Langevin Dynamics

Parameters

Temperature kT = 0.30 Barrier height a = 1.00 N particles = 60 Time step dt = 0.005

Mean position ⟨x⟩—

⟨x²⟩—

Kramers rate Γ—

Crossings/s—

Langevin equation (overdamped):
dx/dt = −V'(x)/γ + √(2kT/γ) η(t)

V(x) = a·x⁴ − 2a·x² (double well, minima at x=±1, barrier at x=0 height a).

Kramers escape rate:
Γ = (ω₀ω_b)/(2πγ) exp(−ΔV/kT)
where ΔV = a, ω₀²=|V''(±1)|=4a, ω_b²=|V''(0)|=4a.

The histogram converges to the Boltzmann distribution P(x) ∝ e^{−V(x)/kT}.