42-EFT.WP.Heat.Decoherence v1.0 | Chapter 6: Heat Transport & Non-Equilibrium — Fourier / Cattaneo / BTE / Landauer

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Chapter 6: Heat Transport & Non-Equilibrium — Fourier / Cattaneo / BTE / Landauer

I. Objectives & Applicability

• Establish a macro–meso–quantum consistent description of heat transport and non-equilibrium: Fourier diffusion law (parabolic), Cattaneo–Vernotte (hyperbolic, finite propagation speed), Boltzmann Transport Equation (BTE, RTA/MC), and the Landauer quantum thermal conductance. Provide mappings and metrology linking temperature fields T(r,t), heat flux q(r,t), interface/size effects, and effective parameters κ(T), α(T), R_K to decoherence modeling.
• All formulas/symbols/definitions are in English and wrapped in backticks; SI units; ω/f and PSD conventions per Chapter 2.

II. Minimal Statements & Principles (S60-*)

• S60-1 (Fourier diffusion)
  q = -κ ∇T, ∂_t T = α ∇^2 T + (Q/ρ c); [κ]=W·m^-1·K^-1, [α]=m^2·s^-1.
• S60-2 (Cattaneo–Vernotte hyperbolic)
  τ_q ∂_t q + q = -κ ∇T; eliminating q gives τ_q ∂_{tt} T + ∂_t T = α ∇^2 T + (Q/ρ c); [τ_q]=s.
• S60-3 (Green–Kubo for effective κ)
  κ = (1/(k_B T^2 V)) ∫_0^∞ ⟨ J_Q(0) · J_Q(t) ⟩ dt, with total heat current J_Q; ensures passivity/causality.
• S60-4 (BTE in RTA)
  ∂_t f + v·∇_r f = -(f - f_eq)/τ(ω,p,T); energy density E = ∫ ħω f D(ω,p) dω dp, heat flux q = ∫ ħω v f D dω dp.
• S60-5 (Diffusive–ballistic–hydrodynamic map)
  Knudsen Kn = l_mfp / L; Kn≪1 diffusion, Kn≫1 ballistic, Kn≈O(1) requires BTE; when momentum-conserving ph–ph grows, phonon hydrodynamics emerges.
• S60-6 (Landauer thermal conductance)
  1D channel G_th(T) = ∫_0^∞ (ħω/2π) 𝒯(ω) (∂ n̄/∂T) dω; quantum limit G_0 = (π^2 k_B^2 / 3h) T · N_modes.
• S60-7 (Kapitza interface resistance)
  ΔT = R_K q_n; spectral/thermal dependence R_K(T,ω) via mismatch/relaxation models; stacked R_K,eff = Σ_i R_{K,i} (series).
• S60-8 (Effective media & size corrections)
  Thin films/nanowires κ_eff = κ_bulk /(1 + Λ/L) (example; Λ characteristic scattering length); membrane/beam modal DOS corrections enter BTE D(ω,p).
• S60-9 (Energy conservation & sources)
  Enforce ∂_t (ρ c T) + ∇·q = Q under any approximation; numerical schemes must preserve discrete energy.

III. Models & Derivations (Macro → Meso → Quantum)

• Macro PDE layer: Fourier/Cattaneo for Kn≪1 and weak non-equilibrium; steady/transient boundary-value problems solved by FEM/FVM; high-frequency/short pulses require τ_q correction.
• Meso BTE layer: RTA τ(ω,p,T) via scattering channels (U/N, interfaces, impurities) with Matthiessen’s rule; Monte Carlo or deterministic DOM/SN for q, κ_eff.
• Quantum channel layer: Landauer for low-D/strongly coherent/ballistic regimes; transmission (ω) from NEGF/cavity spectral |χ_c(ω)|^2; series/parallel with R_K to form thermal networks.
• Cross-layer consistency: validate κ_eff with Green–Kubo; reconcile BTE/NEGF q with PDE q under identical geometry/BCs; use Kn and Pe = V L / α to guide model choice.

IV. Metrology Chain & Data Contract (Required Fields)

unit_system: "SI"

materials:

kappa_T: "<W/m/K>", alpha_T: "<m^2/s>", rho: "<kg/m^3>", c: "<J/kg/K>"

relaxation:

tau_q: "<s>", tau_RTA: {U: "<s>", N: "<s>", imp: "<s>", boundary: "<s>"}

geometry:

dim: "1D|2D|3D", L: "<m>", mesh: "<file|hash>", bc: {type: "Dirichlet|Neumann|Robin", values: "..."}

interfaces:

R_K: [{pair: "A/B", value: "<K m^2/W>", model: "AMM|DMM|fit"}]

bte:

dos: "<D(ω,p)>", velocity: "<v(ω,p)>", rta: "<tau(ω,p,T)>", solver: "MC|DOM|SN"

landauer:

transmission: "<T(ω)>", nmodes: "<N_modes>", range: {ωmin:"", ωmax:""}

sources:

Q: "<W/m^3>", pulses: {type:"rect|gauss", FWHM:"<s>"}

outputs:

fields: ["T(r,t)","q(r,t)"], effective: ["kappa_eff","G_th","R_K_eff"]

uncertainty:

Σ_y: "<blocks>", priors: {kappa:"...", tau_q:"...", R_K:"..."}

references: ["Heat.Decoherence v1.0:Ch.2 S20-*","Ch.5 S50-*","Ch.7 S70-*"]

V. Algorithmic Workflows (M6-*)

• M6-1 (PDE / Fourier–Cattaneo solver)
  Mesh & BCs → assemble q & energy equation → implicit/Crank–Nicolson stepping → outputs T(r,t), q(r,t), κ_eff.
• M6-2 (BTE · RTA/MC)
  Import D(ω,p), v(ω,p), τ(ω,p,T) → choose DOM/MC → compute heat flux & κ_eff → reconcile with macro layer.
• M6-3 (Landauer)
  Given 𝒯(ω) → compute G_th(T) and modular thermal networks (series R_K) → return G_th, R_K_eff.
• M6-4 (Parameter inversion & calibration)
  Fit κ(T), τ_q, R_K to thermal imaging T(r,t) / micro-calorimetric q; output confidence and Fisher conditioning.
• M6-5 (Cross-layer consistency)
  Green–Kubo checks, energy-balance audit, Kn/Pe domain labeling and model-selection report.

VI. Implementation Bindings & Interfaces (I60-*)

• I60-1 solve_fourier(geom, kappa_T, Q, bc) -> {T(r,t), q(r,t), kappa_eff}
• I60-2 solve_cattaneo(geom, kappa_T, tau_q, Q, bc) -> {T(r,t), q(r,t)}
• I60-3 solve_bte(dos, velocity, tau_rta, solver) -> {q, kappa_eff, stats}
• I60-4 compute_landauer(T_trans, N_modes, T) -> {G_th, R_th}
• I60-5 fit_thermal_params(data, model) -> {kappa(T), tau_q, R_K, cov}
• I60-6 consistency_checks(solns) -> {energy_balance, green_kubo, kn_pe_map}

Error codes: E/INPUT (missing), E/UNIT (units), E/NUMERIC (non-convergence/energy non-conservation), E/IDENTIFIABILITY (ill-conditioned).

VII. Quality Gates (This Chapter)

• Q1 Energy conservation: residual ||∂_t (ρ c T) + ∇·q - Q|| below threshold; discrete conservation holds.
• Q2 Units/dimensions: q/κ/α/R_K pass check_dim; report ENBW/time-step stability.
• Q3 Domain selection: label applicability via Kn/Pe; alert and switch to BTE/Landauer if out of domain.
• Q4 Identifiability: conditioning F = J^T Σ^{-1} J controlled; advise bandwidth/excitation upgrades when insufficient.
• Q5 Cross-layer reconciliation: κ_eff and G_th reconcile with Green–Kubo/experiments under matched geometry/BCs; series/parallel R_K rules consistent.

VIII. Cross-References & Anchors (This Chapter)

• Cross-refs (fixed style): Ch. 2 (metrology & conventions), Ch. 5 (FDT & spectral estimation), Ch. 7 (platform-specific channels), Ch. 11 (simulation stack & SBC).
• Anchors: Minimal S60-1—S60-9; Workflows M6-1—M6-5; Interfaces I60-1—I60-6.

IX. Summary
This chapter unifies Fourier/Cattaneo/BTE/Landauer descriptions of heat transport across scales, provides solution & calibration workflows from PDE to BTE/NEGF, and reconciles layers via Green–Kubo and energy conservation. Outputs T(r,t), q(r,t), κ_eff, G_th, R_K feed Chapter 7 platform channels and Chapter 11 simulation stack as traceable thermal inputs.