Abstract
The Airy distribution (AD) describes the probability distribution of the area under a Brownian excursion. The AD is prominent in several areas of physics, mathematics and computer science. Here we use a dilute colloidal system to directly measure, for the first time, the AD in experiment. We also show how two different techniques of theory of large deviations - the Donsker-Varadhan formalism and the optimal fluctuation method - manifest themselves in the AD. We advance the theory of the AD by calculating, at large and small areas, the position distribution of a Brownian excursion conditioned on a given area, and measure its mean in the experiment. For large areas, we uncover two singularities in the large deviation function, which can be interpreted as dynamical phase transitions of third order. For small areas the position distribution coincides with the Ferrari-Spohn distribution, and we identify the reason for this coincidence.
Highlights
Brownian motion came to prominence in physics and other sciences with the theoretical works of Einstein [1], von Smoluchowski [2], and Langevin [3] and the experimental work of Perrin [4]
We show how two different techniques of theory of large deviations, the Donsker-Varadhan formalism and the optimal fluctuation method, manifest themselves in the Airy distribution (AD)
We advance the theory of the AD by calculating, at large and small areas, the position distribution of a Brownian excursion conditioned on a given area and measure its mean in the experiment
Summary
The full position distribution of excursions, conditioned on a given area, requires a considerably larger number of trajectories. For the small-area simulations, in the window A = 0.3 ± 0.0075, we used 1885 trajectories These were extracted from a total of 2 × 106 unconditioned Brownian excursions. We verified the simulation method by measuring the area distribution of the excursions and comparing the results with the theoretical predictions (2) and (3).
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