and Grosberg, who obtain $N_c^F \sim \exp(\beta m L^2/\tau^2)$ when $n_p=1$ (see section 4 of Ref. [@LuaGrosberg05]).

These results verify that the more dissipative process (compression) is the one for which the exponential average converges more rapidly. We can understand why this is the case, without performing explicit quantitative estimates. For the compression process, as mentioned, a dominant realization begins with all the particles in the left half of the box, thus $n_p$ particles must simultaneously satisfy a condition that is not so unusual for any given particle. For the expansion process, on the other hand, half the particles must begin with $v_x \approx 2u$, therefore $n_p/2$ particles must simultaneously satisfy a condition that is very unusual for even a single particle (since $u\gg v_{\rm th}$). The latter situation is the less likely by far. Of course, when $n_p \gg 1$, both $N_c^F$ and $N_c^R$ are extremely large [@Gross.condmat0509648].

In practice, the convergence difficulties associated with Eq. \[eq:nwt0\] are mitigated somewhat if we have data for both the forward and the reverse processes. In that case, Bennett’s [*acceptance ratio method*]{} [@Bennett76; @Crooks00; @Shirts03; @Collin05] converges faster than a direct exponential average of either the forward or reverse work values. It 