Is the usual assumption of perfectly random mixing good enough for the entropy?

EXPERIMENTS.md · lines 919–973

E21 — Configurational entropy computed, not assumed

Date 2026-09-12 · Question Replace the ideal-mixing entropy that makes the 1500 K ranking a count of components. · Provenance …/ce_mc.py, icet + mchammer canonical Monte Carlo on the 8-element CE

Method. Canonical Monte Carlo (atom swaps, fixed composition) on the cluster expansion of E16b, MoNbTaW at 128 sites, down a temperature ladder from 20,000 K to 300 K, annealing continuously. Free energy by thermodynamic integration from the random limit:

beta*F(beta) = -S_ideal/k_B + integral_0^beta <E> dbeta'

At beta = 0 the alloy is random and F = -T*S_ideal exactly, so the integration starts from a known point rather than a fitted one. <E> at beta = 0 measured directly by sampling 400 random occupancies: -135.961 meV/atom, sd 4.234.

Result.

Entropies in meV/K/atom. Ideal value 0.11946 = k_B ln 4, confirmed independently.

Validation. At 20,000 K the integration returns S/S_ideal = 1.000. That is the known limit the integration begins from, and recovering it is the method's internal correctness check.

Two findings at the operating point.

1. At 1500 K the ideal assumption overstates entropy by 3.4%. In energy that is T*dS = 1500 * 0.0034 * 0.11946 = about 6 meV/atom - the same size as the 8-20 meV gaps the project ranks alloys by. Not negligible.
2. Short-range order lowers the energy by 12.5 meV/atom at 1500 K, from -137.5 random to -150.0 equilibrated. That is larger than the composition differences being ranked. How the atoms arrange themselves matters more than which of these alloys was chosen.

At 300 K entropy falls to 50.8% of ideal: the ideal-mixing approximation fails outright in that regime.

Caveats. One composition, one supercell size (128 sites), one CE. Finite-size effects on the ordering transition are not assessed, and the temperature ladder is coarse below 700 K where the entropy is changing fastest. The CE is fitted to MACE-MPA-0, so this is the potential's thermodynamics, inheriting its 6.61 meV/atom error against DFT.