In recent years, non-equilibrium (NEQ) approaches have attracted increasing attention [1-3]. Rather than spending computational resources on sampling intermediate states, non-equilibrium methods focus on the end states given a fixed NEQ protocol. In theory, a locally optimal NEQ protocol with the minimal entropy production can be achieved by setting the switching speed proportional to the inverse of the square root of the alchemical friction [4]. In practice, however, when the alchemical friction differs in the forward and backward transition due to a locally non-positive potential curvature, only a sub-optimal protocol can be found to minimise the average entropy production. 

In this study, we demonstrate that enveloping distribution sampling (EDS) [5] is a potential alternative to the commonly used linear interpolation (LI) potential. The behaviour of the EDS potential is characterised by the smoothing parameter s that modulates the local curvature of the potential energy surface and recovers the LI potential in the limit of s→0. Using a range of systems – from simple harmonic traps and anti-traps to particle insertions and dipole inversion – we demonstrate that the parameter s serves as a promising tuning variable for optimizing NEQ protocols in real-world alchemical transformations.

[1] Wan, S.; Bhati, A. P.; Coveney, P. V. Comparison of Equilibrium and Nonequilibrium Approaches for Relative Binding Free Energy Predictions. J. Chem. Theory Comput. 2023, 19, 7846– 7860,  

[2] Bhati, A. P.; Wan, S.; Coveney, P. V. Equilibrium and nonequilibrium ensemble methods for accurate, precise and reproducible absolute binding free energy calculations. J. Chem. Theory Comput. 2025, 21, 440– 462, 

[3] Albergo, M. S. and Vanden-Eijnden, E. Nets: A non-equilibrium transport sampler. arXiv preprint arXiv:2410.02711, 2024.

[4] D. A. Sivak and G. E. Crooks, Thermodynamic metrics and optimal paths, Phys. Rev. Lett. 108, 190602 2012.

[5] Christ, C. D.; van Gunsteren, W. F. Enveloping distribution sampling: A method to calculate free energy differences from a single simulation. J. Chem. Phys. 2007, 126, 184110,