Stochastic Resonance

Stochastic resonance is the counterintuitive phenomenon where adding noise to a nonlinear system improves its ability to detect a weak subthreshold signal. The system is a particle in a double-well potential V(x) = -ax²/2 + x⁴/4, driven by a weak periodic signal A·cos(ωt) and Gaussian noise σξ(t). The Kramers escape rate is r_K = (a/π)·exp(-a²/2σ²). Resonance occurs when the noise-induced switching rate matches the signal frequency: r_K ≈ ω/2, giving optimal noise σ* = a/√(2ln(a²/πω)). Applications: sensory neurons, quantum tunneling, climate oscillations (Milankovitch), image processing.