to poor convergence with $N$. The aim of this paper is develop an understanding of these rare but important realizations. I will argue that there is a simple description of these dominant realizations, which leads to both quantitative estimates and useful heuristic insights regarding the number of realizations needed for convergence of the average of $e^{-\beta W}$.

The organization of this paper is as follows. In Section \[sec:summary\] the central result is summarized, then illustrated using a simple example. Section \[sec:derivation\] contains a derivation of this result. Section \[sec:numbers\] discusses the number of realizations needed for convergence of the exponential average. Section \[sec:fep\] focuses on the free energy perturbation method (a limiting case of Eq.\[eq:nwt0\]), and Section \[sec:discussion\] concludes with a brief discussion.

Summary and illustration of central result {#sec:summary}
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The average in Eq.\[eq:nwt0\] can be written as $$\label{eq:nwt_int}
\Bigl\langle e^{-\beta W}\Bigr\rangle =
\int dW\,\rho(W)\,e^{-\beta W} \equiv \int dW\,g(W) ,$$ where $\rho(W)$ is the ensemble distribution of work values. Fig.\[fig:rho.and.g\] shows a schematic plot of this distribution, and of the integrand in Eq.\[eq:nwt\_int\], $g(W) = \rho(W) \,e^{-\beta W}$. While $\rho$ is peaked near the mean of the distribution, $$\label{eq:wbar}
\overline{W} = \int dW\,\rho(W)\,W,$$ $g$ is peaked around a lower 