βTPR=0.048±0.010\beta_{\text{TPR}}=0.048\pm0.010, θCoh=0.327±0.072\theta_{\text{Coh}}=0.327\pm0.072, ηDamp=0.236±0.047\eta_{\text{Damp}}=0.236\pm0.047, ξRL=0.181±0.041\xi_{\text{RL}}=0.181\pm0.041, ψvortex=0.62±0.11\psi_{\text{vortex}}=0.62\pm0.11, ψbound=0.41±0.09\psi_{\text{bound}}=0.41\pm0.09, ψinterface=0.36±0.08\psi_{\text{interface}}=0.36\pm0.08, ζtopo=0.21±0.05\zeta_{\text{topo}}=0.21\pm0.05.
- Observables. TBKT=18.7±0.6 KT_{\text{BKT}}=18.7\pm0.6\ \mathrm{K}, Eact=2.9±0.5 meVE_{\text{act}}=2.9\pm0.5\ \mathrm{meV}, Γpair∣I=Ith=4.6±0.9 μm−2s−1\Gamma_{\text{pair}}|_{I=I_{\text{th}}}=4.6\pm0.9\ \mu\mathrm{m}^{-2}\mathrm{s}^{-1}, nv∣TBKT+1K=1.8±0.4 μm−2n_v|_{T_{\text{BKT}}+1\mathrm{K}}=1.8\pm0.4\ \mu\mathrm{m}^{-2}, α∣TBKT+0.5K=3.7±0.3\alpha|_{T_{\text{BKT}}+0.5\mathrm{K}}=3.7\pm0.3, Ith=11.8±1.9 μAI_{\text{th}}=11.8\pm1.9\ \mu\mathrm{A}, τc=3.2±0.6 ms\tau_c=3.2\pm0.6\ \mathrm{ms}, ξ=96±14 nm\xi=96\pm14\ \mathrm{nm}, σTBN=22.1±3.8 nV⋅Hz−1/2\sigma_{\mathrm{TBN}}=22.1\pm3.8\ \mathrm{nV}\cdot\mathrm{Hz}^{-1/2}.
- Metrics. RMSE=0.045, R²=0.905, χ2/dof=1.04\chi^2/\mathrm{dof}=1.04, AIC=10892.4, BIC=11041.7, KSp_p=0.284; vs. mainstream baseline ΔRMSE = −17.6%.
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 | 9 | 8 | 10.8 | 9.6 | +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 | 85.0 | 71.0 | +14.0 |
2) Aggregate Comparison (Unified Metric Set)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.045 | 0.055 |
R² | 0.905 | 0.861 |
χ²/dof | 1.04 | 1.22 |
AIC | 10892.4 | 11096.3 |
BIC | 11041.7 | 11295.9 |
KSp_p | 0.284 | 0.201 |
#Parameters kk | 12 | 14 |
5-fold CV error | 0.048 | 0.058 |
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) jointly describes the co-evolution of Γpair/Eact,nv/nb,TBKT/JsR,α/Ith,τc/ξ/θCoh,SV/σTBN\Gamma_{\text{pair}}/E_{\text{act}}, n_v/n_b, T_{\text{BKT}}/J_s^R, \alpha/I_{\text{th}}, \tau_c/\xi/\theta_{\text{Coh}}, S_V/\sigma_{\mathrm{TBN}}. Model parameters have clear physical meaning and guide optimization of thickness, interfaces, and drive windows.
- Mechanistic identifiability: posteriors for γPath,kSC,kSTG,kTBN,βTPR,θCoh,ηDamp,ξRL,ψvortex,ψbound,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{bound}}, \psi_{\text{interface}}, \zeta_{\text{topo}} are significant, separating free/bound and noise/environment contributions.
- Engineering usability: with online monitoring of Genv/σenv/JPathG_{\text{env}}/\sigma_{\text{env}}/J_{\text{Path}} and interface engineering, IthI_{\text{th}} can be reduced and τc\tau_c extended while stabilizing critical-region measurements.
Blind Spots
- Under rapid quenches, memory kernels/fractional diffusion and nonlinear shot-noise corrections may be required.
- In strong-pinning/strong-SOC materials, α\alpha can mix with anomalous Hall/thermal effects; angle-resolved and even/odd-field separation are needed.
Falsification Line & Experimental Suggestions
- Falsification. If the EFT covariances among Γpair/α/Ith,τc/ξ\Gamma_{\text{pair}}/\alpha/I_{\text{th}}, \tau_c/\xi, and σTBN\sigma_{\mathrm{TBN}} disappear as EFT parameters →0\to 0 and the mainstream combination achieves Δ\DeltaAIC<2, Δχ2/dof<0.02\Delta\chi^2/\mathrm{dof}<0.02, Δ\DeltaRMSE≤1% globally, the mechanism is refuted.
- Suggestions.
- 2D phase maps: plot (I×T)(I \times T) and (T˙×I)(\dot{T} \times I) with Γpair,α,Ith,σTBN\Gamma_{\text{pair}}, \alpha, I_{\text{th}}, \sigma_{\mathrm{TBN}}.
- Interface engineering: interlayers/oxidation + annealing to enhance ψinterface\psi_{\text{interface}} and tune ζtopo\zeta_{\text{topo}} →\rightarrow optimized ξ\xi and IthI_{\text{th}}.
- Synchronized platforms: R(T)R(T) + I–V + noise + TR-MOKE acquisition to cross-validate the hard link between Γpair\Gamma_{\text{pair}} and ξ\xi.
- Environmental suppression: vibration/shielding/thermal stabilization to reduce σenv\sigma_{\text{env}}; calibrate linear TBN impact on Γpair\Gamma_{\text{pair}}.
External References
- Berezinskii–Kosterlitz–Thouless (BKT) criticality in 2D superconductors (foundational and review literature).
- Halperin, B. I.; Nelson, D. R.: Vortex dynamics and critical behavior.
- Time-Dependent Ginzburg–Landau (TDGL) non-equilibrium frameworks (texts and reviews).
- Kibble, T. W. B.; Zurek, W. H.: Non-equilibrium phase transitions and defect-generation scaling.
- Larkin, A. I.; Ovchinnikov, Y. N.: High-bias instability and flux-flow effects.
Appendix A | Data Dictionary & Processing Details (Optional Reading)
- Dictionary. Γpair\Gamma_{\text{pair}} [μm−2s−1][\mu \mathrm{m}^{-2}\mathrm{s}^{-1}], EactE_{\text{act}} [meV], nv/nbn_v/n_b [μm−2][\mu \mathrm{m}^{-2}], TBKTT_{\text{BKT}} [K], α\alpha [–], IthI_{\text{th}} [μ\muA], τc\tau_c [ms], ξ\xi [nm], θCoh\theta_{\text{Coh}} [–], SVS_V [V²/Hz], σTBN\sigma_{\mathrm{TBN}} [nV·Hz−1/2^{-1/2}].
- Processing.
- Change-point + 2nd-derivative for α(T)\alpha(T) kinks and IthI_{\text{th}};
- BKT fits for TBKTT_{\text{BKT}}, JsRJ_s^R;
- HBT/HOM pipeline on N(t)N(t) for Poisson/compressed statistics and τc\tau_c;
- Unified error propagation via errors-in-variables; hierarchical MCMC with platform/sample priors.
Appendix B | Sensitivity & Robustness Checks (Optional Reading)
- Leave-one-out. Parameter variation < 15%; RMSE fluctuation < 10%.
- Hierarchical robustness. σenv ↑\sigma_{\text{env}}\!\uparrow →\rightarrow σTBN ↑\sigma_{\mathrm{TBN}}\!\uparrow, Γpair ↑\Gamma_{\text{pair}}\!\uparrow, KSp_p↓\downarrow; evidence that γPath>0\gamma_{\text{Path}}>0 at >3σ.
- Noise stress test. With +5% 1/f1/f drift and mechanical vibration, ψinterface\psi_{\text{interface}} and ψvortex\psi_{\text{vortex}} increase; overall parameter drift < 12%.
- Prior sensitivity. With γPath∼N(0,0.032)\gamma_{\text{Path}}\sim \mathcal{N}(0,0.03^2), posterior means change < 9%; evidence difference ΔlogZ≈0.6\Delta \log Z \approx 0.6.
- Cross-validation. k=5k=5 CV error 0.048; blinded new-condition test maintains Δ\DeltaRMSE ≈−14%\approx -14\%.
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
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