Lomax, Bates & Whitworth (2018)

The big idea¶

Molecular clouds are well described by fractional Brownian motion (FBM) density fields — self-similar random fields with a well-defined fractal dimension. Since clusters are born from clouds, Lomax+ propose modelling cluster structure with the same FBM fields, rather than with the discrete box-fractal (BF; Goodwin & Whitworth 2004) or radial density-profile (RDP) models used to calibrate the CW04 $\mathcal{Q}$ parameter. An FBM cluster is parametrised by just two numbers: the drift/Hurst exponent $H$ (structural roughness) and the log-density standard deviation $\sigma$. Crucially, FBM can produce both centrally-concentrated and substructured clusters, giving a much better match to real clusters than BF or RDP.

This is the same construction gravoturb uses for its 3-D density field (a Gaussian random field with a power-law power spectrum, exponentiated to a lognormal) — so Lomax+ 2018 is the published, peer-reviewed grounding for our differentiable field method. We differ only in the final marginal map (see Notes).

Core equations (their Section 2)¶

Work on an $E$-dimensional grid ($E=2$ or 3); $\mathcal{U}$ denotes a uniform random variate. Generate the field spectrum with a power-law amplitude (Eqs. 1–2):

with $\mathcal{P} = \sum_{\mathbf{k}}\lVert\mathbf{k}\rVert^{-\beta}$ and $A(\mathbf 0)=0$. Hermitian-symmetric phases (Eq. 3) make the inverse DFT real:

Optionally smooth with a Gaussian kernel of width $h$ (a nuisance resolution parameter; Eq. 4). Finally exponentiate the standardised field $f'$ to a lognormal density (Eq. 5):

where $\sigma$ is the standard deviation of $\ln g$. Stellar positions are then sampled with $g$ as the PDF; 3-D fields are projected to 2-D for analysis.

Fractal dimension. The FBM field has

so in 3-D ($E=3$): $H=1$ gives smooth/centrally-concentrated structure ($D=2$ surface-like), while $H\to0$ gives rough, multi-clump structure ($D\to3$, space-filling sub-clumps).

$\mathcal{Q}$ statistics (their Eq. 8). Lomax+ use the Schmeja & Klessen (2006) normalisation $R=\sqrt{A_\mathrm{hull}}$ for both:

(The $R$ choice differs from CW04’s $R=R_\mathrm{cluster}$ + $A=\pi R^2$, so the absolute $\mathcal{Q}$ scale differs — see Cartwright & Whitworth (2004).)

The key result for us¶

Lomax+ show that $\mathcal{Q}$ analysis is unable to estimate FBM parameters: in the $\mathcal{Q}$ (and $\bar m$–$\bar s$) plane, FBM clusters with different $(H,\sigma)$ overlap badly, and $H\sim1$ FBM clusters occupy the same region as smooth RDP clusters. $\mathcal{Q}$ is therefore a poor predictor of $H$ and/or $\sigma$; they resort to a machine-learning regressor trained on $(\bar m,\bar s)$ to recover $(H,\sigma)$ with uncertainties.

Implication (AC7 risk). Because the gravoturb density field is an FBM-type field, a clean monotonic $\mathcal{Q}(f_\mathrm{sub})$ calibration may be optimistic — $\mathcal{Q}$ may weakly discriminate our GRF-based substructure, exactly as Lomax+ found. This caution is recorded for the P3 / AC7 calibration.

Use in progenax¶

• experimental/gravoturb/realization/gaussian_field.py (P2) — the GRF + exponentiation engine is the Lomax-2018 FBM construction; we map $(H,\sigma)\leftrightarrow(\beta,\sigma_s)$ via $\beta = E+2H$ and $\sigma_s$ from BM19.

• Cartwright & Whitworth (2004) — the $\mathcal{Q}$ estimator Lomax+ analyse (different $R$ convention).

Notes¶

• What we keep / change. We keep Lomax’s spatial engine (GRF, $\beta=E+2H$, Hermitian → iFFT, differentiable in $H$). We replace the plain exponentiation (Eq. 5, a lognormal of width $\sigma$) with a rank copula to the BM19 lognormal+power-law marginal, so the field carries the physically-motivated dense tail rather than a pure lognormal. Both maps are differentiable in the cloud parameters.

• $H$ (Hurst) ↔ fractal dimension $D=E-H$ ↔ power-spectrum slope $\beta=E+2H$ are three views of the same roughness parameter.