  For systems in an externally controllable time-dependent potential, the
optimal protocol minimizes the mean work spent in a finite-time transition
between two given equilibrium states. For overdamped dynamics which ignores
inertia effects, the optimal protocol has been found to involve jumps of the
control parameter at the beginning and end of the process. Including the
inertia term, we show that this feature not only persists but that even delta
peak-like changes of the control parameter at both boundaries make the process
optimal. These results are obtained by analyzing two simple paradigmatic cases:
First, a Brownian particle dragged by a harmonic optical trap through a viscous
fluid and, second, a Brownian particle subject to an optical trap with
time-dependent stiffness. These insights could be used to improve free energy
calculations via either thermodynamic integration or "fast growth" methods
using Jarzynski's equality.

Statistical Mechanics;Soft Condensed Matter