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

Arthur Genthon|HAL (Le Centre pour la Communication Scientifique Directe)|Jun 9, 2020

Philosophy and History of Science38 references4 citations

TL;DR

This paper traces the evolution of the concept of velocity in Brownian motion from physics to mathematics, analyzing how Einstein, Langevin, and Smoluchowski developed physical theories, while Wiener and Ornstein-Uhlenbeck later formalized stochastic processes. It reveals a fundamental conflict: Wiener's non-differentiable trajectories deny velocity, whereas Ornstein-Uhlenbeck define it via a mean-reverting process, showing how mathematical idealization and physical intuition diverged yet complemented each other.

ABSTRACT

Interest in Brownian motion was shared by different communities: this phenomenon was first observed by the botanist Robert Brown in 1827, then theorised by physicists in the 1900s, and eventually modelled by mathematicians from the 1920s, while still evolving as a physical theory. Consequently, Brownian motion now refers to the natural phenomenon but also to the theories 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 experiments 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 the period straddling the two `half-histories' just mentioned, in order to highlight continuity, to investigate the domain-shift from physics to mathematics, and to survey the enhancements of later physical theories. We study the works of Einstein, Smoluchowski, Langevin, Wiener, Ornstein and Uhlenbeck from 1905 to 1934 as well as experimental results, using the concept of Brownian velocity as a leading thread. We show how Brownian motion became a research topic for the mathematician 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.

Motivation & Objective

• To bridge the gap between the physical and mathematical histories of Brownian motion, which are often studied in isolation.
• To investigate how and why Brownian motion transitioned from a physical phenomenon to a mathematical research topic, focusing on Norbert Wiener’s role.
• To analyze the contrasting treatments of velocity in Einstein-Smoluchowski theory, Langevin’s approach, Wiener’s stochastic process, and Ornstein-Uhlenbeck’s velocity-based model.
• To clarify the polysemy of 'Brownian motion' by distinguishing its physical, mathematical, and experimental interpretations.
• To demonstrate how the concept of velocity became a central, contested thread in the development of stochastic processes.

Proposed method

• The study traces the historical development of Brownian motion from 1827 to 1934, focusing on key figures: Einstein, Smoluchowski, Langevin, Wiener, Ornstein, and Uhlenbeck.
• It uses the concept of velocity as a unifying thread to compare physical theories (1905–1910) with mathematical formalizations (1920s–1930s).
• The paper analyzes original texts and translations in English and French, emphasizing conceptual shifts in the treatment of time scales and differentiability.
• It reconstructs Wiener’s construction of Brownian motion via integration theory and non-differentiability, linking it to Perrin’s experimental observations.
• It compares the Ornstein-Uhlenbeck approach, which introduces a velocity variable and a mean-reverting process, with Wiener’s velocity-less model.
• The analysis includes a synthesis of key results in tables, showing the physical ingredients, mathematical tools, and outcomes of each theory.

Figure 1: Example of trajectory where $P_{i}$ are the collision points and $\varepsilon$ is the deviation angle. From [ 44 ] .

Experimental results

Research questions

• RQ1 How did the concept of velocity evolve from experimental observation to theoretical modeling in Brownian motion?
• RQ2 Why did Norbert Wiener choose Brownian motion as a foundation for his mathematical theory of stochastic processes?
• RQ3 What were the key differences between Wiener’s non-differentiable trajectories and Ornstein-Uhlenbeck’s velocity-based model, and how did they coexist without communication?
• RQ4 How did the failure to measure velocity in experiments influence the development of mathematical models of Brownian motion?
• RQ5 In what way did the Ornstein-Uhlenbeck process resolve the limitations of Einstein’s theory at short time scales?

Key findings

• Wiener’s 1921–1933 work established that Brownian trajectories are continuous but nowhere differentiable, proving that velocity does not exist in the classical sense.
• Ornstein and Uhlenbeck showed that at short times, the mean squared displacement scales as ⟨x²⟩ ∝ t², implying a well-defined velocity, contradicting Einstein’s t-proportional scaling.
• While Einstein’s theory assumes Gaussian displacement for all times, Wiener’s model extends this to all times, including the short-time limit, but without defining velocity.
• The Ornstein-Uhlenbeck process introduced a velocity variable with a mean-reverting dynamics, linking it to physical parameters like friction and temperature.
• Despite parallel developments, there was no meaningful dialogue between mathematicians and physicists, leading to conflicting interpretations of velocity.
• The paper concludes that 'Brownian motion' is a polysemic term, referring to the physical phenomenon, the Einstein-Smoluchowski model, the Wiener process, and the Ornstein-Uhlenbeck process, each with distinct definitions of velocity.

Figure 2: Example of a path going through the windows of a particular interval $i(I_{3})$ . The image has been rotated by 90 ° compared to the one in the original article.