$p_\lambda$, $Z_\lambda$, and $F_\lambda$ on temperature will be left implicit, and equilibrium states will be identified by the parameter value $\lambda$.

To perform the forward process ($F$), we first prepare the system in the equilibrium state $A$, then we remove the heat reservoir. Then, from $t=0$ to $t=\tau$ we let the system evolve under Hamilton’s equations as we vary $\lambda$ from $A$ to $B$ according to a pre-determined schedule, $\lambda_t^F$. Let $H_t^F(\Gamma)=H(\Gamma;\lambda_t^F)$ denote the time-dependent Hamiltonian obtained when $\lambda$ is varied in this manner, and let $\gamma^F = [\Gamma_t^F]_0^\tau$ denote a phase space trajectory evolving under this Hamiltonian. The notation indicates that the trajectory $\gamma^F$ passes through the set of points $\Gamma_t^F$, for $0\le t\le\tau$. Such a trajectory describes the microscopic history of the system during a single realization of the forward process.

By repeating this process infinitely many times, we generate a sequence of trajectories, $\{\gamma_1^F,\gamma_2^F,\cdots\}$. These can be viewed as random samples from a probability distribution ${\cal P}^F[\gamma^F]$, defined on the set of all possible trajectories generated by $H_t^F$. Because the dynamics are deterministic, the probability of observing a given trajectory is simply that of sampling its initial conditions from a canonical distribution [@Sun03]: $$\label{eq:pa}
{\cal P}^F[\gamma^F] = 
