When dissipative steady states admit thermodynamic occupation laws

Abstract

Non-equilibrium steady states (NESSs) generally lack thermodynamic occupation laws because finite stationary circulation and a globally exact rate-ratio field cannot coexist for the same Markov generator. Here we construct a sector-separated geometry that overcomes this incompatibility without arresting dissipation. Entropy-production exposure-and-separation excludes the entropy-producing state~$0$ from the conditional occupation manifold while retaining it in the dissipative full graph; physical returns $i\to0\to0^\ast$ become effectively Markovian in the strong-bias/rapid-reset (SR) limit. For a thermodynamically complete conditional manifold, autonomous redistribution (AR) eliminates residual futile circulation, making the rate-ratio one-form exact. Thermodynamic calibration gives $X_i=\beta(\Delta\mu- \mathcal F_i^{\mathrm{cost}})$ and $p_i=e^{X_i}/Z_\mathcal{C}$, with $Z_\mathcal{C}=1+\sum_i e^{X_i}$. Full-graph probabilities factorize exactly as $P_\alpha=(1-P_0)p_\alpha$. In the SR limit, the kinetic factor tends to unity while $p_\alpha\to e^{X_\alpha}/Z_\mathcal C$, yielding $P_\alpha\to p_\alpha$ while finite dissipation persists. Near AR, integrability is lost linearly in residual cycle current whereas dissipation begins quadratically. In the binary zero-cycle-rank limit, occupation redistributes autonomously under maintained $\Delta\mu$ bias, yielding the inverted Fermi--Dirac law, which is applied to thermal smearing in quantum-dot lasers. The framework provides constructive acquisition conditions and failure diagnostics for thermodynamic occupation laws in dissipative NESSs.