[L'équation de Langevin]
L'existence des atomes a été prédite depuis longtemps par savants et philosophes. Le développement de la thermodynamique et son interprétation par la mécanique statistique, à la fin du XIXe et au début du XXe siècle, ont rendu possible le comblement de l'écart entre l'échelle spatiale du monde macroscopique et celle des atomes. En 1905 et 1906, Einstein et Smoluchowski montrent, selon deux approches complètement différentes, que le mouvement brownien de particules de taille mésoscopique mesurable directement est une manifestation du mouvement incessant des atomes dans le fluide environnant. Peu après, en 1908, Langevin montre comment mettre dans un cadre cohérent l'effet des fluctuations aléatoires du monde atomique, qui font mouvoir la particule brownienne, et la viscosité du fluide régissant les mouvements macroscopiques de cette même particule brownienne pour les ralentir. Nous examinons les méthodes de déduction d'Einstein et de Smoluchowski ainsi qu'un article antérieur de Sutherland sur la diffusion d'une solution de sphéres mésoscopiques dans un liquide. Nous présentons ensuite la note de Langevin aux Comptes rendus de l'Académie des sciences, en insistant sur la division fondamentale entre force aléatoire et force visqueuse. Ceci nous amène à différentes questions, telles que les contraintes à satisfaire par la force de Langevin et les généralisations de l'équation de Langevin. Nous insistons sur les contraintes issues de la réversibilité en temps des fluctuations d'équilibre. Nous discutons aussi une remarque de Lorentz montrant que, si la particule brownienne n'est pas très dense, on ne peut utiliser la formule de la traînée de Stokes pour une vitesse constante dans l'équation de Langevin. Finalement, nous examinons ce qu'on appelle la théorie de Schrödinger–Langevin (ou de Heisenberg–Langevin) en mécanique quantique.
The existence of atoms has been long predicted by philosophers and scientists. The development of thermodynamics and of the statistical interpretation of its concepts at the end of the nineteenth century and in the early years of the twentieth century made it possible to bridge the gap of scales between the macroscopic world and the world of atoms. Einstein and Smoluchowski showed in 1905 and 1906 that the Brownian motion of particles of measurable size is a manifestation of the motion of atoms in fluids. Their derivation was completely different from each other. Langevin showed in 1908 how to put in a coherent framework the subtle effect of the randomness of the atomic world, responsible for the fluctuating force driving the motion of the Brownian particle and the viscosity of the “macroscopic” flow taking place around the same Brownian particle. Whereas viscous forces were already well understood at this time, the “Langevin” force appears there for the first time: it represents the fluctuating part of the interaction between the Brownian particle and the surrounding fluid. We discuss the derivation by Einstein and Smoluchowski as well as a previous paper by Sutherland on the diffusion coefficient of large spheres. Next we present Langevin's short note and explain the fundamental splitting into a random force and a macroscopic viscous force. This brings us to discuss various points, like the kind of constraints on Langevin-like equations. We insist in particular on the one arising from the time-reversal symmetry of the equilibrium fluctuations. Moreover, we discuss another constraint, raised first by Lorentz, which implies that, if the Brownian particle is not very heavy, the viscous force cannot be taken as the standard Stokes drag on an object moving at uniform speed. Lastly, we examine the so-called Langevin–Heisenberg and/or Langevin–Schrödinger equation used in quantum mechanics.
Mots-clés : Fluctuations, Équations stochastiques, Mouvement brownien
Yves Pomeau 1 ; Jarosław Piasecki 2
@article{CRPHYS_2017__18_9-10_570_0,
author = {Yves Pomeau and Jaros{\l}aw Piasecki},
title = {The {Langevin} equation},
journal = {Comptes Rendus. Physique},
pages = {570--582},
publisher = {Elsevier},
volume = {18},
number = {9-10},
year = {2017},
doi = {10.1016/j.crhy.2017.10.001},
language = {en},
}
Yves Pomeau; Jarosław Piasecki. The Langevin equation. Comptes Rendus. Physique, Science in the making: The Comptes rendus de l’Académie des sciences throughout history, Volume 18 (2017) no. 9-10, pp. 570-582. doi : 10.1016/j.crhy.2017.10.001. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2017.10.001/
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