p_A(\Gamma_0^F) =
\frac{1}{Z_A}\exp[-\beta H(\Gamma_0^F;A)].$$ Since the system is thermally isolated (not in contact with a heat reservoir) as $\lambda$ is varied from $A$ to $B$, the work performed on the system is equal to the net change in its energy: $$\label{eq:WFdef}
W^F[\gamma^F] = H(\Gamma_\tau^F;B) - H(\Gamma_0^F;A).$$

Similar remarks and notation apply to the reverse process ($R$). The system is prepared in equilibrium state $B$, then $\lambda$ is varied from $B$ to $A$ (Eq. \[eq:for\_rev\_schedules\]), generating a Hamiltonian trajectory $\gamma^R = [\Gamma_t^R]_0^\tau$, with probability $$\label{eq:pb}
{\cal P}^R[\gamma^R] = 
p_B(\Gamma_0^R) =
\frac{1}{Z_B}\exp[-\beta H(\Gamma_0^R;B)].$$ The work performed is $$\label{eq:WRdef}
W^R[\gamma^R] = H(\Gamma_\tau^R;A) - H(\Gamma_0^R;B).$$

As mentioned in Section \[sec:summary\], every forward trajectory $\gamma^F$ has a conjugate twin $\gamma^R$ (Fig. \[fig:conjugatePair\]), that is a solution of Hamilton’s equations when the parameter is varied according to the reverse schedule. In the remainder of this paper, whenever $\gamma^F$ and $\gamma^R$ appear together (e.g. in the same equation or sentence) it will be understood that these two trajectories form such a conjugate pair.

From Eqs.\[eq:tri\], \[eq:WFdef\] and \[eq:WRdef\], we get $$\label{eq:wfr}
W^F[\gamma^F] = -W^R[\gamma^R] \quad;$$ the work during a forward realization is the opposite of that during its conjugate twin. In what follows it will be convenient 