varying the work parameter during the forward process, from $\lambda_0^F=A$ to $\lambda_\tau^F=B$, then the reverse process is defined by the schedule $$\label{eq:for_rev_schedules}
\lambda_t^R = \lambda_{\tau-t}^F,$$ where $\tau$ is the duration of either process.

Note that the nonequilibrium work theorem applies to both processes: $$\label{eq:both}
\Bigl\langle e^{-\beta W}\Bigr\rangle_F = e^{-\beta\Delta F} \qquad,\qquad
\Bigl\langle e^{-\beta W}\Bigr\rangle_R = e^{+\beta\Delta F} ,$$ where $\langle\cdots\rangle_{F/R}$ denotes an average over realizations of the forward/reverse process; and by convention $\Delta F\equiv F_B-F_A$ in both equations. As with the forward process, it is useful to distinguish between [*typical*]{} realizations of the reverse process, and the [*dominant*]{} realizations that contribute the most to $\langle e^{-\beta W}\rangle_R$.

Throughout this paper, the evolution of the system is modeled as a Hamiltonian trajectory in phase space, where the Hamiltonian is made time-dependent by externally varying the work parameter. (See Section \[sec:discussion\] for a brief discussion of other, e.g. stochastic, models.) In the absence of magnetic fields [^1] – more precisely, under the assumption of time-reversal invariance, Eq.\[eq:tri\] – forward and reverse realizations come in [*conjugate pairs*]{} related by time-reversal, as illustrated in Fig.\[fig:conjugatePair\]: if a phase space trajectory $\Gamma_t^F$ represents a possible realization of the forward process (a solution of Hamilton’s equations 