  Transition paths are rare events occurring when a system, thanks to the
effect of fluctuations, crosses successfully from one stable state to another
by surmounting an energy barrier. Even though they are of great significance in
many mesoscale processes, their direct determination is often challenging due
to their short duration as compared to other relevant time-scales. Here, we
measure the local average velocity along transition paths of a colloidal bead
embedded in a glycerol/water mixture that hops over a barrier separating two
optical potential wells. Owing to the slow dynamics of the bead in this viscous
medium, we can spatially resolve the mean velocity profiles of the transition
paths for distinct potentials, which agree with theoretical predictions of a
model for the motion of a Brownian particle traversing a parabolic barrier.
This allows us to experimentally verify various expressions linking the
behavior of such mean velocities with equilibrium and transition path position
distributions, mean transition-path times and mean escape times from the wells.
We also show that artifacts in the mean velocity profiles arise when reducing
the experimental time resolution, thus highlighting the importance of the
sampling rate in the characterization of the transition path dynamics. Our
results confirm that mean transition path velocity establishes a fundamental
relationship between mean transition path times and equilibrium rates in
thermally activated processes of small-scaled systems.
