2 F eb 2 00 5 100 Years of Brownian motion

In the year 1905 Albert Einstein published four papers that raised him to a giant in the history of science of all times. These works encompass the photon hypothesis (for which he obtained the Nobel prize in 1921), his first two papers on (special) relativity theory and, of course, his first paper on Brownian motion, entitled “Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen” (submitted on May 11, 1905). Thanks to Einstein intuition, the phenomenon observed by the Scottish botanist Rober Brown in 1827 – a little more than a naturalist’s curiosity – becomes the keystone of a fully probabilistic formulation of statistical mechanics and a well-established subject of physical investigation which we celebrate in this Focus issue entitled – for this reason – : “100 Years of Brownian Motion”. Although written in a dated language, Einstein’s first paper on Brownian motion already contains the cornerstones of the modern theory of stochastic processes. The author starts out by using arguments of thermodynamics and the concept of osmotic pressure of suspended particles to evaluate a particle diffusion constant by balancing a diffusion current with a drift current (through Stokes’ law). In doing so, he obtains a relation between two transport coefficients: the particle diffusion constant and the fluid viscosity, or friction. This relation, known as the Einstein relation, was generalized later on in terms of the famous fluctuation-dissipation theorem by Callen and Welton, and with the linear response theory by Kubo. A much clearer discussion of Einstein’s arguments can be found in his thesis work, accepted by the University of Zurich in July 1905, which he submitted for publication on August 19, 1905. The second part of his 1905 paper contains a heuristic derivation of the (overdamped) diffusion equation, from which he deduces his famous prediction that the root mean square displacement of suspended particles is proportional to the square root of time. Moreover, the trajectories of a Brownian particle can be regarded as memory-less and non-differentiable, so that its motion is not ballistic (a bold statement that troubled mathematicians for half a century!). The latter also explained why earlier attempts to measure the velocity of Brownian particles yielded puzzling results, and consequently were doomed to fail.