# ⚛  L1 Principle — Hydrodynamic Semiconductor Transport

**ID:** `L1-489` · **Status:** ⊙ Testnet (genesis catalog)

> **🌐 Domain:** Semiconductor Physics — *Energy-balance transport*
> **🎯 Problem class:** nonlinear inverse · **🧮 Solution space:** n v Te hydrodynamic fields
> **📡 Carrier:** electron · **🌫 Noise:** gaussian
> **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554141

---

## 🧠 1. Introduction

**Hydrodynamic Semiconductor Transport** is a **nonlinear inverse problem** whose unknown lives in **n v Te hydrodynamic fields** space, within the **Energy-balance transport** sub-domain of **Semiconductor Physics**.

Measurements consist of electrons collected by an electron detector via a **hot carrier transport** sensing mechanism.

The forward operator applies, in order: E · continuity operator; E · momentum balance operator; E · energy balance operator; a fixed-point or gradient iteration on the unknown; O · hot carrier IV operator.

Observations are corrupted by additive Gaussian noise. Stiff hyperbolic-parabolic system; shocks at hot-carrier fronts; closure model breaks down in ballistic regime.

## ⚙ 2. Forward Model

Physical chain: **x** → E · continuity → E · momentum balance → E · energy balance → O · hot carrier IV → **y** (detector).

```
y = `O.hot_carrier_IV` `E.energy_balance` `E.momentum_balance` `E.continuity` x + n,    n ~ 𝒩(0, σ²)
```

**Measurement DAG:**

| Primitive | What it does |
|---|---|
| `E.continuity` | E · continuity operator |
| `E.momentum_balance` | E · momentum balance operator |
| `E.energy_balance` | E · energy balance operator |
| `O.hot_carrier_IV` | O · hot carrier iv operator |

**🛠 Solver components** _(used inside the solver, not in the forward equation)_:

| Primitive | What it does |
|---|---|
| `O.iter` | A fixed-point or gradient iteration on the unknown |

## 🔬 3. Physics Fingerprint

| Property | Value |
|---|---|
| Domain | Semiconductor Physics |
| Sub domain | Energy-balance transport |
| Carrier | electron |
| Problem class | nonlinear_inverse |
| Solution space | n_v_Te_hydrodynamic_fields |
| Noise model | gaussian |
| Integration axis | spatial |
| Difficulty delta | 5 |
| L dag | 4.2 |

## 📡 4. Measurement Model

Stiff hyperbolic-parabolic system; shocks at hot-carrier fronts; closure model breaks down in ballistic regime.

| Metric | Value |
|---|---|
| Metric | IV_log_relative_error_high_field |
| Secondary | Te_RMSE_K |

## 📏 5. Operating Range (Ω)

**Center problem class:** `hydrodynamic_parameter_inversion` · **Forward operator:** `hydrodynamic_forward`

**Center point:**

| Parameter | Unit | Value |
|---|---|---|
| T k | — | 300 |
| Vds | — | 1 |
| Mesh n | N | 128 |
| N d cm3 | — | 1000000000000000000 |
| L channel nm | nm | 50 |

**Allowed bounds:**

| Parameter | Unit | Range |
|---|---|---|
| T k | — | 77 – 500 |
| Vds | — | 0.1 – 5.0 |
| Mesh n | N | 64 – 1024 |
| N d cm3 | — | 1000000000000000.0 – 100000000000000000000 |
| L channel nm | nm | 5 – 500 |

## 🎯 6. Tolerance (ε)

**Center tolerance:** log10(I) error <= 0.04

| Metric | Range |
|---|---|
| Iv log relative error high field | 0.005 – 0.4 |

## ⚖ 7. Hardness Function

Hardness scales as **`epsilon_fn`** on **IV_log_relative_error_high_field**, with κ = `1500` and δ = `5`.

## 💾 8. Reference Dataset

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

## 9. On-chain Registration

- **Chain hash:** `0x8220575f93ccdf71eee8bc722509f0448e22644fce14de69112a5c889ef29a6d`
- **Chain tx hash:** `0x73c570112e7a00f6e9bacf3974366679ba50482a62227a1b6c9b334c805ac5e6`
- **Chain block:** `41554141`

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## File Mapping

This bundle consists of: `L1-489.md`, `L1-489.json`.

| File | Role | How to regenerate |
|------|------|-------------------|
| `L1-489.md` | Source of truth — edit this | Human or LLM |
| `L1-489.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 `L1-489`._
_Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._