The Brownian Motion






Jean-François COLONNA
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CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, École polytechnique, Institut Polytechnique de Paris, CNRS, France

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[en français/in french]


Keywords: Brownian Motion, Mouvement Brownien.



The Brownian motion was discovered in 1827 by the English botanist Robert Brown while observing pollen grains dispersed on the surface of a liquid under a microscope. It originates from the countless collisions between the molecules of the liquid and the grains. This phenomenon was particularly studied by Albert Einstein (1905) and later, thanks to the experiments of Jean Perrin, it became essential proof of the atomic structure of matter. These studies were then pursued from a mathematical perspective, notably by Paul Lévy [01] and one of his students, Benoît Mandelbrot.

The Brownian motion of a particle consists of a series of random displacements (in direction and magnitude) that are independent of each other. The resulting structures give rise to generally difficult mathematical problems. For example, Benoît Mandelbrot conjectured that the fractal dimension of the envelope —white curve— of two-dimensional Brownian motion —colored curve— was equal to 4/3, and this was proven in 1999 by Greg Lawler, Oded Schramm, and Wendelin Werner [02].

This phenomenon is actually omnipresent: in Physics, Chemistry, Biology [03], certain industrial processes, and even in the world of finance, where it serves as a model for the evolution of asset prices over time.

It is possible to study it in two and three dimensions on our computers in the form of virtual experiments. To do so, we will place ourselves inside a rectangular (or parallelepipedic) domain and arrange particles on a square (or cubic) lattice. These particles will be of two types: on one hand, so-called light particles, initially animated with random velocities in direction but of equal magnitude; they will be represented by small colored spheres. On the other hand, so-called heavy particles, initially immobile; they will be represented by small white spheres. Over time, collisions [04] will occur between all these particles, and gradually, the heavy particles will be set in motion.



We then observe that the heavier the white particles (Robert Brown's "pollen grains"), the less easily they are set in motion by the colored particles (Albert Einstein's "molecules") [05].





Copyright © Jean-François COLONNA, 2025-2025.
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