Stochastic Resonance

A subthreshold periodic signal becomes detectable when the right amount of noise is added. Too little noise: no crossings. Too much: pure noise. Optimal noise reveals the signal.

0.70
0.030
0.80
1.00
—
SNR (dB)
—
Crossings/cycle
—
Optimal σ*
—
% above threshold

Stochastic Resonance (Benzi 1981, Gammaitoni 1998)

System: ẋ = −dV/dx + A·sin(2πft) + ξ(t)
Double well: V(x) = −x²/2 + x⁴/4, threshold at ±1
Kramers rate: r_K = (1/2π)·exp(−ΔV/D)
SNR peaks at D* = ΔV / ln(√2·ΔΩ/Aω)
Optimal noise: Kramers rate ≈ signal frequency