What is Brownian motion?

Physical origin and properties

Brownian motion, or pedesis, is the randomized motion of molecular-sized particles suspended in a fluid. It results from the stochastic collisions of the particles with the fast-moving molecules in the fluid (energized due to the internal thermal energy). The aforementioned fluid is supposed to be at the so-called thermal equilibrium, where no preferential direction of flow exists (as opposed to various transport phenomena).

Discovery and early explanations

Brownian motion is named after the Scottish botanist Robert Brown, who first described the phenomenon in 1827 while observing pollens (from the Clarkia pulchella plant) immersed in water, through a microscope.

For a long time, the scientific community did not think much of it. Then, in 1905, a 26-year old Swiss patent clerk changed the world of physics by analyzing the phenomena with the help of the laws of thermodynamics.

Image source: Wikipedia

Albert Einstein publisheda seminal paper where he modeled the motion of the pollen, influenced by individual water molecules, and depending on the thermal energy of the fluid. Although this paper is relatively less celebrated than his other 1905 papers, it is one of his most cited publications. In fact, Einstein’s explanation of Brownian motion served as the first mathematically sound evidence that molecules exist.

You can read this enjoyable article commemorating the 100-year of Einstein’s paper.

Brownian motion as a stochastic process

Many-body interaction

The many-body interactions, that yield the intricate yet beautiful pattern of Brownian motion, cannot be solved by a first-principle model that accounts for the detailed motion of the molecules. Consequently, only probabilistic macro-models applied to molecular populations can be employed to describe it.

This is the reasoning behind the description of Brownian motion mostly as a purely stochastic process in its modern form. Almost all practical application also adopts this approach.

Wiener process

Mathematical properties of the one-dimensional Brownian motion was first analyzed American mathematician Norbert Wiener. The resulting formalism is a real-valued continuous-time stochastic process, called the Wiener process

It is one of the best known stochastic processes with attractive properties like stationarity and independent increments. Consequently, it finds frequent applications in a wide range of fields covering pure and applied mathematics, quantitative finance, economic modeling, quantum physics, and even evolutionary biology.

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