---
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 \[R.C. Lua and A.Y. Grosberg, J. Phys. Chem. B [**109**]{}, 6805 (2005)\] is used to illustrate the general discussion.'
author:
- Christopher Jarzynski
title: Rare events and the convergence of exponentially averaged work values
---

The nonequilibrium work theorem, $$\label{eq:nwt0}
\Bigl\langle e^{-\beta W}\Bigr\rangle = e^{-\beta\Delta F},$$ relates the work performed on a system during a nonequilibrium process, to the free energy difference between two equilibrium states of that system. The angular brackets denote an average over an ensemble of realizations (repetitions) of a thermodynamic process, during which a system evolves in time as a control parameter $\lambda$ is varied from an initial 