From Potentials to Ensembles: Boltzmann Sampling from Learned Energy Surfaces via Stochastic Optimal Control

Equilibrium ensembles governed by the Boltzmann distribution are fundamental to understanding the thermodynamics of ligand binding and conformational landscapes. While generative models have proven effective at sampling molecular configurations, they do not inherently recover the underlying physical energy landscape. Bridging this gap between empirical density estimation and physical thermodynamics remains an open challenge. We introduce *Entropica*, a training framework that guides flow- and diffusion-based molecular generators toward equilibrium distributions determined by physical energies and temperature. *Entropica* identifies the optimal transport path over cumulative informational and physical cost and anchors the predicted energy to the potential energy surface (PES). Supervision therefore comes from energies and forces rather than from sample density alone. We first validate the framework on a two-dimensional prototype system: trained on its PES alone, without access to sample data points, the model recovers both the terminal energy landscape and the Boltzmann distribution it defines, reproducing the reference data. Trained on molecules and density functional theory obtained energies and forces, *Entropica* recovers the PES more accurately than a classical force field, while simultaneously learning to sample the associated equilibrium ensemble. Generated conformations lie in low-energy regions of the learned surface and the sampled density matches the Boltzmann statistics of molecular dynamics ground truth. *Entropica* opens a route toward physics based molecular generation via PES guidance, with potential applications from equilibrium ensemble generation to scoring of molecular interactions.