Thermodynamic Interpretation

NSD maps structural and stress distributions to statistical mechanical concepts to mathematically evaluate global stability.

1. Systemic Entropy (\(H\))

Entropy quantifies the disorder and diffusion of stress across the network. Let the instantaneous stress probability mass of node \(i\) be \(p_i(t) = \frac{S_i(t)}{\sum S_j(t)}\). \(H(t) = - \sum_{i=1}^{N} p_i(t) \ln(p_i(t))\) A sharp acceleration in entropy (\(\frac{dH}{dt} \gg 0\)) serves as a computable early-warning indicator of impending topological phase transitions.

2. Normative Free Energy (\(\mathcal{F}_{NSD}\))

We introduce \(\mathcal{F}_{NSD}\) as a Lyapunov candidate function representing the total usable structural resilience of the system. Let systemic “Temperature” (\(\tau\)) be the variance of the exogenous load (\(\text{Var}(L)\)). \(\mathcal{F}_{NSD}(t) = \sum_{i=1}^{N} C_i(t) - \tau H(t)\) Under bounded load injection and strictly positive dissipation, the system is subcritical and globally stable if and only if: \(\frac{d\mathcal{F}_{NSD}}{dt} \le 0\) Collapse is triggered when \(\mathcal{F}_{NSD}(t) < \sum S_i(t)\).

3. The Burnout Bifurcation (Saddle-Node Transition)

Setting \(\frac{dC_i}{dt} = 0\) reveals the steady-state equilibrium: \(\mu(C_i^* - C_i) = \eta \kappa \left( S_i - C_i \right) \sigma(a(S_i - C_i))\) As \(S_i\) increases, a saddle-node bifurcation occurs. The stable high-capacity (“Healthy”) equilibrium collides with an unstable intermediate equilibrium and vanishes. The node is deterministically pulled into a degraded, low-capacity attractor (“Burnout”, \(C_i \to 0\)).