Kolmogorov Forward Equation (Fokker-Planck)

The Fokker-Planck (Kolmogorov forward) equation ∂_t p = -∂_x[μ(x)p] + ½∂²_x[σ²(x)p] governs probability density evolution of a stochastic process dX = μ dt + σ dW. Drift μ advects the density; diffusion σ spreads it. The stationary distribution p_∞(x) ∝ exp(2∫μ/σ² dx) is the Boltzmann distribution.

Potential V(x)

Diffusion D = σ²/2: 0.30 Drift strength μ₀: 1.00 Tilt (constant force): 0.00 Initial width σ₀: 0.30

t: 0.00

⟨x⟩: —

⟨x²⟩-⟨x⟩²: —

FP: ∂_t p = ∂_x[V'p] + D∂²_x p
where drift μ = −V'(x)

Stationary: p∞ ∝ e^{−V/D}
Harmonic well → Gaussian
Double well → bimodal

Solved via finite-difference scheme (upwind + Crank-Nicolson)

Fokker-Planck / Kolmogorov Forward Equation

Potential V(x)