The concept of velocity in the history of Brownian motion - From physics to mathematics and back

Interest in Brownian motion is shared by different com-
munities: it was first observed by the botanist
Brown in 1827, then theorised by physicists in the 1900s, and eventu-
ally modelled by mathematicians from the 1920s, while still evolving
as a physical theory. Thus, Brownian motion now refers to both the
natural phenomenon and the theories accounting for it. There
is no 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.
We tackle the period straddling the two ‘half-histories’ just mentioned, in order
to highlight continuity and to investigate the domain-shift from physics to
mathematics. We study the works of Einstein, Smoluchowski, Langevin, Wiener,
Ornstein and Uhlenbeck from 1905 to 1934, using the concept of Brownian velocity as a leading thread.
We show how Brownian motion became a research topic for the math-
ematician Wiener in the 1920s, why his model was an idealization of
physical experiments, what Ornstein and Uhlenbeck added to Einstein’s
results, and how Wiener, Ornstein and Uhlenbeck developed in parallel
contradictory theories concerning Brownian velocity.

Genthon, EPJ H 45, 49–105 (2020)