Estimating thermodynamic expectations and free energies in expanded ensemble simulations: systematic variance reduction through conditioning

摘要

Markov chain Monte Carlo methods are primarily used for sampling from a givenprobability distribution and estimating multi-dimensional integrals based onthe information contained in the generated samples. Whenever it is possible,more accurate estimates are obtained by combining Monte Carlo integration andintegration by numerical quadrature along particular coordinates. We show thatthis variance reduction technique, referred to as conditioning in probabilitytheory, can be advantageously implemented in \emph{expanded ensemble}simulations. These simulations aim at estimating thermodynamic expectations asa function of an external parameter that is sampled like an additionalcoordinate. Conditioning therein entails integrating along the externalcoordinate by numerical quadrature. We prove variance reduction with respect toalternative standard estimators and demonstrate the practical efficiency of thetechnique by estimating free energies and characterizing a structural phasetransition between two solid phases.