Microcanonical sampling assumption

For the regular-shell branch below, assume AC as required by Regular Hamiltonian shell preparation data and its coarea-measure supplier. No choice assumption is needed for the separate finite-window branch.

For a finite microstate model and a declared energy value $E_0$ in joules and resolution $\delta \ge 0$ in joules, define the admitted window ${\mathrm{\Omega }}_{E_0,\delta }:=\{x\in \mathrm{\Omega }:\mathrm{\mid }E(x)-E_0\mathrm{\mid }\le \delta \}\mathrm{.}$ When this set is nonempty, adopt the uniform preparation $P(x)=\mathrm{\mid }{\mathrm{\Omega }}_{E_0,\delta }{\mathrm{\mid }}^{-1}$ on it and zero outside. Its dimensionless Boltzmann entropy is $\log \mathrm{\mid }{\mathrm{\Omega }}_{E_0,\delta }\mathrm{\mid }$, and its physical entropy convention is $k_{\mathrm{B}}\log \mathrm{\mid }{\mathrm{\Omega }}_{E_0,\delta }\mathrm{\mid }$ in $\mathrm{J}{\mathrm{K}}^{-1}$. A zero-width window is allowed only when it contains at least one microstate.

For a continuous classical model, when the regular-band, nonempty-shell, and normalization hypotheses in Regular Hamiltonian shell preparation data hold, the microcanonical preparation is the normalized shell measure ${\nu }_{\widehat{E}}$ defined there. The shell preparation does not require a globally defined flow. These finite-window and regular-shell rules are adopted model preparations. They do not assert Hamiltonian ergodicity, equality with long-time averages, canonical sampling, or equilibration.