L1 ⚛ L1-497 ⊙ Testnet

Phonon Boltzmann Transport Equation

Semiconductor Physics · Thermal phonon transport

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⬇ L1-497.md

# ⚛ L1 Principle — Phonon Boltzmann Transport Equation **ID:** `L1-497` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Semiconductor Physics — *Thermal phonon transport* > **🎯 Problem class:** linear inverse · **🧮 Solution space:** phonon distribution n q s > **📡 Carrier:** phonon · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554142 --- ## 🧠 1. Introduction **Phonon Boltzmann Transport Equation** is a **linear inverse problem** whose unknown lives in **phonon distribution n q s** space, within the **Thermal phonon transport** sub-domain of **Semiconductor Physics**. Measurements consist of lattice phonons probed indirectly via a **thermal conductivity measurement** sensing mechanism. The forward operator applies, in order: E · phonon dispersion operator; E · scattering rates operator; O · composite method operator; O · kappa operator. Observations are corrupted by additive Gaussian noise. Well-posed; inversion for branch-resolved tau is ill-conditioned (only integrated kappa measured). ## ⚙ 2. Forward Model Physical chain: **x** → E · phonon dispersion → E · scattering rates → O · composite method → O · kappa → **y** (detector). ``` y = `O.kappa` `O.composite_method` `E.scattering_rates` `E.phonon_dispersion` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `E.phonon_dispersion` | E · phonon dispersion operator | | `E.scattering_rates` | E · scattering rates operator | | `O.composite_method` | O · composite method operator | | `O.kappa` | O · kappa operator | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Semiconductor Physics | | Sub domain | Thermal phonon transport | | Carrier | phonon | | Problem class | linear_inverse | | Solution space | phonon_distribution_n_q_s | | Noise model | gaussian | | Integration axis | wavevector | | Difficulty delta | 5 | | L dag | 3.6 | ## 📡 4. Measurement Model Well-posed; inversion for branch-resolved tau is ill-conditioned (only integrated kappa measured). | Metric | Value | |---|---| | Metric | kappa_relative_error | | Secondary | kappa_T_curve_RMSE | ## 📏 5. Operating Range (Ω) **Center problem class:** `phonon_bte_inversion` · **Forward operator:** `phonon_bte_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | T k | — | 300 | | N q grid | — | 16 | | Material | — | Si | | Grain size um | µm | 10 | | Isotope naturality | — | 1 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | T k | — | 10 – 1500 | | N q grid | — | 8 – 64 | | Grain size um | µm | 0.001 – 1000 | | Isotope naturality | — | 0.0 – 1.0 | ## 🎯 6. Tolerance (ε) **Center tolerance:** kappa rel error <= 0.03 | Metric | Range | |---|---| | Kappa relative error | 0.01 – 0.4 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **kappa_relative_error**, with κ = `500` and δ = `5`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0xa90c41129839d479a6f9f714c56d115fb02474e988cafe143aff0d0f20a41892` - **Chain tx hash:** `0xe42c883903794eac325d3f907d95f786b069bc9352d658bbd7df71d98e1ecd80` - **Chain block:** `41554142` --- ## File Mapping This bundle consists of: `L1-497.md`, `L1-497.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-497.md` | Source of truth — edit this | Human or LLM | | `L1-497.json` | Structured metadata for the registry | LLM regenerates from the sections above | **Prompt for your LLM after editing this Markdown:** > Read the attached Markdown. Regenerate the sibling `.json` so every field matches. > Preserve the schema documented in the rows above. > Output each file in its own fenced code block tagged with the filename. > Output only the JSON object. _This Markdown was auto-synthesized from the catalog row for `L1-497`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Read the attached Markdown (L1-497.md).
Regenerate L1-497.json so every field matches the Markdown.
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Output each file in its own fenced code block tagged with the filename.

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L2-497-001 Phonon Boltzmann Transport Equation Mismatch-Only Spec

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📋 JSON metadata

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  "artifact_id": "L1-497",
  "chain_block": 41554142,
  "chain_hash": "0xa90c41129839d479a6f9f714c56d115fb02474e988cafe143aff0d0f20a41892",
  "chain_tx_hash": "0xe42c883903794eac325d3f907d95f786b069bc9352d658bbd7df71d98e1ecd80",
  "domain": "Semiconductor Physics",
  "hardness_fn": {
    "delta": 5,
    "kappa": 500,
    "metric": "kappa_relative_error",
    "type": "epsilon_fn"
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  "initiator_dataset": [
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  "layer": "L1",
  "observable_profile": {
    "metric": "kappa_relative_error",
    "regime": "Well-posed; inversion for branch-resolved tau is ill-conditioned (only integrated kappa measured).",
    "secondary": "kappa_T_curve_RMSE"
  },
  "physics_fingerprint": {
    "L_DAG": 3.6,
    "carrier": "phonon",
    "difficulty_delta": 5,
    "domain": "Semiconductor Physics",
    "integration_axis": "wavevector",
    "noise_model": "gaussian",
    "primitives": [
      "E.phonon_dispersion",
      "E.scattering_rates",
      "O.composite_method",
      "O.kappa"
    ],
    "problem_class": "linear_inverse",
    "sensing_mechanism": "thermal_conductivity_measurement",
    "solution_space": "phonon_distribution_n_q_s",
    "sub_domain": "Thermal phonon transport",
    "title": "Phonon Boltzmann Transport Equation"
  },
  "size_tiers": {
    "allowed_forward_operators": [
      "phonon_bte_forward"
    ],
    "allowed_omega_dimensions": [
      "material",
      "T_K",
      "grain_size_um",
      "N_q_grid",
      "isotope_naturality"
    ],
    "allowed_problem_classes": [
      "phonon_bte_inversion"
    ],
    "center_spec": {
      "epsilon_fn_center": "kappa rel error \u003c= 0.03",
      "forward_operator": "phonon_bte_forward",
      "input_format": "measurement_only",
      "omega": {
        "N_q_grid": 16,
        "T_K": 300,
        "grain_size_um": 10,
        "isotope_naturality": 1.0,
        "material": "Si"
      },
      "problem_class": "phonon_bte_inversion"
    },
    "epsilon_bounds": {
      "kappa_relative_error": [
        0.01,
        0.4
      ]
    },
    "omega_bounds": {
      "N_q_grid": [
        8,
        64
      ],
      "T_K": [
        10,
        1500
      ],
      "grain_size_um": [
        0.001,
        1000
      ],
      "isotope_naturality": [
        0.0,
        1.0
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Thermal phonon transport",
  "title": "Phonon Boltzmann Transport Equation"
}

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