Langevin Dynamics & Kramers Escape

dx = −V'(x)dt + σ dW in a double-well V(x) = x⁴ − ax². Multiple Brownian particles escape from wells via thermal fluctuations at the Kramers rate r ∝ exp(−ΔV/σ²).

Parameters

r_K = —
Left well: —
Right well: —
Escapes: 0
Kramers Escape Rate

r_K = (ω_min ω_max)/(2πγ) · e^(−ΔV/k_BT)

ΔV = V(x=0) − V(x=±√(a/2))

Stationary distribution:
ρ(x) ∝ exp(−2V(x)/σ²)

= Boltzmann with "temperature" k_BT = σ²/2.