two sets to be related by conjugate pairing: $S^F$ is an arbitrary region in the upper box of Fig. \[fig:pathSpace\], and $S^R$ is its mirror image in the lower box. Using Eq.\[eq:qFpR.qRpF\] we then get $$\label{eq:central_gen}
\Psi^F\{S^F\} = \Omega^R\{S^R\}.$$ In words: the relative contribution of a set of forward trajectories to $\langle e^{-\beta W}\rangle_F$, is equal to the statistical weight of the conjugate set of reverse trajectories. Of course, the converse is true as well: $\Psi^R\{S^R\} = \Omega^F\{S^F\}$.

Eq. \[eq:central\] is a special case of Eq.\[eq:central\_gen\]: if $\zeta_{\rm typ}^R$ contains 95% of the statistical weight of all reverse trajectories (a reasonable definition of the peak region of ${\cal P}^R$), then the conjugate set of forward trajectories provides 95% of the contribution to $\langle e^{-\beta W}\rangle_F$.

As an illustration of Eq. \[eq:central\_gen\] using the piston-and-gas example, consider the set ${\cal S}_n^F$ of all forward realization for which there are exactly $n$ piston-particle collisions, and the conjugate set ${\cal S}_n^R$ of reverse trajectories. Then $$\label{eq:central_gen.example}
\Psi^F\{S_n^F\} = \Omega^R\{S_n^R\},$$ i.e. the relative contribution of $n$-collision realizations to $\langle e^{-\beta W}\rangle_F$, is exactly the probability of observing $n$ collisions when performing the reverse process, for any $n\ge 0$. Since $\Omega^R\{S_n^R\}$ is peaked around $n\approx 