Abstract
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.
- Received 4 June 2020
- Revised 11 May 2021
- Accepted 18 May 2021
DOI:https://doi.org/10.1103/PhysRevX.11.031022
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
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Highlights in Experimental Statistical, Biological, and Soft-Matter Physics
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Popular Summary
Any experimentally observed trajectory of a diffusing particle is just one realization of the innumerable possible trajectories that could occur. While it is possible to mathematically quantify the relative likelihood for two such trajectories, these predictions cannot be tested directly because, strictly speaking, the probability of any single trajectory is zero. We provide the missing link between theory and experiment, by establishing a protocol to extract ratios of path probabilities from measured time series.
In our experiments, we observe the motion of a single colloidal particle in a microchannel, in which we create a double-well potential-energy landscape using modulated laser light. From time series of the particle position, recorded 1000 times per second, we infer the rate at which the particle first exits a notional finite-radius tube surrounding a given reference path. For a pair of reference paths, we then obtain the corresponding ratio of path probabilities by extrapolating measured finite-radius exit-rate differences to the limit of zero radius. Using this protocol, we directly determine the most probable path for a barrier crossing of the colloidal particle.
Our results open a route to experimentally addressing questions related to individual stochastic trajectories and provide a link, until now missing, between mathematically defined path probabilities and experimentally measured trajectories. Apart from their wide applicability, our findings are particularly relevant to stochastic thermodynamics and the study of irreversibility based on single trajectories.
