Brownian Motion , “ Diverse and Undulating ”

Truly man is a marvelously vain, diverse, and undulating object. It is hard to found any constant and uniform judgment on him. Michel de Montaigne, Les Essais, Book I, Chapter 1: “By diverse means we arrive at the same end”; in The Complete Essays of Montaigne, Donald M. Frame transl., Stanford University Press (1958). Pour distinguer les choses les plus simples de celles qui sont compliquées et pour les chercher avec ordre, il faut, dans chaque série de choses où nous avons déduit directement quelques vérités d’autres vérités, voir quelle est la chose la plus simple, et comment toutes les autres en sont plus, ou moins, ou également éloignées. René Descartes, Règles pour la direction de l’esprit, Règle VI. In order to distinguish what is most simple from what is complex, and to deal with things in an orderly way, what we must do, whenever we have a series in which we have directly deduced a number of truths one from another, is to observe which one is most simple, and how far all the others are removed from this-whether more, or less, or equally. René Descartes, Rules for the Direction of the Mind, Rule VI. Car, supposons, par exemple que quelqu’un fasse quantité de points sur le papier à tout hasard, comme font ceux qui exercent l’art ridicule de la géomance. Je dis qu’il est possible de trouver une ligne géométrique dont la notion soit constante et uniforme suivant une certaine règle, en sorte que cette ligne passe par tous ces points, et dans le même ordre que la main les avaient marqués. ... Mais quand une règle est fort composée, ce qui luy est conforme, passe pour irrégulier. G. W. Leibniz, Discours de métaphysique, H. Lestienne ed., Félix Alcan, Paris (1907). Thus, let us assume, for example, that someone jots down a number of points at random on a piece of paper, as do those who practice the ridiculous art of geomancy.2 I maintain that it is possible to find a geometric line whose notion is constant and uniform, following a certain rule, such that this line passes through all the points in the same order in which the hand jotted them down. ... But, when the rule is extremely complex, what is in conformity with it passes for irregular. G. W. Leibniz, Discourse on Metaphysics. Mens agitat molem. Virgil, AEneid. lib. VI. Un coup de dés jamais n’abolira le hasard. Stéphane Mallarmé, Cosmopolis, 1897. A throw of the dice never will abolish chance. 1Expanded and updated version (13 May 2007). 2Note: From géomance, a way to foretell the future; a form of divination. 202 B. Duplantier Poincaré Seminar 2005 L’antimodernisme, c’est la liberté des modernes. Antoine Compagnon, about his book “Les antimodernes : de Joseph de Maistre à Roland Barthes,” Bibliothèque des Idées, Gallimard, March 2005. Antimodernism is the liberty of modern men. Here we briefly describe the history of Brownian motion, as well as the contributions of Einstein, Sutherland, Smoluchowski, Bachelier, Perrin and Langevin to its theory. The always topical importance in physics of the theory of Brownian motion is illustrated by recent biophysical experiments, where it serves, for instance, for the measurement of the pulling force on a single DNA molecule. In the second part, we stress the mathematical importance of the theory of Brownian motion, illustrated by two chosen examples. The by-now classic representation of the Newtonian potential by Brownian motion is explained in an elementary way. We conclude with the description of recent progress seen in the geometry of the planar Brownian curve. At its heart lie the concepts of conformal invariance and multifractality, associated with the potential theory of the Brownian curve itself. 1 A brief history of Brownian motion Several classic works give a historical view of Brownian motion. Amongst them, we cite those of Brush, Nelson, Nye, Pais, Stachel and Wax. We also cite a number of essays in mathematics, physics, 11 especially those which have appeared very recently for the centenary of Einstein’s 1905 articles, and in biology. 3Stephen G. Brush, The Kind of Motion We Call Heat, Book 2, p. 688, North Holland (1976). 4E. Nelson, Dynamical Theories of Brownian motion, Princeton University Press (1967), second ed., August 2001, http://www.math.princeton.edu/∼nelson/books.html . 5Mary Jo Nye, Molecular Reality: A Perspective on the Scientific Work of Jean Perrin, NewYork: American Elsevier (1972). 6Abraham Pais, “Subtle is the Lord...,” The Science and Life of Albert Einstein, Oxford University Press (1982). 7John Stachel, Einstein’s Miraculous Year (Princeton University Press, Princeton, New Jersey, 1998); Einstein from ‘B’ to ‘Z’, Birkhäuser, Boston, Basel, Berlin (2002). 8N. Wax, Selected Papers on Noise and Stochastic Processes, New-York, Dover (1954). It contains articles by Chandrasekhar, Uhlenbeck and Ornstein, Wang and Uhlenbeck, Rice, Kac, Doob. 9J.-P. Kahane, Le mouvement brownien : un essai sur les origines de la théorie mathématique, in Matériaux pour l’histoire des mathématiques au XXème siècle, Actes du colloque à la mémoire de Jean Dieudonné (Nice, 1996), volume 3 of Séminaires et congrès, pages 123-155, French Mathematical Society (1998). 10M. D. Haw, J. Phys. C 14, pp. 7769-7779 (2002). 11R. M. Mazo, Brownian Motion, Fluctuations, Dynamics and Applications, International Series of Monographs on Physics 112, Oxford University Press (2002). 12B. Derrida and É. Brunet in Einstein aujourd’hui, edited by M. Leduc and M. Le Bellac, Savoirs actuels, EDP Sciences/CNRS Editions (2005); P. Hänggi et al., New J. Phys. 7 (2005); J. Renn, Einstein’s invention of Brownian motion, Ann. d. Phys. (Leipzig) 14, Supplement, pp. 23-37 (2005); D. Giulini & N. Straumann, Einstein’s Impact on the Physics of the Twentieth Century, arXiv:physics/0507107; N. Straumann, On Einstein’s Doctoral Thesis, arXiv:physics/0504201; S. N. Majumdar, Brownian functionals in Physics and Computer Science, Current Science 89, pp. 2075-2092 (2005); J. Bernstein, Einstein and the existence of atoms, Am. J. Phys. 74, pp. 863-872 (2006). 13E. Frey and K. Kroy, Brownian Motion: a Paradigm of Soft Matter and Biological Physics, Ann. d. Phys. (Leipzig) 14, pp. 20-50 (2005), arXiv:cond-mat/0502602. Vol. 1, 2005 Brownian Motion 203