Proof — Brownian motion
Statement
Brownian motion is the random motion of particles suspended in a fluid, caused by collisions with molecules. Einstein (1905) showed it follows from kinetic theory and derived the diffusion coefficient D = kT/(6πηr), providing evidence for atoms. Wiener (1923) gave the rigorous mathematical formulation as a continuous stochastic process.
Formula
⟨x²⟩ = 2Dt (Einstein relation)
Equation
dX_t = μ dt + σ dW_t (Wiener process)
Source
R. Brown (1827 observation), A. Einstein, Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen, 1905
Domains
physics, mathematics