Estimating Free Energy Differences with Virtually Escorted Trajectories

For a process in which a system is driven irreversibly from equilibrium state $A$ toward equilibrium state $B$, the free energy difference $\Delta F = F_B-F_A$ can be estimated using the work fluctuation theorem $\langle e^{-W/T}\rangle = e^{-\Delta F/T}$, where $W$ and $T$ denote work and temperature. The estimate often suffers from poor convergence with the number of trajectories used to calculate the average. Borrowing ideas from escorted free energy estimation, and from diffusion models of machine learning, we show how to construct infinitely many work-like quantities, $W_\theta$, that satisfy $\langle e^{-W_\theta/T}\rangle = e^{-\Delta F/T}$, for the same underlying dynamics. Our method involves a virtual control field ${\boldsymbol u}_\theta$ that does not modify these dynamics. We show how to choose parameter values $\theta$ to optimize convergence of the free energy estimate, for a fixed set of trajectories. We identify conditions under which our method provides a zero-variance estimator of $\Delta F$. We use numerical simulations of model systems to illustrate the gains in convergence that our method can achieve.