02-EFT.WP.Core.Equations v1.1 | Chapter 8 — Statistics and Coarse-Graining

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Chapter 8 — Statistics and Coarse-Graining

I. Aims & Scope

• Unify rigorous definitions, window parameters, and exchange conditions for avg_t[•], avg_V[•], and avg_gamma[•].
• Derive time-/space-averaged minimal equations S80-* from the strong conservation law, and provide flux decomposition and closure recipes.
• Interface seamlessly with Chapter 2 (arrival time), Chapters 4/5 (governing equations), and Chapter 7 (weak-form templates); all dimensional rules follow Core.Terms §6.

II. Averaging & Filtering Operators (S80-1)

• Time average
  avg_t[f; Δt](t) def= ( 1 / Δt ) * ( ∫_{t-Δt/2}^{t+Δt/2} f(τ) d τ )。
  Requirement: Δt > 0, symmetric window; measure d τ.
• Volume average
  avg_V[f; V](x) def= ( 1 / |V| ) * ( ∫_{V} f(ξ) d V )，|V| def= ( ∫_{V} 1 d V )。
• Path average
  L_gamma def= ( ∫_{gamma(ell)} 1 d ell )；
  avg_gamma[f] def= ( 1 / L_gamma ) * ( ∫_{gamma(ell)} f d ell )。
• S80-1 (normalization axioms)
  avg_t[1]=1，avg_V[1]=1，avg_gamma[1]=1；linearity and positivity preserved；window parameters must be explicitly annotated in-form.

III. Reynolds Decomposition & Scale Separation (S80-2)

• Decomposition
  For any scalar or component q, define bar_q def= avg_t[q; Δt] or avg_V[q; V]; disturbance tilde_q def= q - bar_q。
  Then avg_t[tilde_q]=0、avg_V[tilde_q]=0。
• Product average and subgrid term
  avg[ a * b ] = bar_a * bar_b + τ(a,b)，where τ(a,b) def= avg[ (a - bar_a) * (b - bar_b) ] is the subgrid covariance (closure term).

IV. Averaged Conservation Mother Form (S80-3)

• Strong mother form (see Chapter 5)
  ∂_t q + div( J_q ) = S_q on Ω。
• S80-3 (averaged conservation form)
  Under time averaging:
  ∂_t bar_q + div( bar_J_q ) = bar_S_q + R_t，with bar_q def= avg_t[q; Δt]，bar_J_q def= avg_t[J_q; Δt]，
  R_t def= avg_t[∂_t q; Δt] - ∂_t avg_t[q; Δt] (time-exchange residual).

Under volume averaging:
∂_t bar_q + div( bar_J_q ) = bar_S_q + ( 1 / |V| ) * ( ∫_{∂V} ( J_q • nu ) d A ) + R_x，
R_x def= avg_V[ div(J_q); V ] - div( avg_V[J_q; V] )。

V. Flux Decomposition & Closure (S80-4)

• Decomposition
  bar_J_q = J_q( bar_vars ) + J_sgs^q，where J_q( bar_vars ) is the prototype flux with averaged variables inserted;
  J_sgs^q def= bar_J_q - J_q( bar_vars )。
• Minimal gradient-diffusion closure
  J_sgs^q approx - K_sgs^q * grad[ bar_q ]，K_sgs^q ≥ 0，dim(K_sgs^q) = [L]^2 / [T]。
  The coefficient K_sgs^q may be empirical or given by an auxiliary model (see Methods.), and must not be confused with Chapter 4’s K_T.

VI. Exchange Conditions for Averages & Derivatives (P80-1, S80-5)

• P80-1 (sufficient exchange conditions)
  With constant windows, q continuous and integrable within the window, and boundary terms satisfying Chapter 6 (zero-flux or periodic),
  avg_t[∂_t q; Δt] = ∂_t avg_t[q; Δt]，avg_V[ grad[q]; V ] = grad[ avg_V[q; V] ]。
• S80-5 (general residual representations)
  R_t = ( 1 / Δt ) * ( q(t+Δt/2) - q(t-Δt/2) ) - ∂_t bar_q；
  R_x = ( 1 / |V| ) * ( ∫_{∂V} ( ( q - bar_q ) * u_n ) d A )（when J_q = u * q, u_n def= u • nu）。

VII. Path Averaging & Arrival Time (S80-6)

• Equivalent rewrite
  T_arr = ( ∫_{gamma(ell)} ( n_eff / c_ref ) d ell ) = L_gamma * avg_gamma[ n_eff / c_ref ]。
  Hence S20-* additivity corresponds directly to segment-wise averaging for gamma = ⋃ gamma_k:
  T_arr = ( ∑_k L_{gamma_k} * avg_{gamma_k}[ n_eff / c_ref ] )。
• Discrete realization (binding to I20-4)
  For discrete segments ds_i and samples n_eff_i:
  T_arr = ( ∑_i ( n_eff_i / c_ref ) * ds_i ) = L_gamma * ( ∑_i ( ( n_eff_i / c_ref ) * ds_i ) / L_gamma )。

VIII. Parameter Cards for Windows & Domains (S80-7)

• Time window: Δt = α_t * t0，recommend α_t ∈ [10, 100]。
• Spatial volume: |V| = α_x * L0^d (d is spatial dimension)，recommend α_x ∈ [10, 10^3]。
• Path subsegments: ds_max ≤ β * L0，recommend β ∈ [10^{-3}, 10^{-2}]。
  All parameters must be explicitly stated in equations or tables; avoid implicit defaults.

IX. Dimensions & Non-Dimensionalization (S80-8)

• Dimensional closure
  dim( avg_t[q] ) = dim(q)，dim( avg_V[q] ) = dim(q)，dim( avg_gamma[q] ) = dim(q)。
• Non-dimensional mapping
  bar_t := ( t / t0 )，bar_x := ( x / L0 )，bar_q := ( q / q0 )，
  then avg_{bar_t}[ bar_q ; Δbar_t ] = ( 1 / Δbar_t ) * ( ∫ bar_q d bar_t )，isomorphic to the dimensional form.

X. Interface to Weak Forms (building on Chapter 7, S80-9)

XI. Numerical Implementation Binding (I20-aligned)*

• Time-average discretization
  avg_t[q; Δt] ≈ ( 1 / N_t ) * ( ∑_{k=1}^{N_t} q^{n-k} )；N_t = round( Δt / Δt_num )。
• Volume-average discretization
  avg_V[q; V] ≈ ( 1 / |V| ) * ( ∑_{c ∈ V} q_c * |V_c| )。
• Path-average discretization
  avg_gamma[f] ≈ ( ∑_{i} f_i * ds_i ) / ( ∑_{i} ds_i )，then verify T_arr via propagate_time( n_eff_path, ds, c_ref )。
• Interface mapping
  Pass K_sgs^q and window parameters as coeffs to assemble_operator; use compare_solutions(..., metrics=["L2","L_inf","T_arr"]) to assess coarse-graining error.

XII. Lint Rules & Forbidden Items

• Do not omit windows or domains: avg_t[•] must specify Δt; avg_V[•] must specify V。
• Do not use non-canonical arrival time: write ( ∫_{gamma(ell)} ( n_eff / c_ref ) d ell ) and declare gamma(ell), d ell。
• Do not mix symbols: n vs n_eff, T_fil vs T_trans must never substitute for each other。
• Do not leave closures open: if τ(a,b) or J_sgs^q appear, provide an explicit closure or parameter path.

XIII. Registrable Items & Examples

• register_equation("S80-1","definitions of avg_t/avg_V/avg_gamma (normalized integrals)","definition",anchors=["§II"],depends=[])
• register_equation("S80-3","∂_t bar_q + div(bar_J_q) = bar_S_q + R_t (+ boundary term for V)","strong",anchors=["§IV"],depends=["S50-*","S60-*"])
• register_equation("S80-4","bar_J_q = J_q(bar_vars) + J_sgs^q ; J_sgs^q approx - K_sgs^q * grad[bar_q]","model",anchors=["§V"],depends=["S50-*"])
• register_equation("S80-6","T_arr = L_gamma * avg_gamma[n_eff / c_ref]","identity",anchors=["§VII"],depends=["S20-*"])
• register_equation("S80-9","time-averaged weak form with J_sgs^q and R_t","weak",anchors=["§X"],depends=["S70-*","S50-*"])