How Do Equilibrium Samples Become Free Energies?

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 observable-estimator discussion by asking how equilibrium samples become free-energy estimates once a collective variable, normalization, binning rule, and uncertainty model are chosen. Corrections and replication issues should be tracked in sungsoo-ahn/kups-md-tutorials.

Introduction

Free energy is not read directly from a trajectory. It is inferred from a probability distribution over a chosen coordinate. That coordinate may be a distance, an angle, a density, a coordination number, or a learned collective variable, but the estimator changes with the choice.

For ML researchers working with MLIPs, the important shift is from frames to probability. A histogram can be converted into a potential of mean force, but only after asking what was sampled, which bins were empty, how much smoothing or binning bias was introduced, and what uncertainty should accompany the derived barrier.

This draft demonstrates the executable slice of the eighth tutorial with a controlled double-well distribution and a synthetic RDF-derived PMF. It is a small diagnostic for the estimator mechanics before the final argon/kUPS free-energy observable is added.

• smoke configuration
• full configuration
• free-energy notebook
• smoke summary
• full summary
• full provenance manifest
• self-review note

What Is The Free Energy Of?

The current diagnostic keeps the answer key available:

The true double-well barrier is 1.0, so the diagnostic can separate estimator error from the physical free-energy definition.

What Should The Diagnostic Show?

The full run checks three estimator questions. The first panel compares the true PMF, a direct histogram PMF, and a reweighted PMF. The second panel shows that bin width changes the estimated barrier even for equilibrium samples. The third panel shows how an RDF-like (g(r)) can become a shifted PMF through -kT log g(r).

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 08 --profile smoke
uv run kups-tutorial verify 08 --profile smoke
uv run kups-tutorial run 08 --profile full
uv run kups-tutorial verify 08 --profile full
uv run jupyter execute notebooks/post-08-free-energies.ipynb --inplace

The notebook is deliberately not the implementation source. It imports the configuration loader, free-energy 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 free-energy workflows
• committed compact PMF curve and summary 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 article prose
• citations for PMFs, histogram estimators, reweighting, and RDF-derived potentials of mean force
• rendered desktop and mobile page snapshots
• argon/kUPS RDF-derived PMF diagnostics linked back to post 07 observables

The rule for this post is that free energy is a property of an estimator over a chosen coordinate. Changing the coordinate, bins, weights, or sampled support changes what can be claimed.

References

• Frenkel, D. & Smit, B. (2001). Understanding Molecular Simulation: From Algorithms to Applications. Academic Press.
• Tuckerman, M. E. (2010). Statistical Mechanics: Theory and Molecular Simulation. Oxford University Press.
• 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.