Can the slow free-energy calculation be replaced by a quick calibrated shortcut?

EXPERIMENTS.md · lines 1026–1080

E23 — A composition-level free energy fast enough to be an objective

Date 2026-09-12 · Question The generative search proposes compositions, not occupancies, so it needs F(x, T) per composition — and it needs it thousands of times. Monte Carlo costs ~30 s per composition. Can the short-range-order correction be calibrated once and then applied for free? · Provenance forager/thermo.py, …/thermo_test.py

Method. Three tiers in one object (CEThermo), all on the 8-element CE of E16b:

1. Random limit — <E> over random occupancies at fixed composition, with S = S_ideal. Exact at infinite temperature, ~10,000 compositions/s.
2. Reference — canonical Monte Carlo plus thermodynamic integration from that random limit (the E21 procedure), ~30 s per composition.
3. Calibrated fast path — run tier 2 on a training set of compositions, regress the corrections (energy lowering dE and entropy ratio S/S_ideal) on the ideal entropy, then apply those corrections to tier 1.

Training compositions were drawn by Latin hypercube over the composition simplex, not multinomially: a uniform draw on the simplex concentrates near the equiatomic centroid and would leave the dilute corners unsampled. 20 training compositions at 1500 K, 54 sites; 6 held-out compositions for the test.

Result. Both corrections are close to linear in S_ideal:

dE       = -94.54 * S_ideal + 5.14 meV/atom   (residual scatter 3.15)
S/S_ideal = -88.75 * S_ideal + 0.9933

MAE 1.60 meV/atom, max 3.12, against a reference that is itself only as good as the CE (cross-validated RMSE 5.67) and the potential under it (6.61 against DFT). The approximation is not the limiting error anywhere in the chain.

Speed. 2.76 ms per composition — 362 compositions/s, against ~0.03/s for the full integration. Four orders of magnitude, for an error a quarter of the CE's own.

Why the linearity holds. S_ideal measures how much the composition is mixed at all. A nearly-pure composition has little to order and little entropy to lose; an equiatomic one has the most of both. The regression is picking up that single degree of freedom, which is why 20 training points suffice — and why the residual scatter (3.15 meV/atom) is the honest uncertainty on any one composition, not the fit quality.

Caveats. Calibrated at one temperature (1500 K) and one cell size (54 sites); both would need re-fitting elsewhere. The correction is fitted across the whole simplex, so a composition with unusual ordering chemistry — a strongly ordering binary pair in a dilute matrix — is exactly where it would be worst. That is the case to check before trusting a top-ranked candidate, and it is cheap to check: rerun tier 2 on the winners.