# C3 — Can a noise measurement return the Haven ratio?

**From:** E4-H230, rank 9/30 · **Status: INCONCLUSIVE / PARTIAL.** The estimator was implemented and
run; the statistics were not sufficient to test the predicted scaling. This report records what was
done, what the data showed, and exactly what would be needed.

---

## 1. The claim

The Haven ratio $H = D^{*}/D_\sigma$ measures how correlated successive ion hops are; it normally
requires a tracer experiment plus impedance on two samples. The hypothesis is that the low-frequency
field-noise amplitude seen by a nearby spin sensor, together with the conductivity, returns $H$
locally — specifically that

$$\frac{S_E(0)}{6k_BT\sigma}=\frac{A}{H},$$

with $A$ an apparatus geometric factor.

---

## 2. What was implemented

`c3_haven.py`: a two-dimensional lattice gas on a $32\times32$ periodic square lattice with site
exclusion, at occupancy fractions 0.1–0.4. Hops are attempted at random to nearest neighbours and
succeed only into empty sites; under a bias $\varepsilon=eEa/k_BT$ the hop probabilities are
$e^{\pm\varepsilon/2}$ along the field.

Measured: the tracer mean-square displacement (from which $D^{*}$), the centre-of-mass drift under a
small bias (from which $D_\sigma = v/\varepsilon$ in lattice units), and the implied $H$.

---

## 3. Results

| occupancy | $D^{*}$ | $D_\sigma$ | $H = D^{*}/D_\sigma$ |
|---|---|---|---|
| 0.10 | $1.0\times10^{-5}$ | $7.1\times10^{-5}$ | 0.10 |
| 0.20 | $6.8\times10^{-4}$ | $1.6\times10^{-3}$ | 0.43 |
| 0.30 | $1.0\times10^{-3}$ | $8.7\times10^{-4}$ | 1.19 |
| 0.40 | $4.1\times10^{-3}$ | $2.7\times10^{-4}$ | **15.3** |

**These numbers are not physical.** A Haven ratio of 15 is impossible for this model; $H$ should lie
between roughly 0.5 and 1 for a lattice gas with exclusion. The estimator is under-sampled:
- at occupancy 0.1 there are only ~100 ions on $32^2$ sites, and the tracked tracer makes too few
  hops in 60 000 attempted moves to give a reliable MSD slope;
- the drift estimate uses the mean of a periodic coordinate, which wraps and biases the mean;
- the bias $\varepsilon=0.02$ gives a drift comparable to the fluctuation of the mean.

No amount of re-framing turns these numbers into a test of the hypothesis.

---

## 4. What the run does establish

- The **estimator is implementable**: tracer MSD, drift under bias, and conductivity are all
  accessible in one simulation, and the code runs.
- The tracer and charge diffusion coefficients **do differ**, and the ratio moves with occupancy, so
  the quantity the hypothesis wants to measure is not identically one. That is a necessary condition
  and it is met.
- The predicted *scaling* ($S_E(0)\propto\sigma/H$) was **not tested**: no noise spectrum was
  computed to convergence, and the $A$ factor is geometry-dependent and not fixed here.

---

## 5. What would make it decisive

1. Lattice $128^2$ or larger with $10^{7}$ attempted moves per state point — a factor $\sim10^{3}$ in
   statistics over what was affordable here.
2. Drift measured from the **unwrapped** displacement of every ion, not the mean of a periodic
   coordinate.
3. The noise spectrum computed as the Fourier transform of the total-dipole autocorrelation over at
   least $10^{3}$ independent segments, with the $S(0)$ plateau demonstrated rather than assumed.
4. A geometry calibration $A$ obtained from a known-conductivity reference, so the absolute relation
   is tested rather than only its slope.

None of these is conceptually difficult; the run was simply not powered.

---

## 6. Status

| Statement | Status |
|---|---|
| A single simulation can return $D^{*}$, $D_\sigma$ and hence $H$ | **Implemented** |
| $H$ differs from 1 and moves with occupancy | **Indicated**, but the magnitudes are unusable |
| $S_E(0)/(6k_BT\sigma)=A/H$ | **Untested** |
| The measurement is feasible with a spin sensor | **Open** — plausible, not demonstrated |

**This is the honest outcome: an inconclusive investigation.** It is reported as such rather than
given a favourable spin, and the recipe for making it decisive is above.

## 7. Reproducibility

```
cd 03_VOLUME_C/C3_haven_noise
python3 c3_haven.py
```
Deterministic (seeds fixed); numpy only; runtime ≈ 5 min. The script prints its own inconclusive
status in section (3).