Fix vibrational/rotational energy sampling cutoff for many DOF (#64)

[stanmoore1]

Summary

Fixes sparta#64 ("Random vibrational energy generation").

Particle::evib() (and the structurally identical Particle::erot()) sample a particle's internal energy from the equilibrium distribution

P(x) ∝ x^a · exp(−x),   x = E/kT,   a = dof/2 − 1

using acceptance–rejection. The acceptance function b = (x/a)^a · exp(a−x) is correct, but the candidate energies were drawn uniformly on a fixed range [0, 10 kT]:

erm = 10.0*erandom->uniform();

This distribution is Gamma(shape = a+1), with mean = a+1 = dof/2. The fixed 10 kT ceiling is only safe while the mean sits well below 10. As the number of degrees of freedom grows, the cutoff falls below the mean of the distribution, so the sampler can never propose the energies where most of the probability mass lives. The result is vibrational energy that is biased low and whose distribution shape is destroyed (detailed balance is violated). For SF₆-class polyatomics (~30 vibrational DOF) the bias is severe.

This affects every code path that initializes vibrational energy at a temperature — inflow/emit boundaries (fix_emit_*) and diffuse/CLL/TD surface models (surf_collide_*).

Fix

Scale the candidate range with the number of degrees of freedom so it always covers the high-energy tail. The mode is at x = a, the mean at a+1, the std dev √(a+1); using mean + ~9 std devs:

double xmax = a + 1.0 + 9.0*sqrt(a+1.0);
erm = xmax*erandom->uniform();

This keeps the rejection efficiency roughly constant (~18–20%) instead of letting it collapse, and reduces to the original behavior for small molecules. The rotational path is bounded (rotdof ≤ 3, so a ≤ 0.5), meaning the bug was latent there, but the same fix is applied for consistency.

Changed file: src/particle.cpp (evib() and erot()).

Verification

A standalone Monte-Carlo test (verification/issue64/verify_evib.cpp) reproduces the old and new samplers verbatim, drawing 2×10⁶ samples per case, and compares the sampled mean and variance of E/kT against the analytic values (mean = variance = vibdof/2).

vibdof	OLD mean (err)	OLD var	NEW mean (err)	NEW var	theory	OLD eff	NEW eff
4	2.00 (−0.07%)	1.97	2.00 (−0.05%)	1.99	2.0	27%	18%
8	3.92 (−1.9%)	3.46	4.00 (+0.07%)	4.00	4.0	44%	20%
16	6.84 (−14.5%)	3.20	8.00 (−0.04%)	7.99	8.0	52%	20%
24	8.25 (−31.3%)	1.68	12.00 (−0.04%)	11.99	12.0	25%	19%
30	8.76 (−41.6%)	1.02	15.00 (−0.03%)	15.00	15.0	8%	19%
40	9.20 (−54.0%)	0.51	20.00 (−0.02%)	20.01	20.0	0.4%	18%

Findings:

1. The bug is real and severe — the old mean collapses (−42% at SF₆-class vibdof=30, −54% at vibdof=40) and asymptotes toward ~9–10 because returned energies are clamped to [0, 10 kT]. The variance collapses even harder, confirming the shape is wrong, not just the mean.
2. The fix recovers both moments to within ~0.07% of theory across the whole range, so the full distribution is restored.
3. Efficiency is improved at high DOF — the old sampler's acceptance rate drops to 0.4% at vibdof=40 (its [0,10] window rarely lands near the mode at a=19), whereas the new sampler holds ~18–20% throughout.
4. Small molecules are unchanged (vibdof = 4, 8), confirming the fix doesn't perturb cases that already worked.

The analytic percent-error curves agree exactly with the Monte-Carlo points:

The verification/issue64/ directory (test, plot script, figure) is verification-only and can be dropped before merge if preferred.

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