Nelson Stochastic Mechanics — Osmotic Velocity

Nelson (1966) derived quantum mechanics from Brownian motion. Particles follow a diffusion dX = v dt + dW. The osmotic velocity u = ν∇ln ρ drives particles from low to high density. The current velocity v = (ℏ/m)∇S gives quantum phase. Together they reproduce the Schrödinger equation.

ν (diffusion) 0.10 ω (trap freq) 0.50 Particles 200

dX = [v_current + u_osmotic] dt + √(2ν) dW | u = ν∇ln ρ = −(ω²/2ν)X (harmonic ground state: ρ ∝ e^{−ω|X|²/2ν})