# ⚛  L1 Principle — Carrier Boltzmann Transport Equation

**ID:** `L1-490` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Semiconductor Physics — *Kinetic carrier transport*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** distribution f k of r
> **📡 Carrier:** electron · **🌫 Noise:** poisson
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554141

---

## 🧠 1. Introduction

**Carrier Boltzmann Transport Equation** is a **nonlinear inverse problem** whose unknown lives in **distribution f k of r** space, within the **Kinetic carrier transport** sub-domain of **Semiconductor Physics**.

Measurements consist of electrons collected by an electron detector via a **mobility noise measurement** sensing mechanism.

The forward operator applies, in order: E · band structure operator; E · scattering rates operator; int · stochastic operator; O · moments operator.

Observations are corrupted by Poisson counting noise. Well-posed for given scattering rates; inversion for deformation potentials is ill-conditioned (multiple rates couple to same mobility).

## ⚙ 2. Forward Model

Physical chain: **x** → E · band structure → E · scattering rates → int · stochastic → O · moments → **y** (detector).

```
y = `O.moments` `int.stochastic` `E.scattering_rates` `E.band_structure` x,    measurements ~ Poisson(αy)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.band_structure` | E · band structure operator |
| `E.scattering_rates` | E · scattering rates operator |
| `int.stochastic` | Int · stochastic operator |
| `O.moments` | O · moments operator |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Semiconductor Physics |
| Sub domain | Kinetic carrier transport |
| Carrier | electron |
| Problem class | nonlinear_inverse |
| Solution space | distribution_f_k_of_r |
| Noise model | poisson |
| Integration axis | time |
| Difficulty delta | 5 |
| L dag | 3.7 |

## 📡 4. Measurement Model

Well-posed for given scattering rates; inversion for deformation potentials is ill-conditioned (multiple rates couple to same mobility).

| Metric | Value |
|---|---|
| Metric | mobility_relative_error |
| Secondary | noise_S_I_log_error |

## 📏 5. Operating Range (Ω)

**Center problem class:** `kinetic_transport_inversion` · **Forward operator:** `boltzmann_mc_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| T k | — | 300 |
| N d cm3 | — | 100000000000000000 |
| Material | — | Si |
| N particles | — | 100000 |
| E field v cm | — | 10000 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| T k | — | 77 – 500 |
| N d cm3 | — | 100000000000000.0 – 100000000000000000000 |
| N particles | — | 10000 – 10000000 |
| E field v cm | — | 10 – 1000000.0 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** mobility relative error <= 0.05

| Metric | Range |
|---|---|
| Mobility relative error | 0.01 – 0.5 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **mobility_relative_error**, with κ = `600` and δ = `5`.

## 💾 8. Reference Dataset

- **primary** · weight 1.0 · IPFS _(not pinned yet)_

## 9. On-chain Registration

- **Chain hash:** `0x90b87c96eb25800edc6d30ddb99f6c0eac7c38b58d4bc07f49d884e55e13010b`
- **Chain tx hash:** `0x3104ae52719cb42e7e16b4d1349748a0d9457e6d5551b72d641ca58c2f2b5461`
- **Chain block:** `41554141`

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## File Mapping

This bundle consists of: `L1-490.md`, `L1-490.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-490.md` | Source of truth — edit this | Human or LLM |
| `L1-490.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 `L1-490`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._