Brownian Motion and Statistical Physics

In 1905 — the same annus mirabilis that produced special relativity and the photoelectric effect — a paper entitled Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen provided a quantitative theory of the erratic jiggling of microscopic particles suspended in liquid, first observed by botanist Robert Brown in 1827. The key insight was to treat the suspended particle as subject to a vast number of random collisions from solvent molecules, and to ask only statistical questions about the resulting displacement. The analysis yielded a definite prediction: the mean squared displacement of a particle is proportional to both the elapsed time and to the ratio of the thermal energy kT to the viscous drag coefficient — what is now called the Einstein relation for diffusion.

This result had profound consequences beyond the description of colloids. At the turn of the twentieth century, the reality of atoms remained contested; the kinetic theory of gases had been developed by Clausius, Maxwell, and Boltzmann, but critics such as Ernst Mach questioned whether atoms were anything more than a convenient bookkeeping device. By connecting the diffusion coefficient to Avogadro’s number, the theory provided a route to determining the latter from purely macroscopic, observable quantities. Jean Perrin’s painstaking experimental verification of these predictions (for which he was awarded the 1926 Nobel Prize in Physics) effectively silenced the last serious opposition to atomic theory.

The theoretical methods developed in this work — the connection between fluctuations and dissipation, the Fokker–Planck equation, the Langevin approach — became the foundation of the modern theory of stochastic processes and have found applications ranging from financial mathematics and polymer physics to the modeling of biological molecular motors. The fluctuation–dissipation theorem, which generalizes the Einstein relation, remains one of the central results of statistical mechanics out of equilibrium.