dorsal/arxiv
View SchemaThe origin of the Langevin equation and the calculation of the mean squared displacement: Let's set the record straight
| Authors | K. Razi Naqvi |
|---|---|
| Categories | |
| ArXiv ID | physics/0502141 |
| URL | https://arxiv.org/abs/physics/0502141 |
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.
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"abstract": "Ornstein and his coauthors, who constructed a dynamical theory of Brownian\nmotion, taking the equation $mdv/dt =-\\zeta v+X$ as their starting point,\nusually named the equation after Einstein alone or after both Einstein and\nLangevin; furthermore, Ornstein, who was the first to extract from this\nequation the correct expression for $\\bar{\\Delta^2}$, the mean-squared distance\ncovered by a Brownian particle, credited de Haas-Lorentz, rather than Langevin,\nfor finding the stationary limit of $\\bar{\\Delta^2}$. A glance at Einstein\u0027s\n1907 paper, titled ``Theoretical remarks on Brownian motion\u0027\u0027, should suffice\nto convince one that it is not unfair to attribute the {\\it conception} of the\nabove equation, now universally known as the Langevin equation, to Einstein.\nLangevin\u0027s avowed aim in his 1908 article was to recover, through a route that\nwas `infinitely more simple\u0027, Einstein\u0027s 1905 expression for the diffusion\ncoefficient, but a careful reading of Langevin\u0027s paper shows that--depending on\nhow one interprets his description of the statistical behavior of the random\nforce $X$ appearing in the above equation--his analysis is at best incomplete,\nand at worst a mere tautology. Since textbook accounts are based on the\ninterpretation that renders the proof fallacious, alternative derivations,\nwhich are adaptations of those given by de Haas-Lorentz and Ornstein, are\npresented here. Some neglected aspects of the contents of Ornstein\u0027s early\npapers on Brownian motion are also brought to light.",
"arxiv_id": "physics/0502141",
"authors": [
"K. Razi Naqvi"
],
"categories": [
"physics.chem-ph"
],
"title": "The origin of the Langevin equation and the calculation of the mean squared displacement: Let\u0027s set the record straight",
"url": "https://arxiv.org/abs/physics/0502141"
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