Machine Learning the Entropy to Estimate Free Energy Differences without Sampling Transitions

TL;DR

This work addresses the challenge of estimating free-energy differences between metastable phases separated by high barriers without sampling transitions. It introduces MICE, a multi-scale entropy estimation framework that writes entropy as a sum of mutual-information contributions across subdivided volumes and uses the Mutual Information Neural Estimator (MINE) to learn these contributions from short, phase-separated MD runs. With area-law extrapolation to bulk, MICE yields accurate entropy differences and, combined with enthalpy, accurate melting temperature predictions for Na and Al, outperforming entropy estimates based on $s_2$ and mitigating biases in standard metadynamics. The approach broadens the toolkit for phase-stability calculations by eliminating the need to identify slow collective variables or sample transition pathways, and it holds promise for other systems with large free-energy barriers.

Abstract

Thermodynamic phase transitions, a central concept in physics and chemistry, are typically controlled by an interplay of enthalpic and entropic contributions. In most cases, the estimation of the enthalpy in simulations is straightforward but evaluating the entropy is notoriously hard. As a result, it is common to induce transitions between the metastable states and estimate their relative occupancies, from which the free energy difference can be inferred. However, for systems with large free energy barriers, sampling these transitions is a significant computational challenge. Dedicated enhanced sampling algorithms require significant prior knowledge of the slow modes governing the transition, which is typically unavailable. We present an alternative approach, which only uses short simulations of each phase separately. We achieve this by employing a recently developed deep learning model for estimating the entropy and hence the free energy of each metastable state. We benchmark our approach calculating the free energies of crystalline and liquid metals. Our method features state-of-the-art precision in estimating the melting transition temperature in Na and Al without requiring any prior information or simulation of the transition pathway itself.

Figures (4)

• Figure 1: MI estimations for liquid and crystalline Na (left) Training curves for solid (blue, upper plot) and liquid (orange, bottom plot) Na close to $T_m$. Bold lines represent running averages over $5000$ steps and dashed line are the converged values over last $10^4$ steps. (middle) MI estimation for a system of fixed physical size as a function of the spatial resolution. The estimate plateaus at high resolutions. (right) Estimation of MI between two halves of a cubic subsystem, as a function of the interface area. The dashed lines show a linear trendline, showing that large systems obey an area law. The corresponding figures for Al data are presented in \ref{['sec:aluminum']}.
• Figure 2: MI density across successive partitions. Starting from the rightmost panel, the system is halved along a different dimension at each step, and MI is computed across the resulting interface. Because the three cut directions cycle, the interface area is unchanged every third partition while the subsystem volume halves at every step (e.g., iterations 2-3). Consequently, the MI density doubles between such iterations.
• Figure 3: (a-b) Gibbs free energy of melting vs temperature. The data for classical MD and MICE are generated from sampled $\Delta h$ and the entropy estimates $\Delta s_2$ and $\Delta s_{MICE}$ correspondingly. WT-MetaD data is shown by the black points, and the linear fit through them produces estimates of $\Delta h$ and $\Delta s$. (c-d) $\Delta h$ and $T_m\Delta s$ calculated from classical MD, WT-MetaD and MICE (enthalpy estimations for MICE and MD are identical by construction).
• Figure 4: MI estimations for liquid and crystalline Aluminum (top) MI estimation for a system with a fixed physical size as a function of the spatial resolution. The estimate plateaus at high resolutions. (middle) Estimation of MI between two halves of a cubic subsystem, as a function of the interface area. The dashed lines show a linear trendline, showing that large systems obey an area law. (bottom) MI density across successive partitions. Starting from the rightmost panel, the system is halved along a different dimension at each step, and MI is computed across the resulting interface.