IMF + equipartition recovery (B2)

This is the flagship demo: it couples a cluster’s mass function to its internal kinematics through a single physical parameter, and shows that parameter is recoverable from both channels at once.

The self-consistent physics¶

A multimass LIMEPY cluster Gieles & Zocchi, 2015 is built from an initial mass function. progenax bins the IMF into $J=4$ mass groups; each group $j$ has a representative mass $m_j$ and a population weight set by the IMF. Two-body relaxation drives the cluster toward energy equipartition — heavier stars sink and slow down — encoded in the multimass DF by a per-group velocity scale,

with $\bar m$ the central-density-weighted mean mass and $\delta$ the equipartition degree Peuten et al., 2017. Real clusters reach only partial equipartition ($\delta < \tfrac12$); the heaviest stars approach $\sigma\propto m^{-1/2}$ while the light stars saturate at an escape-speed ceiling — the Bianchini relation, derived directly from this same DF on the multimass equipartition theory page.

The high-mass IMF slope. The observed masses follow a Maschberger (2013) IMF whose high-mass behaviour is the power law

(full smooth form and characteristic mass: Maschberger (2013)). Here is the coupling that makes the demo self-consistent: the same $\alpha$ that sets the slope of the observed mass histogram also sets the masses $m_j$ and weights of the equipartition groups in (1) — so $\alpha$ is constrained by the masses (the histogram) and by the kinematics (the group-by-group $\sigma_j(r)$). The three recovered parameters are $\theta = (\alpha,\ \delta,\ W_0)$, with $W_0$ the central dimensionless concentration.

The joint likelihood¶

Two channels, summed (scripts/demo_delta_recovery.py):

The kinematic predictor is the differentiable Engine A multimass oracle rebuilt from $\theta$ inside the traced loss (the find_alpha_for_masses eigenvalue solve is differentiable in $\alpha,\delta,W_0$); the mass term is the analytic, normalized Maschberger log-pdf — fully differentiable in $\alpha$.

Inputs and assumptions¶

The fit recovers three parameters $(\alpha, \delta, W_0)$; the truncation order, the total mass, and the mass range are assumed known. The headline result — that the mass channel pins $\alpha$ and breaks the kinematic $(\alpha,\delta)$ degeneracy — rests on the clean-mock decoupling described above.

Result 1 — joint recovery (freshly run, ALL PASS)¶

Measured 2026-06-11 ($N=10^5$, three dispersed Adam starts; exit 0):

All within $3\sigma$ (the heaviest, sparsest group $j=3$ has occupancy $1153 \ge 300$, so the fit is not starved). The MLE is a robust optimum — the two best of the three dispersed starts agree in loss to $3\times10^{-3}$.

Result 2 — the degeneracy the mass channel breaks¶

Kinematics alone cannot cleanly separate $\alpha$ from $\delta$: a steeper $\sigma(m)$ can come from a larger equipartition degree or from an $\alpha$ that reweights the group masses. The Fisher information quantifies it (freshly measured):

Adding the mass histogram pins $\alpha$ $4.7\times$ tighter (ellipse area ratio 4.67) and collapses the $(\alpha,\delta)$ correlation from -0.265 toward zero. Crucially the mass channel pins $\alpha$ rather than rotating the ellipse — $\delta$’s width barely moves ($0.0357\to0.0344$); the gain is almost entirely in $\alpha$.

Result 3 — full posterior (NUTS)¶

A vendored blackjax No-U-Turn sampler (300 warmup + 600 samples) draws the full posterior. Recorded measured run (--run-nuts, $\sim 52$ min wall):

• 0 divergent transitions;

• posterior mean within $1\sigma$ of the MLE per parameter ($\alpha\ -0.05$, $\delta\ +0.27$, $W_0\ -0.38\ \sigma$);

• the target is made flat in $\theta$ by adding the $\sum\ln(d\theta/dz)$ reparametrization Jacobian, so the corner is a faithful posterior, not a box-edge artifact.

Result 4 — wrong-IMF bias + robustness grid¶

What if you assume the wrong $\alpha$ and refit only the kinematics? Freezing $\alpha$ at a grid of wrong values and refitting $(\delta, W_0)$ from kinematics alone (5 seeds each) gives the bias curve. The recovered $\hat\delta$ peaks at the true $\alpha$ ($\hat\delta = 0.42 \approx \delta_{\rm true}$ at $\alpha_{\rm assumed}=2.3$) and is biased low on either side — assuming the wrong IMF slope corrupts the equipartition measurement. The linear sensitivity $d\hat\delta/d\alpha = -0.005 \pm 0.020$ (seed-ensemble SE) is near zero precisely because the response is peaked, not monotonic — it is reported, not gated (no published reference value to assert against).

The companion robustness grid regenerates fresh truth datasets at $\alpha_{\rm true}\in\{1.9, 2.3, 2.7\}$ and refits the full joint $(\alpha,\delta, W_0)$ — every case recovers within $3\sigma$:

Caveats¶

How to run¶

env -u VIRTUAL_ENV uv run --no-sync python scripts/demo_delta_recovery.py            # MLE + Fisher (minutes)
env -u VIRTUAL_ENV uv run --no-sync python scripts/demo_delta_recovery.py --run-nuts # + NUTS corner (~52 min)
env -u VIRTUAL_ENV uv run --no-sync python scripts/demo_delta_recovery_bias.py       # wrong-IMF curve + grid

References¶

The multimass LIMEPY DF is Gieles & Zocchi (2015) with the equipartition $m$-convention of Peuten et al. (2017); the $\sigma(m)$ equipartition relation and its derived equipartition mass are Bianchini et al. (2016); the IMF is Maschberger (2013). The equipartition physics is developed on the multimass equipartition theory page, and the underlying equilibrium is validated at multimass equilibrium.