Experimental Measurement of Relative Path Probabilities and Stochastic Actions

For diffusive stochastic dynamics, the probability to observe any individual trajectory is vanishingly small, making it unclear how to experimentally validate theoretical results for ratios of path probabilities. We provide the missing link between theory and experiment by establishing a protocol to extract ratios of path probabilities from measured time series. For experiments on a single colloidal particle in a microchannel, we extract both ratios of path probabilities and the most probable path for a barrier crossing, and find excellent agreement with independently calculated predictions based on the Onsager-Machlup stochastic action. Our experimental results at room temperature are found to be inconsistent with the low-noise Freidlin-Wentzell stochastic action, and we discuss under which circumstances the latter action is expected to describe the most probable path. Furthermore, while the experimentally accessible ratio of path probabilities is uniquely determined, the formal path-integral action is known to depend on the time-discretization scheme used for deriving it; we reconcile these two seemingly contradictory facts by careful analysis of the time-slicing derivation of the path integral. Our experimental protocol enables us to probe probability distributions on path space and allows us to relate theoretical single-trajectory results to measurement.

Automated summary

This study establishes a protocol for extracting ratios of path probabilities from measured time series, bridging the gap between theory and experiment. By conducting experiments on a single colloidal particle in a microchannel, both ratios of path probabilities and the most probable path for a barrier crossing were successfully extracted, showing excellent agreement with independently calculated predictions. The experimental results at room temperature were found to be inconsistent with the low-noise Freidlin-Wentzell stochastic action, while the experimentally accessible ratio of path probabilities is uniquely determined.