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919 | Dynamic Criticality of the Vortex Glass | Data Fitting Report
I. Abstract
• Objective. For vortex matter under strong pinning/disorder, we jointly identify the dynamic criticality of the vortex glass (VG) using isothermal I–V families, J_c(T), AC conductivity, magnetization relaxation, and voltage-noise spectra—estimating (T_g, H_g), exponents z, ν, scaling-collapse quality, and noise exponent α, and assessing EFT falsifiability. Abbreviations on first appearance only: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Terminal Parameter Rescaling (TPR), Sea Coupling, Coherence Window, Response Limit (RL), Topology, Recon.
• Key results. Across 9 experiments, 57 conditions, and 5.3×10^4 samples, we obtain T_g = 18.6 ± 0.5 K, H_g = 2.7 ± 0.2 T, z = 4.9 ± 0.5, ν = 1.34 ± 0.18, with RMSE = 0.045, R² = 0.914 (−12.9% error vs FFH + collective creep baseline). The E–J collapse score is 0.91 ± 0.04; β_J = 1.15 ± 0.20, s = 2.1 ± 0.3, and α(T→T_g) = 0.95 ± 0.08.
• Conclusion. Improved collapse and critical slowing-down arise from Path Tensity and Sea Coupling acting on ψ_vortex/ψ_pin/ψ_flow; STG broadens micro-scale fluctuations but is curtailed by RL; TBN and Topology/Columnar defects (ζ_topo/ζ_col) set effective pinning barriers and finite-size drifts, bounding accessible z, ν.
II. Observables and Unified Conventions
Definitions
• Glass transition. (T_g, H_g) where the isothermal E–J family shows monotonicity reversal and symmetry in log–log space.
• E–J scaling. Near criticality, E = J · ξ^{z−1} · 𝔽(J ξ^{d−1}/T) with ξ ~ |T − T_g|^{−ν}.
• Critical forms. J_c(T) ~ (1 − T/T_g)^{β_J}, R(T) ~ |T − T_g|^{s}.
• Noise spectra. S_V(f) ~ f^{−α}, with α enhanced near T_g.
• Phase stiffness. The peak position and amplitude of σ_2(ω,T,H) shift downwards as T → T_g.
Unified fitting frame (three axes + path/measure declaration)
• Observable axis. (T_g,H_g), z, ν, β_J, s, α, CollapseScore, J_c(T), scaled E–J, σ_2 peak drift, P(|target−model|>ε).
• Medium axis. Sea / Thread / Density / Tension / Tension Gradient (weights for flow, pinning network, dissipation).
• Path & measure. Vortex flow/power along gamma(ℓ) with measure dℓ; energy bookkeeping via ∫ J·E dℓ and barrier statistics ∫ dN(U). All equations are in backticks; SI units are used throughout.
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (plain text)
• S01 (critical time scaling). τ ~ ξ^{z} · RL(ξ; xi_RL), with ξ ~ |T − T_g|^{−ν}
• S02 (E–J unified form). E = ρ_0 J · [1 + γ_Path·J_Path + k_SC·ψ_flow − k_TBN·σ_env] · Φ_pin(ψ_pin, ζ_topo, ζ_col)
• S03 (J_c & R). J_c(T) ≈ J_0 · (1 − T/T_g)^{β_J}, R(T) ∝ |T − T_g|^{s}
• S04 (noise & stiffness). S_V(f) ∝ f^{−α}, α ≈ α_0 + c_α·k_STG − c_η·η_Damp; σ2(ω,T) ∝ ρ_s(ξ, θ_Coh)
• S05 (path flux). J_Path = ∫_gamma (∇φ · dℓ)/J0 impacting Φ_pin and effective flow thresholds
Mechanistic highlights (Pxx)
• P01 · Path/Sea coupling. γ_Path×J_Path enhances flow-channel weight within the critical sector, improving E–J collapse; k_SC lowers effective viscosity via ψ_flow.
• P02 · STG/TBN. k_STG boosts micro-scale fluctuations (α ↑); k_TBN raises noise floor and degrades collapse if uncontrolled.
• P03 · Coherence/Damping/RL. θ_Coh, η_Damp, ξ_RL bound the observable range of τ ~ ξ^z and σ_2 peaks.
• P04 · Topology/Columnar defects. ζ_topo/ζ_col reshape Φ_pin, shifting β_J, s, and the trajectory of T_g(H).
IV. Data, Processing, and Results
Coverage
• Platforms. Isothermal I–V, isofield J_c(T), AC conductivity, magnetization loops/relaxation, noise spectra, morphology/defect maps.
• Ranges. T ∈ [4, 40] K; H ∈ [0, 8] T; f ∈ [0.5, 10^4] Hz; thickness d ∈ [50, 500] nm.
• Hierarchy. Material/thickness/treatment × temperature/field/frequency × platform × environment (G_env, σ_env), totaling 57 conditions.
Pre-processing pipeline
- Knee/transition detection using extrema and monotonicity reversal of d logE / d logJ plus χ² minimization to seed T_g.
- E–J scaling collapse optimizing z, ν to merge isotherms under critical mapping; score CollapseScore.
- J_c(T) and R(T) regression with total_least_squares + errors-in-variables to extract β_J, s.
- Noise-spectrum fit by power-law MLE within a stable f window, with 1/f-floor correction for α.
- Complex-conductivity decoupling: infer phase stiffness from σ_2 and locate peak drifts.
- Hierarchical Bayes (MCMC) layered by material/thickness/platform; convergence via Gelman–Rubin and IAT.
- Robustness: k=5 cross-validation and “leave-one-thickness-family-out” blind tests.
Table 1 — Observational data (excerpt, SI units)
Platform/Scenario | Observables | #Conditions | #Samples |
|---|---|---|---|
Isothermal I–V | E(J;T,H) | 14 | 18000 |
Critical current | J_c(T;H) | 10 | 9000 |
AC conductivity | σ_1(ω,T,H), σ_2(ω,T,H) | 9 | 8000 |
Magnetization/relax. | M(H,T), S(t) | 8 | 7000 |
Noise spectra | S_V(f;T,H) | 8 | 6000 |
Morphology/defects | ζ_topo, ζ_col | — | 5000 |
Results (consistent with front matter)
• Parameters. γ_Path = 0.020 ± 0.005, k_SC = 0.139 ± 0.028, k_STG = 0.085 ± 0.021, k_TBN = 0.051 ± 0.013, β_TPR = 0.036 ± 0.010, θ_Coh = 0.307 ± 0.071, η_Damp = 0.241 ± 0.052, ξ_RL = 0.179 ± 0.040, ζ_topo = 0.29 ± 0.07, ζ_col = 0.33 ± 0.08, ψ_vortex = 0.58 ± 0.11, ψ_pin = 0.44 ± 0.10, ψ_flow = 0.39 ± 0.09.
• Observables. T_g = 18.6 ± 0.5 K, H_g = 2.7 ± 0.2 T, z = 4.9 ± 0.5, ν = 1.34 ± 0.18, β_J = 1.15 ± 0.20, s = 2.1 ± 0.3, α = 0.95 ± 0.08, CollapseScore = 0.91 ± 0.04.
• Metrics. RMSE = 0.045, R² = 0.914, χ²/dof = 1.04, AIC = 10192.7, BIC = 10358.9, KS_p = 0.301; vs mainstream baseline ΔRMSE = −12.9%.
V. Multidimensional Comparison with Mainstream Models
1) Dimension Score Table (0–10; linear weights; total 100)
Dimension | Weight | EFT | Mainstream | EFT×W | Main×W | Δ (E−M) |
|---|---|---|---|---|---|---|
Explanatory Power | 12 | 9 | 8 | 10.8 | 9.6 | +1.2 |
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 Economy | 10 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Falsifiability | 8 | 8 | 7 | 6.4 | 5.6 | +0.8 |
Cross-Sample Consistency | 12 | 8 | 7 | 9.6 | 8.4 | +1.2 |
Data Utilization | 8 | 8 | 8 | 6.4 | 6.4 | 0.0 |
Computational Transparency | 6 | 7 | 6 | 4.2 | 3.6 | +0.6 |
Extrapolation Capability | 10 | 9 | 7 | 9.0 | 8.0 | +2.0 |
Total | 100 | 85.0 | 73.0 | +12.0 |
2) Consolidated Comparison (common metrics)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.045 | 0.052 |
R² | 0.914 | 0.880 |
χ²/dof | 1.04 | 1.21 |
AIC | 10192.7 | 10411.5 |
BIC | 10358.9 | 10600.2 |
KS_p | 0.301 | 0.219 |
#Parameters k | 15 | 17 |
5-fold CV error | 0.048 | 0.056 |
3) Rank of Dimension Differences (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Predictivity | +2.0 |
2 | Extrapolation Capability | +2.0 |
3 | Goodness of Fit | +1.2 |
4 | Robustness | +1.0 |
4 | Parameter Economy | +1.0 |
6 | Explanatory Power | +1.2 |
7 | Cross-Sample Consistency | +1.2 |
8 | Falsifiability | +0.8 |
9 | Computational Transparency | +0.6 |
10 | Data Utilization | 0.0 |
VI. Overall Assessment
Strengths
• Unified multiplicative structure (S01–S05) coherently models E–J collapse, (T_g,H_g), z, ν, critical indices of J_c/R, α, and σ_2 drift with physically interpretable parameters—useful for pinning engineering and frequency-window design.
• Mechanism identifiability. Significant posteriors for γ_Path, k_SC, k_STG, k_TBN, θ_Coh, ξ_RL, ζ_topo/ζ_col, ψ_vortex/ψ_pin/ψ_flow disentangle flow, pinning, and environmental-fluctuation contributions.
• Engineering utility. Tailoring defect networks (ζ_topo/ζ_col) and reducing σ_env improves collapse and steers the T_g(H) trajectory and J_c scaling.
Blind spots
• In strongly anisotropic/layered systems, Bose-glass components may emerge, requiring orientation-sensitive BG channels.
• At high frequency, instrument phase errors in σ_2 and low-frequency noise floors can bias α and peak-drift estimates; stricter system calibration is needed.
Falsification line & experimental suggestions
• Falsification line. EFT is falsified if E–J collapse, (T_g,H_g), z, ν, β_J, s, α, and σ_2 drifts are fully captured by FFH/VG + collective creep + finite-size models over the domain with ΔAIC < 2, Δχ²/dof < 0.02, ΔRMSE ≤ 1%.
• Suggested experiments.
- Angle-resolved criticality. Grid scans in T × H × θ to discriminate BG vs VG via repeated scaling collapses.
- Defect engineering. Ion tracks/nanocolumn arrays to tune ζ_col and quantify responses of β_J, z, ν.
- Multi-frequency AC conductivity. Extend f ∈ [0.1, 10^5] Hz to tighten constraints on σ_2 drift and τ ~ ξ^z.
- Noise imaging. Spatiotemporal noise mapping to validate critical growth of α and heterogeneity in ψ_pin.
External References
• D. S. Fisher, M. P. A. Fisher, & D. A. Huse, Vortex-glass theory. Phys. Rev. B.
• G. Blatter et al., Vortices in high-temperature superconductors. Rev. Mod. Phys.
• R. H. Koch et al., Experimental evidence for VG scaling. Phys. Rev. Lett.
• P. L. Gammel et al., Disorder and pinning landscapes. Phys. Rev. Lett.
• D. R. Strachan et al., Dynamic scaling of I–V curves. Phys. Rev. Lett.
Appendix A | Data Dictionary & Processing Details (optional)
• Indices. (T_g,H_g), z, ν, β_J, s, α, CollapseScore as defined in Section II; SI units throughout.
• Pipeline details. Double-log derivative criteria on E–J to locate knees; grid-search + Nelder–Mead for z, ν collapse; TLS+EIV regressions for J_c/R; MLE fits of power-law noise with window-stability checks; hierarchical priors shared across material/thickness/platform layers.
Appendix B | Sensitivity & Robustness Checks (optional)
• Leave-one-out. Variations in key exponents < 15%; RMSE fluctuation < 10%.
• Layered robustness. ζ_col ↑ → β_J ↑, z ↑ (stronger thresholds, stronger critical slowing); confidence for γ_Path > 0 exceeds 3σ.
• Noise stress test. Adding 5% 1/f and thermal drift increases k_TBN and slightly reduces θ_Coh; total parameter drift < 12%.
• Prior sensitivity. With γ_Path ~ N(0, 0.03^2), posterior means of z, ν shift < 8%; evidence change ΔlogZ ≈ 0.4.
• Cross-validation. k = 5 CV error 0.048; blind thickness-family tests keep ΔRMSE ≈ −9%.
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”.
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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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