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.

中文翻译：

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相对路径概率和随机动作的实验测量

对于扩散随机动力学，观察到任何单个轨迹的概率微乎其微，因此不清楚如何通过实验验证路径概率比的理论结果。我们通过建立一个协议来从测量的时间序列中提取路径概率的比率，从而提供理论和实验之间缺失的联系。对于微通道中单个胶体粒子的实验，我们提取了路径概率的比率和障碍穿越的最可能路径，并发现与基于 Onsager-Machlup 随机作用的独立计算预测非常吻合。发现我们在室温下的实验结果与低噪声 Freidlin-Wentzell 随机作用不一致，并且我们讨论在什么情况下后一个动作应该描述最可能的路径。此外，虽然可通过实验获得的路径概率比率是唯一确定的，但已知正式的路径积分作用取决于用于推导它的时间离散化方案；我们通过仔细分析路径积分的时间切片推导来调和这两个看似矛盾的事实。我们的实验协议使我们能够探测路径空间上的概率分布，并允许我们将理论单轨迹结果与测量联系起来。我们通过仔细分析路径积分的时间切片推导来调和这两个看似矛盾的事实。我们的实验协议使我们能够探测路径空间上的概率分布，并允许我们将理论单轨迹结果与测量联系起来。我们通过仔细分析路径积分的时间切片推导来调和这两个看似矛盾的事实。我们的实验协议使我们能够探测路径空间上的概率分布，并允许我们将理论单轨迹结果与测量联系起来。