we have used Eqs.\[eq:ratio\] and \[eq:central1\], and on the second we have used the fact that the trajectories $\gamma^R\in\zeta_{\rm typ}^R$ constitute most of the probability distribution ${\cal P}^R$. Thus an estimate of the number of realizations needed to obtain a single trajectory in $\zeta_{\rm dom}^F$ is:

\[eq:Nc\] $$\label{eq:NcF}
N_c^F = P^{-1} \sim \exp\Bigl(\beta \overline W_d^R\Bigr).$$ The same argument gives us the expected number of reverse realizations needed to obtain a trajectory in $\zeta_{\rm dom}^R$: $$\label{eq:NcR}
N_c^R \sim \exp\Bigl(\beta \overline W_d^F\Bigr).$$

These results suggest that the number of realizations required for convergence grows exponentially in the average dissipated work, in agreement with the findings of Gore [*et al.*]{} [@Gore03], and therefore exponentially with system size (assuming dissipated work is an extensive property), as concluded by Lua and Grosberg [@LuaGrosberg05]. Interestingly, however, it is the average amount of work dissipated during the [*reverse*]{} process that determines the convergence of $\langle e^{-\beta W}\rangle$ for the [*forward*]{} process (Eq.\[eq:NcF\]), and vice-versa (Eq.\[eq:NcR\]). This implies that, of the two processes, the more dissipative one is the one for which $\langle e^{-\beta W}\rangle$ converges more rapidly. We can understand this counterintuitive conclusion with the following plausibility argument. Fig. \[fig:crossing\] depicts the work distributions $\rho^F(W)$ and $\rho^R(-W)$, 