n_p/2$ (during compression we almost always observe roughly $n_p/2$ collisions, Fig. \[fig:piston\_Rtyp\_Fdom\]a), it follows that the greatest contribution to $\langle e^{-\beta W}\rangle_F$ comes from realizations of the expansion process for which $n \approx n_p/2$ (Fig. \[fig:piston\_Rtyp\_Fdom\]b).

Number of realizations needed for convergence {#sec:numbers}
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When using Eq.\[eq:estDF\] to evaluate $\Delta F$, how many realizations do we need to obtain a reasonable estimate? While a precise answer depends on the desired accuracy, a back-of-the-envelope estimate can be derived as follows.

Imagine that we repeatedly carry out the forward process, either in a laboratory experiment or using numerical simulations. In doing so we generate trajectories $\{\gamma_1^F, \gamma_2^F, \cdots \}$ sampled from ${\cal P}^F[\gamma^F]$. Since the most important contribution to $\langle e^{-\beta W}\rangle_F$ comes from the region $\zeta_{\rm dom}^F$, we must sample this region to get a decent estimate of the average. The probability that a single randomly sampled trajectory falls within $\zeta_{\rm dom}^F$ is $$\begin{aligned}
P &=&
\int_{\zeta_{\rm dom}^F} d\gamma^F\, {\cal P}^F[\gamma^F] =
\int_{\zeta_{\rm typ}^R} d\gamma^R\, {\cal P}^R[\gamma^R] \, \exp\Bigl(-\beta W_d^R[\gamma^R]\Bigr) \\
&\sim& \exp\Bigl(-\beta \overline W_d^R\Bigr) \int_{\zeta_{\rm typ}^R} d\gamma^R\, {\cal P}^R[\gamma^R]
\sim \exp\Bigl(-\beta \overline W_d^R\Bigr).\end{aligned}$$ Here $\overline W_d^R$ is the average work dissipated when performing the reverse process. On the first line 