Reformulation of Ensemble Averages via Coordinate Mapping

A general framework is established for reformulation of the ensemble averages commonly encountered in statistical mechanics. This "mapped-averaging" scheme allows approximate theoretical results that have been derived from statistical mechanics to be reintroduced into the underlying formalism, yielding new ensemble averages that represent exactly the error in the theory. The result represents a distinct alternative to perturbation theory for methodically employing tractable systems as a starting point for describing complex systems. Molecular simulation is shown to provide one appealing route to exploit this advance. Calculation of the reformulated averages by molecular simulation can proceed without contamination by noise produced by behavior that has already been captured by the approximate theory. Consequently, accurate and precise values of properties can be obtained while using less computational effort, in favorable cases, many orders of magnitude less. The treatment is demonstrated using three examples: (1) calculation of the heat capacity of an embedded-atom model of iron, (2) calculation of the dielectric constant of the Stockmayer model of dipolar molecules, and (3) calculation of the pressure of a Lennard-Jones fluid. It is observed that improvement in computational efficiency is related to the appropriateness of the underlying theory for the condition being simulated; the accuracy of the result is however not impacted by this. The framework opens many avenues for further development, both as a means to improve simulation methodology and as a new basis to develop theories for thermophysical properties.