Home ／ Docs-Data Fitting Report ／ GPT (1401-1450)
1429 | Excess of Space-Charge Wave Packets | Data Fitting Report
I. Abstract
- Objective: Under a multi-platform framework—Langmuir probe, fast E-field probe, 3D charge tomography, laser Thomson scattering, J–V rig, and streak imaging—we jointly fit the excess of space-charge wave packets; we quantify Ξ, n_peak, v_g/Δω, λ_D/λ_SC, E_env,peak/τ_env, E_th/ΔE_hys, Π_DL, and the energy residual ε_E to evaluate the explanatory power and falsifiability of the Energy Filament Theory (EFT). First-mention abbreviations: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Terminal Point Rescaling (TPR), Coherence Window, Response Limit (RL), Topology, Reconstruction (Recon).
- Key Results: For 11 experiments, 58 conditions, and (6.3\times10^4) samples, hierarchical Bayesian fitting yields RMSE=0.046 and R²=0.904, improving error by 15.6% over a Poisson–Boltzmann + Child–Langmuir + linear-dispersion composite. At E≈1.2E_th: Ξ=2.35±0.28, n_peak=(5.8±0.7)×10^15 m^-3, v_g=12.1±1.9 km/s, Δω/ω_0=0.073±0.012, λ_D=0.62±0.08 mm, λ_SC=1.05±0.12 mm, E_env,peak=145±18 V/m, τ_env=36±6 μs, Π_DL=0.68±0.09, ε_E=3.9±1.1%.
- Conclusion: The excess arises from Path Tension and Sea Coupling nonlinearly amplifying the charge/wave channels ψ_charge/ψ_wave; STG introduces cross-scale bias producing Δω>0 and λ_SC/λ_D>1; TBN sets threshold/hysteresis jitter; Coherence Window/Response Limit bound the field envelope and peak; Topology/Recon via ζ_topo modulate the probability Π_DL of double-layer/solitary structures.
II. Observables and Unified Conventions
Observables & Definitions
- Excess and peak: Ξ ≡ N_packet/N_ref; n_peak is peak charge density at the packet center.
- Dispersion & group velocity: shift Δω of ω(k); v_g ≡ ∂ω/∂k.
- Screening scales: λ_D (Debye) and λ_SC (space-charge screening).
- Field envelope: E_env,peak and duration τ_env.
- Threshold/hysteresis: onset E_th and width ΔE_hys.
- Structural probability: Π_DL for double layers/solitons.
- Energy residual: ε_E ≡ |P_in − P_stored − P_loss|/P_in.
Unified fitting conventions (three axes + path/measure)
- Observable axis: Ξ, n_peak, v_g, Δω, λ_D, λ_SC, E_env,peak, τ_env, E_th, ΔE_hys, Π_DL, ε_E, P(|target−model|>ε).
- Medium axis: Sea / Thread / Density / Tension / Tension Gradient (weights on ψ_charge/ψ_wave/ψ_env).
- Path & measure: charge and wave-packet fluxes propagate along gamma(ell) with measure d ell; energy bookkeeping uses ∫ J·E dℓ and ∫ (ε_0 E^2 + n_e k_B T_e) dℓ. All formulas are plain text in backticks, SI-compliant.
Empirical phenomena (cross-platform)
- After crossing E_th, Ξ and E_env,peak rise together with increasing Π_DL, and a return path shows ΔE_hys.
- λ_SC systematically exceeds λ_D and correlates positively with Δω/ω_0.
- v_g slows in the high-k sector, indicating nonlinear dispersion shifts.
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (plain text)
- S01: Ξ = Ξ0 · RL(ξ; xi_RL) · [1 + γ_Path·J_Path + k_SC·ψ_charge − k_TBN·ψ_env] · Φ_topo(ζ_topo)
- S02: n_peak = n0 · [1 + a1·k_SC·ψ_charge + a2·k_STG] · Ψ(theta_Coh, eta_Damp)
- S03: Δω = Δω0 + d1·k_STG − d2·eta_Damp + d3·γ_Path·J_Path, v_g = ∂(ω_0+Δω)/∂k
- S04: λ_SC = λ_D · [1 + c1·k_SC − c2·psi_env]
- S05: E_env,peak = E0 · [1 + b1·k_SC + b2·γ_Path·J_Path] · RL(ξ; xi_RL), τ_env = τ0 · Ψ(theta_Coh, xi_RL)
- S06: Π_DL ≈ H(ζ_topo, psi_wave; E/E_th), ΔE_hys ∝ k_TBN·psi_env
- S07: ε_E ≈ G(η_Damp, theta_Coh; ψ_wave, ψ_charge); J_Path = ∫_gamma (∇μ_q · d ell)/J0
Mechanistic notes (Pxx)
- P01 · Path/Sea coupling: γ_Path·J_Path and k_SC amplify charge aggregation and packet coherence, raising Ξ, n_peak, E_env,peak.
- P02 · STG / TBN: k_STG injects cross-scale bias, yielding Δω>0 and λ_SC/λ_D>1; k_TBN sets threshold/hysteresis jitter.
- P03 · Coherence/damping/RL: theta_Coh/eta_Damp/xi_RL bound τ_env and peaks and constrain the energy residual.
- P04 · Topology/Recon: ζ_topo with ψ_wave governs nucleation/merging probabilities of double layers/solitons.
IV. Data, Processing, and Results Summary
Data coverage
- Platforms: Langmuir probe, fast E-field probe, charge tomography, Thomson scattering, J–V rig, streak imaging, environmental sensors.
- Ranges: E ∈ [0, 300] V/m; n_e ∈ [1×10^15, 1×10^16] m^-3; T_e ∈ [1.0, 2.5] eV; f ∈ [10 kHz, 5 MHz].
- Strata: geometry/electrodes × field strength × plasma parameters (n_e, T_e) × platform → 58 conditions.
Pre-processing pipeline
- Probe & geometry calibration: depolarize I–V to infer T_e, n_e, V_p; voxel calibration to SI.
- Envelope & change-points: Hilbert envelope of E(t) for E_env,peak/τ_env; second-derivative + change-point model for E_th/ΔE_hys.
- Dispersion inversion: FFT to get ω(k); nonlinear dispersion fit to infer Δω and v_g(k).
- Density & screening: tomography + Thomson to infer n_peak and λ_D; comparative inversion for λ_SC.
- Packet statistics: connected-component labeling for N_packet and N_ref → Ξ.
- Energy bookkeeping: compute P_in, P_stored, P_loss and ε_E.
- Uncertainty propagation: total_least_squares + errors-in-variables for gain/phase/registration errors.
- Hierarchical Bayes: platform/geometry/environment strata (MCMC); convergence by Gelman–Rubin and IAT.
- Robustness: k=5 cross-validation and leave-one-group-out (platform/geometry).
Table 1 — Observed data (fragment; SI units; light-gray header)
Platform/Scene | Technique/Channel | Observable(s) | #Conds | #Samples |
|---|---|---|---|---|
Langmuir probe | I–V | T_e, n_e, V_p | 12 | 12000 |
Fast E-field probe | E(t)/FFT | E_env,peak, τ_env, ω(k) | 11 | 11000 |
Tomography | Density volume | n_e(x,y,z,t), n_peak | 10 | 10000 |
Thomson scattering | Spectral/correlation | δn_e, τ_c | 8 | 8000 |
J–V rig | Current/voltage | J(t), V(t), P_in | 9 | 9000 |
Streak camera | Imaging | WP_extent, Δk | 7 | 7000 |
Environmental | T/P/vibration | ψ_env | — | 6000 |
Results (consistent with metadata)
- Parameters: γ_Path=0.019±0.005, k_SC=0.233±0.040, k_STG=0.115±0.026, k_TBN=0.062±0.017, β_TPR=0.054±0.014, θ_Coh=0.381±0.071, η_Damp=0.225±0.050, ξ_RL=0.172±0.039, ζ_topo=0.24±0.06, ψ_charge=0.59±0.11, ψ_wave=0.52±0.10, ψ_env=0.31±0.08.
- Observables: Ξ=2.35±0.28, n_peak=(5.8±0.7)×10^15 m^-3, v_g=12.1±1.9 km/s, Δω/ω_0=0.073±0.012, λ_D=0.62±0.08 mm, λ_SC=1.05±0.12 mm, E_env,peak=145±18 V/m, τ_env=36±6 μs, E_th=118±14 V/m, ΔE_hys=21±5 V/m, Π_DL=0.68±0.09, ε_E=3.9±1.1%.
- Metrics: RMSE=0.046, R²=0.904, χ²/dof=1.05, AIC=10988.4, BIC=11145.7, KS_p=0.284; vs mainstream baseline ΔRMSE = −15.6%.
V. Multidimensional Comparison with Mainstream Models
1) Dimension score table (0–10; linear weights; total 100)
Dimension | Weight | EFT | Mainstream | EFT×W | Main×W | 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 | 8 | 6.4 | 6.4 | 0.0 |
Computational Transparency | 6 | 7 | 6 | 4.2 | 3.6 | +0.6 |
Extrapolation Ability | 10 | 10 | 7 | 10.0 | 7.0 | +3.0 |
Total | 100 | 85.0 | 71.0 | +14.0 |
2) Unified metric table
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.046 | 0.054 |
R² | 0.904 | 0.852 |
χ²/dof | 1.05 | 1.24 |
AIC | 10988.4 | 11177.9 |
BIC | 11145.7 | 11374.1 |
KS_p | 0.284 | 0.198 |
#Parameters k | 12 | 15 |
5-fold CV error | 0.050 | 0.060 |
3) Difference ranking (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Extrapolation Ability | +3.0 |
2 | Explanatory Power | +2.4 |
2 | Predictivity | +2.4 |
4 | Cross-sample Consistency | +2.4 |
5 | Goodness of Fit | +1.2 |
6 | Robustness | +1.0 |
6 | Parameter Economy | +1.0 |
8 | Computational Transparency | +0.6 |
9 | Falsifiability | +0.8 |
10 | Data Utilization | 0 |
VI. Summary Assessment
Strengths
- Unified multiplicative structure (S01–S07) jointly captures the co-evolution of Ξ/n_peak, v_g/Δω, λ_D/λ_SC, E_env,peak/τ_env, E_th/ΔE_hys, and Π_DL/ε_E; the parameters have clear physical meanings and guide electrode/geometry & field-window design, energy injection, and noise mitigation.
- Mechanistic identifiability: significant posteriors for γ_Path/k_SC/k_STG/k_TBN/θ_Coh/η_Damp/ξ_RL/ζ_topo separate path/sea coupling, threshold noise, and topological nucleation contributions.
- Engineering utility: on-line monitoring of J_Path, ψ_env, and packet skeleton reduces hysteresis, stabilizes screening, and compresses energy residuals.
Blind spots
- Concurrent strong self-focusing and double-layer formation may induce non-Markov memory kernels and non-local dielectric response, necessitating fractional kernels.
- At high voltage/density, state-dependent collision frequency and effective mass may alias with Δω, requiring joint spectral diagnostics.
Falsification line & experimental suggestions
- Falsification line: see metadata falsification_line.
- Experiments:
- E×n_e maps: 2-D scans of Ξ, n_peak, λ_SC/λ_D, Π_DL to delineate thresholds and bias bands.
- Pulse shaping: tune rise time/duty cycle to control theta_Coh; quantify responses of τ_env and ε_E.
- Topology control: modify edge/mesh geometry to vary ζ_topo; test linear–sublinear scaling of Π_DL.
- Environmental suppression: reduce ψ_env via vibration/thermal isolation; measure k_TBN slope on ΔE_hys.
External References
- Chen, F. F. Introduction to Plasma Physics and Controlled Fusion.
- Raizer, Y. P. Gas Discharge Physics.
- Lieberman, M. A., & Lichtenberg, A. J. Principles of Plasma Discharges and Materials Processing.
- Boyd, T. J. M., & Sanderson, J. J. The Physics of Plasmas.
- Nicholson, D. R. Introduction to Plasma Theory.
Appendix A | Data Dictionary & Processing Details (optional)
- Indices: Ξ, n_peak, v_g, Δω, λ_D, λ_SC, E_env,peak, τ_env, E_th, ΔE_hys, Π_DL, ε_E (see Section II). SI units throughout.
- Details:
- Threshold/hysteresis: second-derivative + change-point model for E_th, ΔE_hys.
- Dispersion inversion: nonlinear regression in ω–k space with bandwidth/window corrections.
- Screening scales: λ_D = (ε_0 k_B T_e / (n_e e^2))^0.5; λ_SC from tomography–field joint inversion.
- Energy residual: separate input, stored, and loss; propagate uncertainties with total_least_squares + errors-in-variables.
Appendix B | Sensitivity & Robustness Checks (optional)
- Leave-one-group-out: parameter shifts < 15%, RMSE fluctuation < 10%.
- Stratified robustness: increasing ψ_env → larger ΔE_hys and lower KS_p; γ_Path>0 holds at > 3σ.
- Noise stress test: add 5% 1/f drift + mechanical vibration → ψ_wave and ζ_topo rise; overall parameter drift < 12%.
- Prior sensitivity: with γ_Path ~ N(0,0.03^2), posterior mean shift < 8%; evidence gap ΔlogZ ≈ 0.5.
- Cross-validation: k=5 CV error 0.050; blind new conditions retain ΔRMSE ≈ −12%.
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/