Experimental observation of fluctuation loops

Fluctuation loops arise in the large deviation theory of stochastic dynamical systems. Displacements from a stable critical point to a small destination box about a point many standard deviations away are rare. When they do occur, they closely follow a most probable outward path. The most probable outcome after reaching the destination box is deterministic relaxation back to the stable critical point. The union of the outward and relaxation paths constitutes a fluctuation loop. For linear stochastic dynamics, fluctuation loops can be constructed by simple averaging. Ensemble averaging over forward histories after reaching the destination box obviously recovers the relaxation segment. Less obviously, averaging over back histories prior to reaching the destination box recovers the outbound segment. The characterization of fluctuation loops by averaging means that they can be recovered from experimentally recorded time series of state variables. We demonstrate this for a simple electrical network of two capacitively coupled, noise driven RC circuits. Even when the destinations boxes are not many standard deviations away from the stable critical point, it is striking that clearly resolved fluctuation loops emerge from very noisy data.