Appendix B Parameter Registry & Default Priors (OT)
I. Abstract & Scope
This appendix defines the unified specification for OT (Oriented Tension) parameter registry and default priors: covering orientation–constitutive (Λ_{ijkl}, A, K, tau_relax, D_Q), couplings (χ_* , D1, α_*, κ_s), interfaces/defects (κ_a, κ_b, K, r_core), media & effective tensors (ε_ij, μ_ij, C_eff, D_eff), EDX energy/flux kernels (K_E, etc.), and (if present) ToA-related fields. All parameters are registered in English notation with SI units; priors must respect physical feasibility and dimensional closure.
II. Dependencies & References
- Notation & units: Ch.2 P80-1/4/5/6; orientation & geometry: Ch.3 S80-1/2; constitutive & dynamics: Ch.4 S80-3/4; couplings: Ch.6 S80-5/6; EDX: Ch.7 S80-7/8; interfaces/defects: Ch.8 S80-9/10/11; numerics & APIs: Ch.10/12 (SimStack-OT, I80-*).
III. Scope & Naming Rules (mandatory)
- Naming: English snake_case/camelCase aligned with anchors; cross-chapter parameters are centrally registered.
- Minimum fields: {name, meaning, unit, dim, transform, bounds, prior, hyper, scope, see}.
- Transforms: identity | log | logit; use log for positive, logit for bounded.
- Prior families: U[a,b], N(μ,σ), LogU[a,b], LN(μ,σ), Beta(α,β), HalfN(σ), etc.
- Feasible domain: enforce positivity/objectivity/symmetry/nonnegative energy alongside unit/dimension closure.
IV. Parameter Registry (grouped)
Tab. B-1 Orientation–Constitutive (Ch.4)
name | meaning | unit | dim | transform | bounds | default prior | see |
|---|---|---|---|---|---|---|---|
Λ_{ijkl} | orientation→tension map | Pa | M·L^-1·T^-2 | identity | symmetry/objectivity set | N(0, σ_Λ) (material-symmetry trimmed) | S80-3 |
A | quadratic energy coeff. | J·m^-3 | M·L^-1·T^-2 | log | (1e-2, 1e9) | LogU[1e0,1e6] | S80-4/7 |
K | gradient-energy coeff. | J·m^-1 | M·T^-2 | log | (1e-6, 1e6) | LogU[1e-4,1e3] | S80-4 |
tau_relax | relaxation time | s | T | log | (1e-6, 1e4) | LN(μ=-2,σ=1) | S80-4 |
D_Q | order-parameter diffusivity | m^2·s^-1 | L^2·T^-1 | log | (1e-16, 1e-6) | LogU[1e-14,1e-8] | S80-4 |
Tab. B-2 Couplings (Ch.6)
name | meaning | unit | dim | transform | bounds | default prior | see |
|---|---|---|---|---|---|---|---|
χ_E | E-field–orientation | J·m^-3·(V·m^-1)^-2 | — | identity | feasible | N(0, σ_χE) | S80-5 |
χ_B | B-field–orientation | J·m^-3·T^-2 | — | identity | feasible | N(0, σ_χB) | S80-5 |
χ_u | strain-rate–orientation | Pa·s | M·L^-1·T^-1 | identity | feasible | N(0, σ_χu) | S80-5 |
D1 | anisotropic diffusion gain | m^2·s^-1 | L^2·T^-1 | identity | [−D0, +D0] | U[−0.9D0, 0.9D0] | S80-6 |
α_E | dielectric anisotropy gain | 1 | 1 | identity | [−1, 1] | U[−0.2, 0.2] | S80-6 |
α_B | magnetic anisotropy gain | 1 | 1 | identity | [−1, 1] | U[−0.2, 0.2] | S80-6 |
κ_s | acoustic/elastic anisotropy gain | m^2·s^-2 | L^2·T^-2 | identity | (−c_s0^2, +c_s0^2) | N(0, 0.1 c_{s0}^2) | S80-6 |
Tab. B-3 Media & Effective Tensors (Ch.6)
name | meaning | unit | dim | transform | bounds | default prior | see |
|---|---|---|---|---|---|---|---|
ε_ij | permittivity (abs/rel) | F·m^-1 / 1 | — | identity | SPD | N(ε_ref, Σ_ε) | S80-6 |
μ_ij | permeability (abs/rel) | H·m^-1 / 1 | — | identity | SPD | N(μ_ref, Σ_μ) | S80-6 |
C_eff,ijkl | effective elasticity | Pa | M·L^-1·T^-2 | identity | sym./SPD | N(C_ref, Σ_C) | S80-6 |
D_eff,ij | effective diffusion | m^2·s^-1 | L^2·T^-1 | identity | SPD | N(D_ref, Σ_D) | S80-6 |
Tab. B-4 Interfaces/Defects (Ch.8)
name | meaning | unit | dim | transform | bounds | default prior | see |
|---|---|---|---|---|---|---|---|
κ_a, κ_b | anchoring coefficients | J·m^-2 | M·T^-2 | log | (1e-6, 1e3) | LogU[1e-4,1e2] | S80-9 |
K | gradient penalty | J·m^-1 | M·T^-2 | log | (1e-6, 1e6) | LogU[1e-4,1e3] | S80-10 |
r_core | effective core radius | m | L | log | (1e-9, 1e-3) | LogU[1e-8,1e-4] | S80-10 |
Tab. B-5 EDX & Flux Kernels (Ch.7)
name | meaning | unit | dim | transform | bounds | default prior | see |
|---|---|---|---|---|---|---|---|
K_E | energy-flux diffusivity | (J·m^-3)·m^2·s^-1 | M·L·T^-3 | log | (1e-8, 1e4) | LogU[1e-6,1e2] | S80-7 |
Φ_cpl_scale | coupling-flux scale | W·m^-2 | M·T^-3 | log | (1e-6, 1e3) | LogU[1e-4,1e1] | S80-7/6 |
Tab. B-6 Numerics & Instrumentation (Ch.10/12)
name | meaning | unit | dim | transform | bounds | default prior | see |
|---|---|---|---|---|---|---|---|
Δt | time step | s | T | log | (1e-9, 1e0) | LogU[1e-8,1e-2] | S80-18 |
h | spatial step | m | L | log | (1e-9, 1e-1) | LogU[1e-8,1e-3] | S80-18 |
R_inst | instrument kernel params | — | — | identity | feasible | N(R_ref, Σ_R) | Ch.5/10 |
S_n | noise PSD | (unit y)^2·Hz^-1 | — | log | app.-specific | LogU[a,b] | Ch.5 |
(Optional) Tab. B-7 ToA-Related (if present, Ch.12)
name | meaning | unit | dim | transform | bounds | default prior | see |
|---|---|---|---|---|---|---|---|
n_eff | effective-index model params | 1 | 1 | identity/log | feasible | N(…, …)/LogU[…] | I80-10 |
path_params | path parameterization | — | — | identity | feasible | model prior | I80-10 |
delta_form | ToA form flag | — | — | — | {A,B} | record-only | I80-10 |
V. Hierarchical Priors & Shared Hypers (Ch.9 M80-25/27)
- Orientation–constitutive layer:
log A ~ N(μ_A, σ_A), μ_A ~ N(0,2), σ_A ~ HalfN(1);
log K ~ N(μ_K, σ_K), μ_K ~ N(0,2), σ_K ~ HalfN(1);
log tau_relax ~ N(μ_τ, σ_τ), μ_τ ~ N(−2,2), σ_τ ~ HalfN(1). - Coupling layer:
χ_* | (σ_χ) ~ N(0, σ_χ^2), σ_χ ~ HalfN(s_0); D1 | (σ_D1) ~ N(0, σ_D1^2). - Media-tensor layer: share hypers across material families (isotropic/transversely isotropic, etc.).
VI. Prior Coding & Feasibility (Execution)
- Enforce SPD via Cholesky/eigendecomposition (e.g., D_eff = L L^T); symmetry & traceless constraints in parameter space or via projection.
- Objectivity/symmetry through invariant bases or DoF reduction (e.g., material-symmetry constraints on Λ_{ijkl}).
- Nonnegative energy: A≥0, K≥0; positive diffusion/flux; violations are rejected or penalized.
VII. Card & Workflow Integration (I75-/I80-/M-series)
- ParameterCard: emit per this appendix; embed in ModelCard.params.
- I80 APIs: oriented_tension, couple_transport, edx_balance, simulate_ot, fit_ot read parameters and return unit/dimension-bearing results.
- M workflows: M80-39/40/41/42 and M80-31…38 consume default priors; ModelCard may override with override:true and provenance.
VIII. Dual-Form ToA Parameterization (if used, mandatory)
- Form A: T_arr = ( 1 / c_ref ) * ( ∫_{gamma(ell)} n_eff d ell ) (depends on n_eff model & path params).
- Form B: T_arr = ( ∫_{gamma(ell)} ( n_eff / c_ref ) d ell ).
Parameter cards must list {paths = gamma(ell), measure = d ell, n_eff_model} and the delta_form policy.
IX. Templates & Examples
- Minimal ParameterCard snippet:
{"name":"tau_relax","unit":"s","dim":"T","transform":"log","bounds":[1e-6,1e4],"prior":"LN(-2,1)","see":"S80-4"} - Override example: in ModelCard.params, set override:true and cite the data/literature source when replacing defaults.
— This appendix establishes the baseline for OT parameter registration and priors, ensuring reproducibility, auditability, and physical consistency across chapters and implementation layers.