35-EFT.WP.EDX.OrientedTension v1.0 | Chapter 7 Energy Exchange & Power Partition (EDX)

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Chapter 7 Energy Exchange & Power Partition (EDX)

I. Abstract & Scope
This chapter provides the minimal closure for oriented-energy and power accounting (EDX). With the orientation energy density W_orient(Q_ij, ∇Q_ij, …) as the state variable, we establish the control-volume energy balance, the partition of coupling power terms, and an identifiable split of transport/dissipation terms. We also specify procedures to estimate power terms and to use dominance masks (in energy/frequency) for partitioned accounting. Symbols are in English notation with backticks; SI units apply. No ToA terms appear here.

II. Dependencies & References

• Orientation geometry & distributions: Chapter 3 S80-1/2.
• Axioms & minimal equations: Chapter 4 P80-2/3/9/10, S80-3/4.
• Metrology & inversion: Chapter 5 M80-1…4 (Q_ij, T_fil_ij, tau_relax, D_Q).
• Couplings & media: Chapter 6 S80-5/6 (W_cpl and coupling kernels).
• Numerics & implementation: Chapter 10 (SimStack-OT), Chapter 12 (I80-*).

III. Normative Anchors (added in this chapter, S80-/M80-)

• S80-7 (Control-Volume Energy Balance):
  d/dt ∫_V W_orient dV = P_in − P_diss − P_trans,
  where P_in is external/boundary input power (including coupling work), P_diss ≥ 0 is dissipation, and P_trans = ∮_{∂V} Φ_E · n_hat dA is outward energy flux. Density form:
  ∂_t W_orient + ∇·Φ_E = 𝒫_in − 𝒫_diss.
• S80-8 (Coupling-Power Partition & Consistency):
  𝒫_in = 𝒫_cpl + 𝒫_mech + 𝒫_src,
  with 𝒫_cpl = − ∂W_cpl/∂t (see Chapter 6), 𝒫_mech = T_fil_ij D_{ij} (D_{ij}=(∂_i u_j+∂_j u_i)/2), and 𝒫_src external sources. All must be consistent with S80-3/4 and S80-5/6.
• M80-19 (Power-Term Estimation Flow): estimate {𝒫_in, 𝒫_cpl, 𝒫_mech, 𝒫_diss, Φ_E} from metrology outputs and coupling kernels, then perform closure audit.
• M80-20 (Energy/Frequency-Band Partition): use Chapter 6 dominance masks to accumulate 𝒫_in and W_orient by bands and form a segmented ledger.
• M80-21 (Consistency & Evidence Checks): verify closure of S80-7/8 and apply positive/negative criteria over experimental/simulation windows.

IV. Body Structure

I. Background & Problem Statement

• The energetics of oriented systems depend on order-tensor evolution and oriented tension, as well as coupling channels and transport mechanisms. A conservation law consistent with Chapters 4–6 is required to place W_orient, T_fil_ij, coupling power, and flux in a single ledger.
• Objective: provide measurable, allocable, and testable power expressions and workflows that support parameter identification, model comparison, and engineering prediction.

II. Key Equations & Derivations (S-series)

1. S80-7 (Volume/Density forms):
   • Volume: d/dt ∫_V W_orient dV = ∫_V 𝒫_in dV − ∫_V 𝒫_diss dV − ∮_{∂V} Φ_E · n_hat dA.
   • Density: ∂_t W_orient + ∇·Φ_E = 𝒫_in − 𝒫_diss. A minimal closure for the flux is Φ_E = − K_E ∇W_orient + Φ_cpl with K_E ≥ 0 and a coupling-induced term Φ_cpl.
2. S80-8 (Power-term partition):
   • Mechanical work: 𝒫_mech = T_fil_ij D_{ij}, consistent with T_fil_ij from Chapter 4.
   • Coupling work: 𝒫_cpl = − ∂W_cpl/∂t = − ( ∂W_cpl/∂Q_ij ) ∂_t Q_ij − ( ∂W_cpl/∂field ) · ∂_t field.
   • Dissipation & positivity (quadratic approximation): with W_orient ≈ (1/2) A Q_ij Q_ij + (1/2) K ∂_k Q_ij ∂_k Q_ij (A≥0, K≥0),
     𝒫_diss = (1/τ_relax) A Q_ij Q_ij + D_Q K (∂_k Q_ij)(∂_k Q_ij) ≥ 0.

III. Methods & Flows (M-series)

1. M80-19 Power-Term Estimation
   • Inputs: {Q_ij(t,r), T_fil_ij(t,r), u_vec, fields} with metrology covariances.
   • Estimation: compute 𝒫_mech, 𝒫_cpl via S80-8; fit parameters for 𝒫_diss and Φ_E (e.g., K_E) in spectral/time domains.
   • Closure audit: test ∂_t W_orient + ∇·Φ_E − (𝒫_in − 𝒫_diss) ≈ 0 residuals and CIs.
2. M80-20 Band Allocation
   • Masks: apply η_dom(ê,ω [or E]) from Chapter 6 to obtain m(ê,ω).
   • Allocation: masked integration/windowing of 𝒫_in and W_orient to yield a {Band_k} ledger.
   • Consistency: band sums must agree with the global totals within uncertainty.
3. M80-21 Consistency & Evidence
   • Model sets: {with-coupling, decoupled, no-diffusion}; compare evidence and closure residuals.
   • Robustness: perturb volume/time/frequency windows and verify ledger stability and fidelity.

IV. Cross-References within/beyond this Volume

• Chapter 4: constitutive/dynamics inputs T_fil_ij, Q_ij for 𝒫_mech/𝒫_cpl.
• Chapter 5: tau_relax, D_Q, and tension calibration feed 𝒫_diss and Φ_E estimation.
• Chapter 6: W_cpl and kernels determine the form/sign of 𝒫_cpl.
• Chapters 10/12: power kernels, ledger accumulation, and I80-* exports in SimStack-OT.
• Companion: energy conservation & power partition chapters in Energy Filaments.

V. Validation, Criteria & Counterexamples

1. Positive criteria:
   • Disabling couplings/diffusion (e.g., χ_*→0, D_Q→0) worsens evidence and closure residuals.
   • 𝒫_mech aligns with loading history; 𝒫_cpl sign/magnitude varies consistently with Q_ij and external fields.
   • 𝒫_diss ≥ 0 and increases with τ_relax^{-1} and D_Q; Φ_E direction aligns with −∇W_orient (modulo Φ_cpl).
2. Negative criteria:
   • Removing key terms maintains or improves evidence, or closure residuals unchanged (mechanism falsified/nonessential).
   • Band-sum vs global totals disagree beyond CIs.
   • 𝒫_cpl inconsistent with calibrated data in units/dimensions or sign.
3. Contrasts:
   • Evidence & closure residuals among {full, decoupled, no-diffusion}.
   • Power shares for {mechanical-only, EM-only, mechanical+EM}.
   • {isotropic flux, anisotropic flux} impacts on Φ_E.

VI. Deliverables & Figure List

1. Deliverables:
   • EDXLedger.npz (energy ledger over time/frequency/energy with uncertainties).
   • PowerTerms.nc (spatiotemporal spectra of 𝒫_in, 𝒫_mech, 𝒫_cpl, 𝒫_diss, Φ_E).
   • ClosureReport.md (closure residuals and evidence-ratio assessment).
   • DominanceMasks.npz (consistent with Chapter 6).
2. Figures/Tables (suggested):
   • Tab. 7-1 Definitions, units, and dimensional audits of power terms.
   • Fig. 7-1 Distributions of closure residuals vs time/frequency.
   • Tab. 7-2 Power partition per energy/frequency band with CIs.
   • Fig. 7-2 Evidence/residual comparisons for {with-coupling, decoupled}.
   • Tab. 7-3 Correlations of 𝒫_diss with {tau_relax, D_Q}.