What Do Free-Energy Estimators Assume?

Created on July 14, 2026 by Sungsoo Ahn , last updated on July 14, 2026

2026   ·   molecular-dynamics   free-energy   estimators   bar   kups   ·   science

Note: This is an early draft page for the executable kUPS MD tutorial series. It is intentionally hidden from site navigation while the simulations, notebooks, figures, and review artifacts mature. This post follows the PMF discussion by asking when free-energy estimators can be trusted, especially when overlap and effective sample size are poor. Corrections and replication issues should be tracked in sungsoo-ahn/kups-md-tutorials.

Introduction

Free-energy perturbation is exact as an identity and fragile as an estimator. The difference is overlap. If samples from state A almost never visit configurations that matter for state B, the exponential average is controlled by rare tail events rather than by the typical trajectory frames.

For ML researchers working with MLIPs, this is the useful mental model: free-energy estimators are diagnostics for probability mass, not just formulas for energies. BAR improves on one-sided exponential averages by using samples from both states, and WHAM/MBAR extend the same overlap logic across multiple biased or intermediate states (Zwanzig, 1954; Bennett, 1976; Kumar et al., 1992; Shirts & Chodera, 2008).

This draft demonstrates the executable slice of the ninth tutorial with exactly solvable Gaussian state pairs. The true free-energy difference is known, so the diagnostic can separate estimator assumptions from physical-model error.

What Is Being Estimated?

The current diagnostic compares two unit-variance one-dimensional states. The second state is displaced and assigned a known free-energy offset:

Case Mean shift True Delta F Intended regime
good_overlap 0.5 0.8 many useful samples in both states
marginal_overlap 1.5 0.8 estimator looks plausible but ESS warns
poor_overlap 3.0 0.8 rare tails dominate one-sided FEP

This is not a production alchemical calculation. It is a controlled test for the failure modes that production calculations must diagnose.

What Should The Diagnostic Show?

The full run compares forward FEP, reverse FEP, and BAR against the known answer. It also records overlap coefficients and exponential-weight effective sample sizes. In the poor-overlap case, the forward effective sample size collapses to less than one percent of the nominal sample count even though the simulation contains fifty thousand samples per state.

Estimator diagnostics for the committed full profile. BAR remains close to the known \(\Delta F\) in this controlled example, while the overlap and ESS panels show why one-sided FEP becomes fragile as state overlap disappears.

Reproduction

The current executable path is:

git clone https://github.com/sungsoo-ahn/kups-md-tutorials
cd kups-md-tutorials
uv sync
uv run kups-tutorial run 09 --profile smoke
uv run kups-tutorial verify 09 --profile smoke
uv run kups-tutorial run 09 --profile full
uv run kups-tutorial verify 09 --profile full
uv run jupyter execute notebooks/post-09-estimators.ipynb --inplace

The notebook is deliberately not the implementation source. It imports the configuration loader, estimator diagnostics, and figure generator from src/kups_md_tutorials/.

Current Status

This page is not the final article. The implemented pieces are:

  • smoke and full controlled estimator workflows
  • committed compact estimator summaries and work-sample outputs
  • executable notebook
  • generated SVG/PNG figure and snapshot review
  • self-review note covering code, science, notebook, and figure feedback

The missing pieces are:

  • final 3,500-10,000-word article prose
  • rendered desktop and mobile page snapshots
  • fuller WHAM/MBAR discussion and any final multi-state diagnostic figures
  • final citation pass and production-style diagnostics

The rule for this post is that estimator reliability is an overlap question. More samples help only when they include the configurations that carry the statistical weight.

References

  • Zwanzig, R. W. (1954). High-temperature equation of state by a perturbation method. The Journal of Chemical Physics, 22, 1420-1426. ↩
  • Bennett, C. H. (1976). Efficient estimation of free energy differences from Monte Carlo data. Journal of Computational Physics, 22, 245-268. ↩
  • Kumar, S., Rosenberg, J. M., Bouzida, D., Swendsen, R. H. & Kollman, P. A. (1992). The weighted histogram analysis method for free-energy calculations on biomolecules. Journal of Computational Chemistry, 13, 1011-1021. ↩
  • Shirts, M. R. & Chodera, J. D. (2008). Statistically optimal analysis of samples from multiple equilibrium states. The Journal of Chemical Physics, 129, 124105. ↩