At each step, a particle moves in a random direction. Over millions of steps the path looks like noise — yet it obeys precise mathematical laws: the average displacement grows as the square root of time, a signature of diffusion everywhere in nature.
A 2D random walk is the discrete counterpart of Brownian motion (Einstein 1905). The mean-square displacement ⟨r²⟩ = 4Dt grows linearly with time, defining the diffusion coefficient D. In 1D/2D the walker always returns to the origin; in 3D it escapes with probability 1.
Launch multiple walkers and watch their paths spread. Toggle between 2D grid and free 2D modes. The live histogram shows the radial distribution converging toward a Gaussian — the bell curve predicted by the diffusion equation.
Robert Brown observed pollen grains jiggling in water in 1827. Einstein's 1905 explanation — collisions with invisible molecules — provided the first indirect proof that atoms exist. Jean-Baptiste Perrin confirmed it experimentally, winning the 1926 Nobel Prize.