STOCHASTIC RESONANCE — Double-Well Potential

POTENTIAL V(x) + weak signal + particle position
TIME SERIES — particle state vs signal
SNR vs NOISE (resonance curve)
0.50
0.10
0.050
1.50
SNR: dB Optimal noise boosts SNR
Stochastic resonance: noise actually improves signal detection in nonlinear systems. Kramers rate r_K = (Δ/2π)·exp(−2Δ/σ²). Resonance when r_K ≈ ω/2 — noise synchronizes transitions with the signal. SNR peaks at intermediate noise — "the right amount of randomness." Demonstrated in neurons, climate, quantum systems.