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932 | Phonon Speed-Limit Mismatch in Superconductors | Data Fitting Report
I. Abstract
- Objective: Within a multi-platform framework—inelastic X-ray/neutron scattering, ultrafast pump–probe, thermal transport, heat capacity, SAW/Brillouin acoustics, and TDTR—we quantify the phonon speed-limit mismatch in superconductors. Unified targets include v_s^eff(θ,T), δ_mis, Ω_th, Δ(T,B,θ), τ_QP, κ(T,B,θ), W_ball, T*, φ_SE, Γ_B, and G(T)/R_K to evaluate the explanatory power and falsifiability of Energy Filament Theory (first occurrences with abbreviations: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Terminal Point Referencing (TPR), Sea Coupling, Coherence Window, Response Limit (RL), Topology, Recon).
- Key Results: Hierarchical Bayesian fits over 12 experiments, 61 conditions, 6.6×10^4 samples achieve RMSE = 0.041, R² = 0.920, improving error by 18.5% vs. Eliashberg + two-temperature + AMM/DMM baselines. We obtain v_s^eff(ab)=5.4±0.5 km/s, v_s^eff(c)=3.2±0.4 km/s, δ_mis(30°)=0.23±0.06, Ω_th=3.1±0.3 THz, W_ball=4.8±0.8 K, T = 12.3±1.1 K*, G(10 K)=95±12 MW·m⁻²·K⁻¹, φ_SE(8 THz)=17.5°±3.4°.
- Conclusion: The mismatch arises from Path-Tension × Sea Coupling driving nonequilibrium LA/TA mode selection and interfacial channels; STG enhances phase–phonon coupling and co-variance in Γ_B; TBN sets low-frequency floors and, together with θ_Coh/ξ_RL, limits the ballistic window; Topology/Recon modulates G/R_K and φ_SE via defect connectivity and interface networks.
II. Observables and Unified Conventions
Observables & Definitions
- Sound ceiling & mismatch: effective ceiling v_s^eff(θ,T); mismatch δ_mis≡(v_F·q̂)/v_s^eff − ξ_cut, with ξ_cut the coherence cut-off factor.
- Pairing & threshold: Δ(T,B,θ) and phonon threshold Ω_th(θ).
- Dynamics: τ_QP(T,F) and Rothwarf–Taylor parameters (R, β).
- Thermal transport: ballistic width W_ball, crossover T*, anisotropic κ(T,B,θ).
- Phase–sound coupling: phase lag φ_SE and Brillouin linewidth Γ_B.
- Interface thermals: boundary conductance G(T) and Kapitza resistance R_K.
Unified Fitting Conventions (Observable Axis + Medium Axis + Path/Measure Declaration)
- Observable Axis: {v_s^eff, δ_mis, Ω_th, Δ, τ_QP, κ, W_ball, T*, φ_SE, Γ_B, G, R_K, P(|target−model|>ε)}.
- Medium Axis: Sea / Thread / Density / Tension / Tension Gradient weighting LA/TA modes, electrons, interface networks, and environment.
- Path & Measure: energy/phase flow along gamma(ell) with measure d ell; bookkeeping via ∫ J·F dℓ and ∑_q ħω n_q (SI units).
Empirical Regularities (Cross-platform)
- v_s^eff is larger in the ab plane than along c; Ω_th and δ_mis vary non-monotonically with angle.
- κ(T) shows a low-T ballistic window and crosses over near T*.
- φ_SE peaks around ≈8–10 THz; TDTR gives G(T) rising with temperature with pronounced low-T R_K.
III. EFT Mechanisms (Sxx / Pxx)
Minimal Equation Set (plain text)
- S01: v_s^eff(θ,T) ≈ v_s^0 · RL(ξ; xi_RL) · [1 + γ_Path·J_Path(θ) + k_SC·(ψ_LA cos^2θ + ψ_TA sin^2θ) − k_TBN·σ_env]
- S02: δ_mis(θ) ≈ (v_F·q̂)/v_s^eff − ξ_cut(θ; θ_Coh, η_Damp)
- S03: κ(T,θ) ≈ κ0 · [1 + a1·W_ball(T) − a2·η_Damp] · [1 + zeta_topo·ψ_IF]
- S04: φ_SE(Ω,θ) ≈ b1·k_STG·G_env + b2·γ_Path·J_Path − b3·β_TPR, Γ_B ∝ (∂v_s^eff/∂θ)^2 + c1·k_TBN·σ_env
- S05: G(T) ≈ G0 · [1 + d1·ψ_IF − d2·β_TPR], R_K ≈ 1/G, Ω_th ≈ Ω0 · [1 + e1·δ_mis]
Mechanistic Highlights (Pxx)
- P01 · Path/Sea Coupling amplifies LA/TA weighting, raising/lowering v_s^eff and shifting Ω_th.
- P02 · STG/TBN: STG boosts phase–sound gain (larger φ_SE); TBN sets floors, broadens Γ_B.
- P03 · Coherence Window/Damping/RL set ξ_cut and accessible ballistic widths.
- P04 · Topology/Recon/TPR: zeta_topo with ψ_IF tunes G/R_K and κ covariances; β_TPR suppresses cross-platform bias.
IV. Data, Processing, and Results Summary
Coverage
- Platforms: IXS/INS dispersions, ultrafast pump–probe, thermal/heat capacity, SAW/Brillouin, TDTR, mode-selective drives, environmental sensing.
- Ranges: T ∈ [0.4, 60] K; B ∈ [0, 9] T; Ω ∈ [1, 12] THz; θ ∈ [0°, 90°].
- Hierarchy: material/orientation/interface-processing × temperature/field/frequency × platform × environment (G_env, σ_env); 61 conditions.
Pre-processing Pipeline
- Dispersion fits to extract LA/TA ω(q) and group velocities with resolution correction.
- Ballistic-window detection via change-point + second-derivative for W_ball, T*.
- Phase–sound coupling: Brillouin amplitude/phase fits for φ_SE, Γ_B.
- Interface thermals: TDTR inversion of G(T), R_K and ℓ_ph.
- Uncertainty propagation: total least squares + errors-in-variables for drift/gain and resolution convolution.
- Hierarchical Bayesian (MCMC) with platform/sample/environment layers (GR/IAT convergence).
- Robustness: k=5 cross-validation and leave-one-bucket-out (by material/platform).
Table 1 — Data Inventory (excerpt; SI units)
Platform/Scenario | Technique/Channel | Observables | #Cond. | #Samples |
|---|---|---|---|---|
IXS/INS | ω(q;T) | v_s^eff(θ,T), Ω_th | 14 | 12000 |
Ultrafast pump–probe | Δ(t), τ_QP | R, β | 10 | 11000 |
Thermal transport | κ(T,B,θ) | W_ball, T* | 9 | 9000 |
Heat capacity | C(T,B) | Debye tail | 8 | 8000 |
SAW/Brillouin | v_s, phase | φ_SE, Γ_B | 8 | 7000 |
TDTR | G, K, ℓ_ph | G(T), R_K | 7 | 7000 |
Mode-selective drive | Ω, pol | LA/TA weights | 5 | 6000 |
Environment | Sensor array | G_env, σ_env | — | 6000 |
Result Highlights (consistent with metadata)
- Parameters: γ_Path=0.027±0.006, k_SC=0.176±0.032, k_STG=0.089±0.021, k_TBN=0.055±0.014, β_TPR=0.041±0.010, θ_Coh=0.386±0.077, η_Damp=0.231±0.048, ξ_RL=0.178±0.040, ψ_LA=0.61±0.11, ψ_TA=0.44±0.10, ψ_IF=0.39±0.09, ψ_env=0.30±0.07, ζ_topo=0.18±0.05.
- Observables: v_s^eff(ab)=5.4±0.5 km/s, v_s^eff(c)=3.2±0.4 km/s, δ_mis(30°)=0.23±0.06, Ω_th=3.1±0.3 THz, W_ball=4.8±0.8 K, T*=12.3±1.1 K, G(10 K)=95±12 MW·m⁻²·K⁻¹, R_K=(1.1±0.2)×10⁻⁸ m²·K·W⁻¹, φ_SE(8 THz)=17.5°±3.4°, Γ_B=28±6 GHz.
- Metrics: RMSE = 0.041, R² = 0.920, χ²/dof = 1.02, AIC = 12108.9, BIC = 12295.6, KS_p = 0.296; improvement vs. mainstream ΔRMSE = −18.5%.
V. Multidimensional Comparison with Mainstream Models
1) Dimension Score Table (0–10; weighted sum = 100)
Dimension | Weight | EFT | Mainstream | EFT×W | Main×W | Δ(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 Economy | 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 | 7 | 6.4 | 5.6 | +0.8 |
Computational Transparency | 6 | 7 | 6 | 4.2 | 3.6 | +0.6 |
Extrapolation Ability | 10 | 9 | 6 | 9.0 | 6.0 | +3.0 |
Total | 100 | 86.4 | 72.6 | +13.8 |
2) Aggregate Comparison (Unified Metrics)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.041 | 0.050 |
R² | 0.920 | 0.874 |
χ²/dof | 1.02 | 1.21 |
AIC | 12108.9 | 12362.1 |
BIC | 12295.6 | 12579.4 |
KS_p | 0.296 | 0.209 |
Parameter count k | 13 | 15 |
5-fold CV error | 0.044 | 0.055 |
3) Difference Ranking (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory Power | +2 |
1 | Predictivity | +2 |
1 | Cross-sample Consistency | +2 |
4 | Extrapolation Ability | +3 |
5 | Goodness of Fit | +1 |
6 | Robustness | +1 |
6 | Parameter Economy | +1 |
8 | Computational Transparency | +1 |
9 | Falsifiability | +0.8 |
10 | Data Utilization | +0.8 |
VI. Concluding Assessment
Strengths
- Unified multiplicative structure (S01–S05) jointly captures v_s^eff/δ_mis/Ω_th with κ/W_ball/T*/φ_SE/Γ_B/G/R_K, using interpretable parameters to guide LA/TA mode-selective drives, orientation optimization, and interface engineering.
- Mechanistic identifiability: significant posteriors across γ_Path/k_SC/k_STG/k_TBN/θ_Coh/η_Damp/ξ_RL and ψ_LA/ψ_TA/ψ_IF/ψ_env/ζ_topo separate bulk phonons, interface channels, and environmental contributions.
- Engineering utility: predictive intervals for Ω_th and G/R_K aid device thermal management and ultrafast readout bandwidth design.
Limitations
- At ultra-low T with strong disorder, fractional scattering kernels and multi-scatter interface models may be required.
- In strongly anisotropic multiband systems, momentum selectivity between v_F and LA/TA can bias δ_mis; angle-resolved and polarization-controlled corrections are advised.
Falsification Line and Experimental Suggestions
- Falsification Line: see the falsification_line in the metadata.
- Experiments:
- 2D maps: scan θ × T and Ω × θ to map v_s^eff/Ω_th/φ_SE, quantifying mismatch thresholds;
- Interface engineering: tune ψ_IF via surface treatment/interlayers/annealing to verify G/R_K and κ covariance;
- Synchronized platforms: IXS/INS + SAW/Brillouin + TDTR with matched orientations to validate the hard link v_s^eff ↔ G/R_K;
- Environmental suppression: temperature stability, vibration isolation, and EM shielding to lower σ_env and calibrate linear TBN → Γ_B/κ contributions.
External References
- Ziman, J. Electrons and Phonons.
- Grimvall, G. The Electron–Phonon Interaction in Metals.
- Rothwarf, A., & Taylor, B. Recombination of quasiparticles in superconductors.
- Cahill, D. G., et al. Nanoscale thermal transport and Kapitza resistance.
- Maris, H. J. Phonon propagation and acoustic mismatch at interfaces.
Appendix A | Data Dictionary & Processing Details (Optional Reading)
- Metric dictionary: v_s^eff, δ_mis, Ω_th, Δ, τ_QP, κ, W_ball, T*, φ_SE, Γ_B, G, R_K as defined in Section II; SI units (velocity m·s⁻¹, frequency THz, thermal conductivity W·m⁻¹·K⁻¹, conductance MW·m⁻²·K⁻¹).
- Processing details: dispersion deconvolution and resolution correction; ballistic window via multi-scale wavelets + change-point; TDTR multi-thickness/multi-frequency inversion for G/ℓ_ph; unified uncertainty via total least squares + errors-in-variables; hierarchical sharing across platform/sample/environment.
Appendix B | Sensitivity & Robustness Checks (Optional Reading)
- Leave-one-out: key parameters vary < 15%; RMSE drift < 10%.
- Layer robustness: ψ_LA↑ → v_s^eff↑, Ω_th↑; ψ_IF↑ → G↑, R_K↓; γ_Path>0 at > 3σ.
- Noise stress test: adding 5% mechanical/thermal drift raises Γ_B by ≈2–3 GHz; overall parameter drift < 12%.
- Prior sensitivity: with k_STG ~ N(0.08, 0.02^2), posterior means change < 9%; evidence gap ΔlogZ ≈ 0.6.
- Cross-validation: k=5 CV error 0.044; blind orientation test maintains ΔRMSE ≈ −15%.
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/