Probability density evolution under drift, diffusion, and viscous drag
The Fokker-Planck equation ∂ₜP = −∂ₓ[(μ − γx)P] + D∂²ₓP governs the probability distribution of a Brownian particle under linear drag (Ornstein-Uhlenbeck process). The steady state is a Gaussian with variance σ² = D/γ. The mean drifts toward μ/γ. This equation describes colloidal particles, plasma transport, financial volatility models, and any system with diffusion plus restoring force.