when $\lambda$ is varied from $A$ to $B$), then its conjugate twin, $$\label{eq:conjPair}
\Gamma_t^R = \Gamma_{\tau-t}^{F*},$$ represents a possible realization of the reverse process. The asterisk denotes a reversal of momenta: $({\bf q},{\bf p})^* = ({\bf q},-{\bf p})$. The trajectory $\Gamma_t^R$ depicts the sequence of events that we would observe, if we were to film the forward realization $\Gamma_t^F$ and then run the movie backward.

The central result of this paper, Eq. \[eq:central\] below, states that the [*dominant*]{} realizations of the forward process are the conjugate twins of [*typical*]{} realizations of the reverse process, and vice versa. Thus, the trajectories that contribute the most to $\langle e^{-\beta W}\rangle_F$, are those during which the behavior of the system appears [*as though we had filmed a typical realization of the reverse process, and then run the movie backward*]{}. The existence of such a duality was anticipated by Ritort, who observed that for large systems the work performed during a dominant realization of one process is (minus) the work performed during a typical realization of the conjugate process; see comments following Eq. 58 of Ref. [@Ritort04].

As an illustration of this result, consider an ideal gas of (mutually non-interacting) point particles inside a 