Random Walk & Brownian Motion

About: Brownian motion is the random thermal motion of particles suspended in a fluid. Einstein (1905) showed the mean squared displacement grows linearly with time: ⟨r²⟩ = 2dDt, so the RMS displacement grows as √t. This explains diffusion — each walker here takes discrete random steps, approximating a Wiener process in continuous time. Diffusion coefficient D = σ²/2Δt where σ is step size; the law ⟨r⟩ ~ √t holds universally regardless of step distribution (central limit theorem). Real-world applications: diffusion of molecules, stock price models (geometric BM), polymer physics, and heat equations.