the energy available to do work — the portion of total energy not locked up in entropy
three formulations, one idea: systems spontaneously minimize free energy, and what remains at the minimum is equilibrium
thermodynamic
Helmholtz: $F = E - TS$, where $E$ is internal energy, $T$ is temperature, $S$ is entropy
Gibbs: $G = H - TS$, where $H$ is enthalpy
a system at constant temperature spontaneously evolves toward the state that minimizes $F$. this is the second law of thermodynamics restated: the universe doesn't maximize disorder — it minimizes free energy
variational (Friston)
the free energy principle: biological agents minimize variational free energy to persist
$$F = E_{q_\theta}[\log q_\theta(z) - \log p(s, z)]$$
where $q_\theta(z)$ is the agent's beliefs about hidden states, $p(s,z)$ is the generative model, $s$ is observations. minimizing $F$ simultaneously sharpens beliefs (perception) and selects actions (planning)
see active inference for the computational framework. see Karl Friston for the originator
tri-kernel functional
the tri-kernel fixed point minimizes a unified free energy over the cybergraph:
$$\mathcal{F}(\phi) = \lambda_s\left[\frac{1}{2}\phi^\top L\phi + \frac{\mu}{2}\|\phi-x_0\|^2\right] + \lambda_h\left[\frac{1}{2}\|\phi-H_\tau\phi\|^2\right] + \lambda_d \cdot D_{KL}(\phi \| D\phi) - T \cdot S(\phi)$$
the spring term encodes structural coherence via the graph Laplacian. the heat term penalizes deviation from context-smoothed state. the diffusion term aligns with random walk distribution. the entropy term $S(\phi)$ encourages diversity
the weights $\lambda_s, \lambda_h, \lambda_d$ emerge as Lagrange multipliers — not tuned, but derived from the variational optimization
the solution: $\phi^*_i \propto \exp(-\beta[E_{\text{spring},i} + \lambda E_{\text{diffusion},i} + \gamma C_i])$ — a Boltzmann distribution
the connection
all three formulations share the same structure: an energy term competing with an entropy term, balanced by temperature. the minimum is always a Boltzmann distribution. thermodynamics discovered it for gases. Karl Friston applied it to brains. cyber applies it to knowledge
Δπ in learning incentives is the gradient of $\mathcal{F}$ — creating valuable structure in the cybergraph is literally reducing free energy
see cybics for the full unification. see negentropy vs entropy for the dual thermodynamics framework. see contextual free energy model for the context-dependent extension