Coles & Jones (1991)

The big idea¶

A Gaussian random field can never model the density of a self-gravitating medium: a Gaussian assigns finite probability to $\rho < 0$. Coles & Jones fix this with the lognormal (LN) random field — take a Gaussian field $X(\mathbf{r})$ and exponentiate it,

so $Y = \rho > 0$ everywhere, the field is fully specified statistically (like a Gaussian, by one covariance function), yet it becomes arbitrarily close to Gaussian at early times / small variance. The one-point PDF is the familiar lognormal

Why turbulent density is lognormal (the multiplicative CLT)¶

The deepest part of the paper is its §3.2 argument for why this form should arise physically. The ordinary central-limit theorem says a sum of many independent influences tends to a Gaussian. Coles & Jones build the non-linear analogue: if instead the field is a product of many independent multiplicative influences,

then $\log Y$ is Gaussian, i.e. $Y$ is lognormal. In their words, “the lognormal is a paradigm for non-linear noise just as the Gaussian is for linear noise.” In a turbulent cloud each passing shock multiplies the local density by a random factor; the accumulated product is lognormal. This is the microphysical origin of the lognormal body of the Burkhart & Mocz (2019) density PDF, and of the hierarchical-fragmentation lognormal mass functions of Zinnecker (1984) the paper cites.

One covariance fixes all orders — the Gaussianization seed¶

Because $Y$ is a fixed monotone transform of a Gaussian, every statistic of $Y$ follows from the single Gaussian covariance $\xi_X(r)$. For the pure lognormal, §5 gives the exact transformed two-point function

and analogous closed forms at all higher orders. This is the prototype of “Gaussianization”: the statistics of a non-Gaussian-but-monotone-mapped field are computable analytically from the underlying Gaussian correlation. Szapudi & Pan (Szapudi & Pan (2004)) generalize (4) to an arbitrary monotone map via a Hermite expansion; that generalization is exactly the gaussianized_xi series used to predict $\xi_s(r)$ in progenax.

Coles & Jones also flag a subtlety progenax inherits: the LN field is not fully specified by its moments (the moment problem is indeterminate for the lognormal), so moment-based descriptors are a poor way to test for non-Gaussianity — a point sharpened later by the Neyrinck/Carron information analyses (Neyrinck, Szapudi & Szalay (2009), Carron & Szapudi (2013)).

Use in progenax¶

• Theoretical root of the FDF. The fractal density field is a Gaussian $g$ with $P(k)\propto k^{-\beta}$ exponentiated/copula-mapped to a target marginal — precisely the Coles & Jones construction (1), generalized from a pure lognormal to the BM19 lognormal + power-law marginal via the rank copula (gaussian_field.py).

• The lognormal body. (3) is the physical justification for the lognormal core of the Density PDFs and the freefall-density factor / Burkhart & Mocz (2019) PDF.

• The Gaussianization series. (4) is the $n=1$, pure-exp special case of the general Hermite series in Differentiable inference — natal cloud parameters from cluster substructure; the log-density two-point $\xi_s$ is computed this way, with $\beta$ entering through the Gaussian correlation $\rho_g(r)$.

Notes¶

• The lognormal is a model, not exact for fully evolved fields — Coles & Jones themselves note it produces some spurious skewness. BM19 corrects the high-density end with a self-gravity power-law tail; progenax imposes that fuller marginal through the copula.

• §7’s Monte-Carlo recipe for point patterns with correlations of all orders (1D/2D/3D) is the conceptual ancestor of the FDF star-sampling step (placement.py).