Fix the two-component k-eff estimator

[nuclearkevin]

Description

The two-component combined k-eff estimator (derived by Urbatsch in LA-12658-MS) is used as a fallback in OpenMC for the case where the three-component k-eff estimator (also derived by Urbatsch) becomes numerically unstable. This happens in multi-group transport with survival biasing, where the collision and absorption estimators are identical.

This fallback is very rarely used in practice. I have been attempting to apply the two-component estimator to obtain a lower-variance estimation of k-eff with delta tracking (combining k-collision and k-absorption), but have been noticing some incredibly strange behaviour. Notably, the standard deviation of the two component combined estimate was consistently 2-3 orders of magnitude lower than the individual estimators (which is definitely not correct). After consulting with @pshriwise and digging through Urbatsch's original report, I found that there was an error in the two-component estimator in OpenMC where the elements of the matrix $\mathbf{\Sigma}$ were being used instead of the matrix $\mathbf{S}$ ( $\mathbf{\Sigma} = \mathbf{S} / (n - 1)$ ). After correcting this, the standard deviation estimates predicted by the two-component estimator are far more in line with the other estimators and converge to zero at approximately the same rate. The mean value of the two-component combined estimator was unaffected by the error as the (n - 1) factor cancels out in the derivation of the mean.

For more information, see Eq. 36 and Eq. 40 in: https://digital.library.unt.edu/ark:/67531/metadc665644/m2/1/high_res_d/212564.pdf

Checklist

• I have performed a self-review of my own code
• I have run clang-format (version 18) on any C++ source files (if applicable)
• I have followed the style guidelines for Python source files (if applicable)
• I have made corresponding changes to the documentation (if applicable)
• I have added tests that prove my fix is effective or that my feature works (if applicable)

[nuclearkevin]

After talking about this some more with @pshriwise, I'm closing the PR and shuffling it over to #3971 since its next to impossible to test in continuous-energy transport without delta tracking.