Langevin equation dx = −V'(x)dt + σdW in a double-well potential V(x) = x⁴ − 2x²
The Langevin equation describes Brownian motion in a force field: dx = f(x)dt + σdW, where dW is a Wiener process increment. For a double-well V(x) = ax⁴ − 2ax², the drift f = −V'(x) = −(4ax³ − 4ax). The Fokker-Planck equation governs the probability density ∂P/∂t = −∂(fP)/∂x + (σ²/2)∂²P/∂x². The stationary distribution is P_∞(x) ∝ exp(−2V(x)/σ²), shown in yellow. Kramers escape rate: k ∝ exp(−ΔV/σ²·2).