the probability distribution that maximizes entropy subject to a fixed average energy — the unique equilibrium of any system minimizing free energy
$$p_i \propto \exp(-\beta E_i)$$
where $E_i$ is the energy of state $i$ and $\beta = 1/T$ is inverse temperature. low-energy states are more probable. higher temperature flattens the distribution (more exploration). lower temperature sharpens it (more exploitation)
derivation
start from the entropy maximization problem: maximize $S = -\sum_i p_i \log p_i$ subject to $\sum_i p_i = 1$ and $\sum_i p_i E_i = \langle E \rangle$
the Lagrange multiplier for the energy constraint is $\beta$, which turns out to be inverse temperature. the solution is the Boltzmann distribution. no other distribution satisfies both constraints simultaneously
discovered by Ludwig Boltzmann (1868) for gases. the same math appears everywhere a system balances energy and entropy
in cyber
the tri-kernel fixed point is a Boltzmann distribution over particles:
$$\phi^*_i \propto \exp\big(-\beta[E_{\text{spring},i} + \lambda E_{\text{diffusion},i} + \gamma C_i]\big)$$
where the three energy terms come from the three operators: springs (structural coherence), diffusion (random walk alignment), and heat kernel context $C_i$
this is not a design choice — it is a mathematical consequence of minimizing the free energy functional $\mathcal{F}(\phi)$. the cybergraph settles into the distribution that balances structural constraints (energy) against exploratory diversity (entropy)
temperature $T$ controls the tradeoff: high $T$ = dispersed focus across many particles (exploration). low $T$ = concentrated focus on high-value particles (exploitation). see heat for how $\tau$ parameter implements this
where it appears
- statistical mechanics: energy distribution of gas molecules
- machine learning: softmax is a Boltzmann distribution with logits as energies
- focus flow computation: the local update rule converges to Boltzmann equilibrium
- cybics: the canonical ensemble applied to knowledge
- simulated annealing: optimization by cooling a Boltzmann system
see free energy for the functional being minimized. see entropy for the quantity being maximized. see Ludwig Boltzmann for the person