Lab 9: Brownian Resonance — Equations

Purpose#

To formalize the stochastic and harmonic behavior of Brownian motion within triadic systems, revealing how noise can amplify signal through resonance.

Key Equations#

1. Brownian Displacement#

$$ \Delta x(t) = \sqrt{2 D t} $$ Where ( D ) is the diffusion coefficient and ( t ) is time.

2. Resonant Noise Amplification#

$$ A(t) = \eta(t) \cdot \sin(\omega t) $$ Where ( \eta(t) ) is stochastic noise and ( \omega ) is the resonance frequency.

3. Triadic Brownian Resonance#

$$ R_B(t) = \int_{0}^{t} A(t) \cdot R_{ijk} , dt $$ Where ( R_{ijk} ) is triadic resonance.

4. Signal-to-Noise Ratio (SNR)#

$$ \text{SNR} = \frac{\langle R_B(t)^2 \rangle}{\langle \eta(t)^2 \rangle} $$ Measures the amplification of signal over noise.

Symbol Legend#