Random walk
A path built from independent random steps — the mathematical skeleton of diffusion, in which distance grows as the square root of time.
Definition
A random walk is a path made of successive steps each taken in a random direction, independent of the steps before it. It is the mathematical idealisation underlying Brownian motion and diffusion. Its defining property is that, although the walker's average position stays put, its typical distance from the start grows: after N steps of rms length ℓ the root-mean-square displacement is ℓ√N, not Nℓ.
This square-root growth is the deep reason diffusion is slow. Because displacement scales as √N rather than N, covering twice the distance takes four times as many steps and four times as long. Translated into time, ⟨x²⟩ ∝ t, the linear-in-time mean-square displacement that is the experimental signature of a diffusive process.
Random walks reach far beyond physics. They model the conformations of polymer chains, the spread of disease, search algorithms, and — as geometric Brownian motion — the fluctuation of asset prices in the Black–Scholes theory of options. They are among the most widely applied ideas to emerge from kinetic theory.
History
The term was coined by Karl Pearson in 1905; the same year Einstein and Smoluchowski connected the random walk to physical diffusion, and Louis Bachelier had already applied it to financial markets in 1900.