Thermal noise drives a particle over a potential energy barrier
Potential V(x) with particle ensemble
Escape time histogram (stochastic simulation)
Kramers rate Γ = ? | Mean escape time 1/Γ = ? | Simulated ⟨τ⟩ = ?
A particle in a metastable potential well of depth ΔV, coupled to a thermal bath at temperature T, escapes via thermal fluctuations. H.A. Kramers (1940) derived the escape rate in the overdamped limit (γ large):
Γ = (ω_min · ω_sad) / (2πγ) · exp(−ΔV/kT)
where ω_min = √|V''(x_min)| is the well curvature, ω_sad = √|V''(x_sad)| is the barrier curvature (from the saddle/maximum). The exponential Arrhenius factor e^{−ΔV/kT} dominates: barriers are exponentially hard to cross. The prefactor captures recrossings and reactive flux.
This is the transition state theory in 1D. In chemistry: reaction rates (Eyring equation). In magnetism: thermal reversal of nanomagnets (Néel-Brown theory). In neuroscience: stochastic resonance and threshold crossing.
The simulation uses the overdamped Langevin equation: γ ẋ = −V'(x) + √(2γkT) η(t), where η is white noise. Each particle starts in the well; we record when it crosses the barrier.