Fokker-Planck Equation

Probability flux & steady-state distribution in a drift-diffusion system

Flux J = μP − D ∂P/∂x
Steady state:
P∗ ∝ e^(−V/D)
Fokker-Planck Equation governs the time evolution of probability densities for stochastic processes: ∂P/∂t = −∂/∂x[μ(x)P] + D ∂²P/∂x² The probability flux J(x,t) = μP − D∂P/∂x represents the net flow of probability. At steady state, ∂P*/∂t = 0. The potential V(x) determines the drift force μ = −dV/dx. In equilibrium (detailed balance), J=0 everywhere and P* ∝ exp(−V/D) — the Boltzmann distribution.