42-EFT.WP.Heat.Decoherence v1.0 | Chapter 11: Simulation Stack & Synthetic Data — Langevin / BTE / Stochastic Liouville

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Chapter 11: Simulation Stack & Synthetic Data — Langevin / BTE / Stochastic Liouville

I. Objectives & Applicability

• Build a pluggable simulation stack & synthetic-data framework unifying Langevin (Q/Langevin), BTE (Boltzmann Transport Equation), and Stochastic Liouville paths; jointly generate noise→spectrum→response→rates with thermal-field coupling for algorithm validation, SBC coverage testing, and cross-platform benchmarking.
• All formulas/symbols/definitions are in English and wrapped in backticks; SI units; ω/f & PSD per Ch. 2; spectrum→rate per Chs. 4/5; heat transport & BCs per Ch. 6; platform channels per Ch. 7; engineering constraints per Ch. 8; metrology & inversion per Ch. 9.

II. Minimal Statements & Principles (S110-*)

• S110-1 (Q/Langevin kernel)
  m \ddot x + ∫_0^t γ(t−t') \dot x(t') dt' + ∂V/∂x = ξ(t), with ⟨ξ⟩=0 and FDT: S_{ξξ}(ω) = 2 k_B T Re{γ(ω)} (classical). Quantum form uses symmetric spectrum S_{ξξ}(ω) = ħω coth(ħω/2k_B T) Re{γ(ω)}.
• S110-2 (BTE · RTA)
  ∂_t f + v·∇_r f = −(f−f_eq)/τ(ω,p,T); energy E = ∫ ħω f D(ω,p) dωdp, heat flux q = ∫ ħω v f D dωdp.
• S110-3 (Stochastic Liouville)
  \dot ρ = −i[H_S + H_c(t) + H_{st}(t), ρ] + 𝒟[ρ]; random drive H_{st}(t) parameterized by known spectrum S_{st}(ω) (Gaussian/non-Gaussian); take 𝒟[ρ] as GKSL if desired.
• S110-4 (Spectral consistency)
  Simulation output S_{xx}(ω) must match target S_{tar}(ω) within tolerance and satisfy Parseval: Var[x] = (1/2π) ∫ S_{xx}(ω) dω.
• S110-5 (Thermo–quantum coupling)
  Use Ch. 6 to obtain T(r,t) and q(r,t); feed into n̄(ω,T) and material params (κ(T), R_K(T)); iterate until energy conservation and rate convergence.
• S110-6 (SBC coverage)
  Sample from the prior p(θ) → simulate y → estimate \hat θ with Ch. 9 → rank stats & coverage near nominal.

III. Modules & Architecture (SimStack)

• Trajectory & thermal field: traj(t): x(t); T(r,t) and boundary bc (from Ch. 6).
• Noise/spectrum module: generate target S_{tar}(ω) (Johnson/1/f/TLS/white/custom), or produce via J(ω)+χ(ω) and FDT.
• Langevin solver: explicit/semi-implicit/exponential integrators, Krylov; colored noise via convolution or state-space embedding.
• BTE solver: RTA DOM/SN or MC; libraries for DOS/group velocity/relaxation times.
• Stochastic Liouville: GKSL/Redfield/TCL2 generators + stochastic Hamiltonians; inject control sequence F(ω).
• Observables & rates: unified outputs {S_{xx}(ω), S_{ΔωΔω}(ω), Γ_1, Γ_φ, T_1, T_2, Q} and thermal {T(r,t), q(r,t)}.
• Recording & repro: repro_bundle = {scripts, params, env, anchors, seed, mesh_hash}.

IV. Metrology Chain & Data Contract (Required Fields)

unit_system: "SI"

scenario:

platform: "SC|Semiconductor|Spin|Optomech|Acoustic|custom"

geom_bc: {mesh:"<file|hash>", bc:"Dirichlet|Neumann|Robin"}

thermal:

kappa_T:"<W/m/K>", R_K:"<K m^2/W>", sources:{Q:"<W/m^3>"}, T0:"<K>"

noise_spectra:

S_target: {family:"Johnson|one_over_f|TLS|white|custom", params:{...}, type:"one-sided|two-sided", freq:"ω|f"}

J_omega: {family:"ohmic|sub|super|custom", params:{eta:"...", s:"...", omega_c:"<rad/s>"}}

dynamics:

method: "Langevin|BTE|SLE|hybrid"

langevin: {gamma:"<γ(ω) or kernel>", V:"<V(x)>"}

bte: {dos:"<D(ω,p)>", v:"<v(ω,p)>", tau:"<τ(ω,p,T)>", solver:"MC|DOM|SN"}

sle: {generator:"GKSL|Redfield|TCL2", Hs:"<H_S>", Hst:"<noise model>", control:"<seq>"}

outputs:

spectra: ["S_xx(ω)","S_dw(ω)"], rates: ["Gamma1","Gamma_phi","T1","T2","Q"], thermal:["T(r,t)","q(r,t)"]

uncertainty:

Σ_y:"<blocks>", ci_level:0.95

reproducibility:

bundle:{scripts:"...", env:"...", anchors:["S110-*","M11-*","I110-*"], seed:2025}

references: ["Heat.Decoherence v1.0:Ch.2 S20-*","Ch.4 S40-*","Ch.5 S50-*","Ch.6 S60-*","Ch.7 S70-*"]

V. Algorithmic Workflows (M11-*)

• M11-1 (Thermo–spectrum preprocessing)
  Solve T(r,t); build n̄(ω,T) and material/interface params → assemble S_target(ω) or J(ω), χ(ω).
• M11-2 (Langevin path)
  With γ(ω), V(x), generate colored noise → time-step → output x(t), S_{xx}(ω) and Γ_φ.
• M11-3 (BTE path)
  Choose DOM/MC → update f, q, κ_eff → write back thermal chain and T(r,t); reconcile layers (Green–Kubo/energy conservation).
• M11-4 (SLE path)
  Build generator + stochastic drive → specify control → integrate ρ(t) & observables → compute Γ_1/Γ_φ via filter functions.
• M11-5 (Synthetic data & SBC)
  Sample θ ~ p(θ) → run any path to produce y → infer \hat θ via Ch. 9 → rank/coverage → issue benchmark tables.

VI. Implementation Bindings & Interfaces (I110-*)

• I110-1 prep_thermo_spectra(geom, thermal, spectra) -> {T(r,t), S_target(ω), J(ω), χ(ω)}
• I110-2 simulate_langevin(gamma, V, S_target, dt, T_end) -> {x(t), S_xx(ω), stats}
• I110-3 simulate_bte(dos, v, tau, solver, bc) -> {f, q, kappa_eff, audit}
• I110-4 simulate_sle(generator, Hs, Hst, control) -> {ρ(t), S_dw(ω), rates}
• I110-5 make_synthetic(sim_config) -> {dataset_card.yaml, data.h5, repro_bundle}
• I110-6 run_sbc(prior, sim_config, estimator, K) -> {coverage, ranks, report}

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

VII. Quality Gates (This Chapter)

• Q1 Convention consistency: ω/f, PSD one-/two-sided, and FDT/Parseval match Chs. 2/5; pass check_dim.
• Q2 Energy & conservation: BTE/thermal-chain energy error below threshold; Langevin/SLE power spectra consistent with energy.
• Q3 Spectral consistency: S_xx(ω) matches S_target(ω) within tolerance; validate colored-noise state-space poles/zeros.
• Q4 Coverage: SBC pass@coverage near nominal; rank histograms near uniform.
• Q5 Reproducibility: complete repro_bundle (scripts/params/env/mesh hash/seed/anchors); one-click replay passes.

VIII. Cross-References & Anchors (This Chapter)

• Cross-refs (fixed style): Ch. 2 (conventions), Ch. 4 (open QS), Ch. 5 (FDT & spectra), Ch. 6 (heat transport), Ch. 7 (platform channels), Ch. 9 (metrology & inversion), Ch. 10 (robust strategies).
• Anchors: Minimal S110-1—S110-6; Workflows M11-1—M11-5; Interfaces I110-1—I110-6.

IX. Summary
This chapter provides a pluggable simulation stack and synthetic-data contract across Langevin / BTE / Stochastic Liouville routes, unifying co-generation of thermal–spectral–rate & control effects. With quality gates on energy conservation, spectral consistency, and coverage, it outputs traceable {S_{xx}(ω), Γ_1, Γ_φ, T_1, T_2} and T(r,t), q(r,t), enabling cross-platform validation, SBC tests, and engineered design comparisons.