The concept of velocity in the history of Brownian motion From physics to mathematics and vice versa Arthur Genthon1,* 1Gulliver, ESPCI Paris, PSL University, CNRS, 75005 Paris, France *
[email protected] Brownian motion is a complex object shared by different communities: first ob- served by the botanist Robert Brown in 1827, then theorised by physicists in the 1900s, and eventually modelled by mathematicians from the 1920s. Consequently, it is now ambiguously referring to the natural phenomenon but also to the theo- ries accounting for it. There is no published work telling its entire history from its discovery until today, but rather partial histories either from 1827 to Perrin's exper- iments in the late 1900s, from a physicist's point of view; or from the 1920s from a mathematician's point of view. In this article, we tackle a period straddling the two `half-histories' just mentioned, in order to highlight its continuity, to question the relationship between physics and mathematics, and to remove the ambiguities mentioned above. We study the works of Einstein, Smoluchowski, Langevin, Wiener, Ornstein and Uhlenbeck from 1905 to 1934 as well as experimental results, through the concept of Brownian velocity. We show how Brownian motion became a research topic for the mathematician Wiener in the 1920s, why his model was an idealization of physical reality, what Ornstein and Uhlenbeck added to Einstein's results and how Wiener, Ornstein and Uhlenbeck developed in parallel contradictory theories concerning Brownian velocity. 1. Introduction Brownian motion is in the first place a natural phenomenon, observed by the Scottish botanist Robert Brown in 1827.