  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.
