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For diffusive stochastic dynamics, the probability to observe any individualtrajectory is vanishingly small, making it unclear how to experimentallyvalidate theoretical results for ratios of path probabilities. We provide themissing link between theory and experiment, by establishing a protocol toextract ratios of path probabilities from measured time series. For experimentson a single colloidal particle in a microchannel, we extract both ratios ofpath probabilities, and the most probable path for a barrier crossing, and findexcellent agreement with independently calculated predictions based on theOnsager-Machlup stochastic action. Our experimental results at room temperatureare found to be inconsistent with the low-noise Freidlin-Wentzell stochasticaction, and we discuss under which circumstances the latter is expected todescribe the most probable path. Furthermore, while the experimentallyaccessible ratio of path probabilities is uniquely determined, the formalpath-integral action is known to depend on the time-discretization scheme usedfor deriving it; we reconcile these two seemingly contradictory facts bycareful analysis of the time-slicing derivation of the path integral. Ourexperimental protocol enables us to probe probability distributions on pathspace, and allows us to relate theoretical single-trajectory results tomeasurement.

Experimental measurement of relative path probabilities and stochastic actions