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This article presents a novel computational method for estimating entropy differences between probability distributions using stochastic interpolants, which are mathematical tools that smoothly connect two distributions. The approach calculates entropy differences through the interaction of probability flow velocity and score fields without requiring computationally expensive divergence calculations, and can produce entropy estimates during the training process. The method was validated on four systems of increasing complexity, from a 40-dimensional Gaussian mixture to an active Brownian polymer, demonstrating its applicability to both equilibrium and non-equilibrium thermodynamic systems.
Why it matters
This technique could significantly reduce computational costs in calculating thermodynamic properties of complex molecular and physical systems. The ability to estimate entropies in non-equilibrium systems opens new possibilities for studying active matter, biological systems, and other far-from-equilibrium phenomena where traditional statistical mechanics approaches are limited.
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⚠️ Preprint – Noch nicht peer-reviewed
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Abstract: We present a general method for estimating entropy differences between arbitrary probability distributions using stochastic interpolants. The marginal distributions which bridge the base and target obey a continuity equation, which allows a straightforward calculation of the entropy difference in terms of an inner product of the probability flow velocity and score fields. This formulation has several advantages: (i) no computationally expensive divergence calculations of either field are required, (ii) the score field need not even be learned directly if model transferability is not required, and (iii) on-the-fly estimates are produced nearly for free during training. When tractable base distributions are chosen (e.g. Gaussian chain, ideal gas, etc.), statistical thermodynamic entropies are then immediately recovered. Notably, the analysis relies only on the definition of the Gibbs-Shannon entropy, rather than any particular statistical mechanical ensemble, so that generalized non-equilibrium entropies may be computed. The method is demonstrated on several systems of increasing complexity: (i) a 40-dimensional Gaussian mixture model, (ii) the classical XY model of N spins arranged in one dimension, (iii) the 13-atom Lennard-Jones cluster, and (iv) an active Brownian polymer.