to deal with [*dissipated*]{} work values

\[eq:defWd\] $$\begin{aligned}
W_d^F[\gamma^F] &\equiv& W^F[\gamma^F] - \Delta F \\
W_d^R[\gamma^R] &\equiv& W^R[\gamma^R] + \Delta F = - W_d^F[\gamma^F].\end{aligned}$$

(The term “dissipated work” here means the amount by which the work $W$ exceeds that which would have been performed, had the process been carried out reversibly [*and isothermally*]{}.) In terms of these quantities, Eq. \[eq:both\] becomes $$\label{eq:both_diss}
\Bigl\langle e^{-\beta W_d}\Bigr\rangle_F =  1
\qquad,\qquad
\Bigl\langle e^{-\beta W_d}\Bigr\rangle_R = 1 .$$

We now have the elements in place to investigate the nature of those realizations that dominate the average $\langle e^{-\beta W}\rangle_F$, or, equivalently, $\langle e^{-\beta W_d}\rangle_F$. Combining Eqs.\[eq:conjPair\], \[eq:tri\], \[eq:fdef\]-\[eq:pb\], we obtain a simple relationship between the probability of observing a trajectory $\gamma^F$ during the forward process, and that of observing its twin $\gamma^R$ during the reverse: $$\label{eq:ratio}
\frac{{\cal P}^F[\gamma^F]}{{\cal P}^R[\gamma^R]} =
\exp\Bigl( \beta W_d^F[\gamma^F] \Bigr) =
\exp\Bigl( -\beta W_d^R[\gamma^R] \Bigr).$$ This result, originally obtained by Crooks for stochastic, Markovian dynamics [@Crooks98], has here been derived in the context of Hamiltonian evolution.

(In Eqs.\[eq:pa\] and \[eq:pb\], as in Ref. [@Sun03], the Liouville measure on initial conditions in phase space has implicitly been used to define a measure on the space of trajectories: the “volume” of trajectory 