Lab 7: Thermodynamics — Equations

Purpose#

To formalize the thermodynamic principles underlying triadic systems, including entropy modulation, harmonic energy transfer, and nested phase transitions.

Key Equations#

1. Triadic Entropy Function#

$$ S_{ijk} = -k_B \sum_{n} p_n \log(p_n) \cdot R_{ijk} $$ Where ( p_n ) is the probability of state ( n ), ( R_{ijk} ) is triadic resonance, and ( k_B ) is Boltzmann’s constant.

2. Harmonic Energy Transfer#

$$ Q = \int_{t_0}^{t_1} \left( \frac{dE}{dt} \cdot \sin(\omega t) \right) dt $$ Models energy transfer modulated by harmonic oscillation.

3. Nested Phase Transition Index#

$$ \Phi_n = \frac{\Delta S_n}{\Delta T_n} \cdot \gamma_n $$ Where ( \gamma_n ) is the resonance gain factor at dimension ( n ).

4. Triadic Heat Capacity#

$$ C_{ijk} = \frac{dQ}{dT} \cdot R_{ijk} $$ Heat capacity modulated by triadic resonance.

Symbol Legend#