Experimental measurement of relative path probabilities and stochastic actions

TL;DR

This paper develops an experimental protocol to measure ratios of path probabilities in stochastic systems, validating theoretical predictions and revealing discrepancies with low-noise approximations, thus bridging theory and experiment.

Contribution

It introduces a method to extract path probability ratios from experimental data, connecting theoretical stochastic action calculations with measurable quantities.

Findings

Excellent agreement with Onsager-Machlup predictions.

Discrepancies with Freidlin-Wentzell action at room temperature.

Reconciliation of path probability ratios with path-integral formalism.

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 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.