Abstract
In computational thermodynamics, a sequence of intermediate states is commonly introduced to connect two equilibrium states. We consider two cases where the choice of intermediate states is particularly important: minimizing statistical error in free-energy difference calculations and maximizing average acceptance probabilities in replica-exchange simulations. We derive bounds for these quantities in terms of the thermodynamic distance between the intermediates, and show that in both cases the intermediates should be chosen as equidistant points along a geodesic connecting the end states.
Highlights
INTRODUCTIONMany computational thermodynamics applications involve the sampling of each of a sequence of ensembles lying between two equilibrium states
We derive bounds for these quantities in terms of the thermodynamic distance between the intermediates, and show that in both cases the intermediates should be chosen as equidistant points along a geodesic connecting the end states
Many computational thermodynamics applications involve the sampling of each of a sequence of ensembles lying between two equilibrium states
Summary
Many computational thermodynamics applications involve the sampling of each of a sequence of ensembles lying between two equilibrium states. The Bennett acceptance ratio method provides an asymptotically efficient estimator for ⌬F that uses equilibrium samples at the two states2,3͔, but converges slowly unless the two states overlap significantly in phase space. Previous studies have suggested choosing intermediates so that the free-energy difference4͔ or the entropy difference5͔ between any two adjacent states is approximately equal; other heuristics have been proposed as well6,7͔ These choices do not minimize the total variance. We show here that geodesics play an important role in selecting the intermediate states used in free-energy calculations and RE simulations: the choice of equidistant intermediates along the geodesic connecting the end states both minimizes the variance in the calculated free energyTheorem 1͒ and provides an “almost optimal” RE scheduleTheorem 2͒
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