The origin of the Langevin equation and the calculation of the mean squared displacement: Let's set the record straight

TL;DR

This paper clarifies the historical origins of the Langevin equation, emphasizing Einstein's role, critiques Langevin's derivation, and offers alternative derivations of the mean-squared displacement in Brownian motion.

Contribution

It reexamines the historical attribution of the Langevin equation and provides alternative derivations of Brownian motion metrics, correcting common misconceptions in textbook accounts.

Findings

Einstein conceived the Langevin equation

Langevin's derivation is incomplete or tautological

Alternative derivations of mean-squared displacement are presented

Abstract

Ornstein and his coauthors, who constructed a dynamical theory of Brownian motion, taking the equation $mdv/dt =-\zeta v+X$ as their starting point, usually named the equation after Einstein alone or after both Einstein and Langevin; furthermore, Ornstein, who was the first to extract from this equation the correct expression for $\bar{\Delta^2}$, the mean-squared distance covered by a Brownian particle, credited de Haas-Lorentz, rather than Langevin, for finding the stationary limit of $\bar{\Delta^2}$. A glance at Einstein's 1907 paper, titled ``Theoretical remarks on Brownian motion'', should suffice to convince one that it is not unfair to attribute the {\it conception} of the above equation, now universally known as the Langevin equation, to Einstein. Langevin's avowed aim in his 1908 article was to recover, through a route that was `infinitely more simple', Einstein's 1905 expression for the diffusion coefficient, but a careful reading of Langevin's paper shows that--depending on how one interprets his description of the statistical behavior of the random force $X$ appearing in the above equation--his analysis is at best incomplete, and at worst a mere tautology. Since textbook accounts are based on the interpretation that renders the proof fallacious, alternative derivations, which are adaptations of those given by de Haas-Lorentz and Ornstein, are presented here. Some neglected aspects of the contents of Ornstein's early papers on Brownian motion are also brought to light.