Rare events and the convergence of exponentially averaged work values

TL;DR

This paper explores the challenges in converging exponential averages of work values for free energy calculations, introducing a duality framework between forward and reverse processes to better understand rare event contributions.

Contribution

It provides a simple duality-based description of rare realizations affecting convergence, offering heuristic and quantitative insights for free energy estimation methods.

Findings

Duality between forward and reverse processes explains rare event dominance.

Quantitative estimates for the number of realizations needed for convergence.

Application of results to equilibrium perturbation methods.

Abstract

Equilibrium free energy differences are given by exponential averages of nonequilibrium work values; such averages, however, often converge poorly, as they are dominated by rare realizations. I show that there is a simple and intuitively appealing description of these rare but dominant realizations. This description is expressed as a duality between ``forward'' and ``reverse'' processes, and provides both heuristic insights and quantitative estimates regarding the number of realizations needed for convergence of the exponential average. Analogous results apply to the equilibrium perturbation method of estimating free energy differences. The pedagogical example of a piston and gas [R.C. Lua and A.Y. Grosberg, J. Phys. Chem. B vol. 109, p. 6805 (2005)] is used to illustrate the general discussion.