when $\overline W_d^F > \overline W_d^R$: the mean of $\rho^F$ is displaced farther to the right of $\Delta F$ than the mean of $\rho^R$ is to the left of $\Delta F$, which in turn suggests that $\rho^F$ is wider than $\rho^R$, since the two distributions cross exactly at $W=\Delta F$ [@Collin05]. Thus, as measured in standard deviations, we must reach deeper into the tail of $\rho^R$ to sample the peak region of $\rho^F$, than the other way around. Hence $N_c^R > N_c^F$ when $\overline W_d^F > \overline W_d^R$. This prediction agrees well with recent numerical simulations of an asymmetric object dragged through a hard-disk gas [@Cleuren05].

The piston-and-gas example provides a nice illustration of Eq. \[eq:Nc\]. From Section \[sec:summary\] we have $\overline W^F \approx 0$, $\overline W^R\approx n_pmu^2$, and $\Delta F = -n_p\beta^{-1}\ln 2$, thus $$\label{eq:wd_gas}
\overline W_d^F \approx n_p\beta^{-1}\ln 2
\qquad,\qquad
\overline W_d^R \approx n_p(mu^2 - \beta^{-1}\ln 2).$$ Hence $\overline W_d^R \gg \overline W_d^F$ in the fast piston limit (Eq. \[eq:fastPiston\]); rapid compression is much more dissipative than rapid expansion.

Let us now analyze the convergence of $\langle e^{-\beta W}\rangle_R$. The dominant realizations are those for which essentially all the gas particles begin in the left half of the 