Stochastic dynamics, Kramers escape, and noise-driven phase transitions
Noise-induced transitions describe how stochastic fluctuations drive a system over energy barriers between metastable states — even when the deterministic dynamics would confine it to one well. Kramers (1940) derived the escape rate r ≈ (ω₀ω_b / 2πγ) exp(−ΔU / D) where ΔU is the barrier height and D = σ²/2 is the noise diffusion. Above the top panel: a particle (dot) performs Brownian motion in a double-well potential V(x) = x⁴/4 − x²/2. The lower panel shows the trajectory and state-space probability density accumulating over time.