between “forward” and “reverse”, and Eq. \[eq:central\] is particularly easy to state: the exponential average is dominated by the conjugate twins of typical realizations.

In the analysis leading to Eq. \[eq:central\], it has implicitly been assumed that the distributions ${\cal P}^F[\gamma^F]$ and ${\cal P}^R[\gamma^R]$ are sharply peaked, i.e. that each process is characterized by well-defined “typical behavior”. This assumption is often reasonable for systems with many degrees of freedom (see e.g. Figs. \[fig:piston\_Ftyp\_Rdom\]a, \[fig:piston\_Rtyp\_Fdom\]a). It is useful, however, to formalize and generalize the discussion, so as to avoid reliance on the notion of typicality.

Let $S^F$ denote an arbitrary set of forward trajectories, and define $$\Omega^F\{S^F\} \equiv \int_{S^F} d\gamma^F {\cal P}^F[\gamma^F]
\qquad,\qquad
\Psi^F\{S^F\} \equiv
\int_{S^F} d\gamma^F {\cal Q}^F[\gamma^F] .$$ $\Omega^F$ is the probability of obtaining a trajectory in $S^F$, when carrying out the forward process; I will refer to this as the [*statistical weight*]{} of the set $S^F$ in the ensemble of forward trajectories. In turn, $\Psi^F$ provides a measure of the [*relative contribution*]{} of $S^F$ to the average $\langle e^{-\beta W}\rangle_F$ (see Eq. \[eq:exp\_F.q\]). Note that $\Omega^F=\Psi^F=1$ when $S^F$ includes all forward trajectories. Define analogous quantities $\Omega^R$ and $\Psi^R$ for a set of reverse trajectories $S^R$. Now take these 