Regular full-shell or finite-band preparation

For the full-shell branch, assume the Axiom of Choice required by the mathematical supplier Coarea and invariant probability on compact regular phase shells. Let $H:{\mathrm{\Gamma }}_N\to \mathbb{R}$ have a $C^2$ coordinate representative in every canonical cotangent chart. Require $H(gz)=H(z)$ for every $z\in {\mathrm{\Gamma }}_N$ and every within-species permutation $g\in G$ with the cotangent action specified by Classical particle counting and physical observables. This symmetry condition applies to both the shell and finite-band branches; ambient almost-everywhere symmetry alone would not specify a law on an individual zero-volume shell. Suppose there is an open interval $J\ni E$ on which $dH\mathrm{\ne }0$ and for which $H^{-1}(K)$ is compact whenever $K\subset J$ is compact. For a nonempty regular shell ${\mathrm{\Sigma }}_E=H^{-1}(\{E\})$, the cited theorem gives finite positive coarea mass $0<{\mu }_E(1)<\mathrm{\infty }$; adopt the microcanonical state $P_E={\mu }_E\mathrm{/}{\mu }_E(1)$ as the probability law for the specified preparation. Here ${\mu }_E$ is the shell measure disintegrating the declared phase reference over energy; its construction and hypotheses are exactly those in that theorem. The constant $c_N$ in the mathematical reference is chosen to match the fixed-$N$ physical reference of Classical particle counting and physical observables; the normalization defining $P_E$ also cancels a common positive rescaling of that reference.

For a physical application, choose fixed conjugate reference scales ${\mathrm{\ell }}_j,p_{0j}>0$ in each cotangent chart, of the respective coordinate and conjugate-momentum units, with ${\mathrm{\ell }}_j{p}_{0j}=A>0$ the same action scale for all $j$. For Cartesian positions these units are m and kg m s${}^{-1}$; other generalized coordinates have their own declared conjugate units. Put $Q^j=q^j\mathrm{/}{\mathrm{\ell }}_j$, $P_j=p_j\mathrm{/}{p}_{0j}$ and $\tilde{H}(Q,P)=H(q,p)$. These coordinates are dimensionless, while $\tilde{H}$ and $E$ remain in J. The actual reference Jacobian gives $d{m}_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=\frac{(A\mathrm{/}{a}_0{)}^n}{{\prod }_s{N}_s!}dQdP,n=dN\mathrm{.}$ Use this transformed constant in the shell theorem; in particular ${\mu }_E^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(1)$ has units J${}^{-1}$. The weighted shell area and gradient are computed in these dimensionless coordinates, not by adding squares of SI position and momentum derivatives of different dimensions. The canonical forms pull back to $A{\sum }_j{P}_{j}d{Q}^j$ and $A{\sum }_{j}d{Q}^j\wedge d{P}_j$. Thus physical time $t$ in seconds obeys $\frac{d{Q}^j}{dt}=A^{-1}{\mathrm{\partial }}_{P_j}\tilde{H},\frac{d{P}_j}{dt}=-A^{-1}{\mathrm{\partial }}_{Q^j}\tilde{H}\mathrm{.}$ Equivalently the Hamiltonian for the normalized mathematical two-form and the same time is $\tilde{H}\mathrm{/}A$, in s${}^{-1}$. If one also chooses an energy reference $E_{\ast }>0$ in J, then only with $\widehat{H}=\tilde{H}\mathrm{/}{E}_{\ast }$ and $\tau =E_{\ast }t\mathrm{/}A$ do the fully dimensionless equations become $d{Q}^j\mathrm{/}d\tau ={\mathrm{\partial }}_{P_j}\widehat{H}$ and $d{P}_j\mathrm{/}d\tau =-{\mathrm{\partial }}_{Q^j}\widehat{H}$. This is an explicit reexpression of the same trajectories and physical clock. It changes neither the adopted shell or band preparation nor the physical Hamiltonian. For dimensionless shell energy $e=E\mathrm{/}{E}_{\ast }$, the shell measures transform as ${\mu }_e^{\mathrm{n}\mathrm{u}\mathrm{m}}=E_{\ast }{\mu }_{E_{\ast }e}^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ so that the coarea integral is unchanged.

Alternatively, for a specified compact regular energy band $B\subset J$ with $0<m_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(H^{-1}(B))={\int }_B{\mu }_E^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(1)dE<\mathrm{\infty },$ one may adopt the normalized restricted physical reference $P_B(A)=\frac{m_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(A\cap H^{-1}(B))}{m_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(H^{-1}(B))}$ for each measurable phase event $A$. Here $m_N^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ is the declared fixed-$N$ reference of Classical particle counting and physical observables, and ${\mu }_E^{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ is its coarea measure; choosing this physical positive constant in Classical configuration, cotangent phase and reference measures scales both measures in the coarea identity equally. This is a distinct finite-band preparation, uniform relative to phase volume in the band. Restricting either law to a dynamically accessible invariant subset or imposing further conserved quantities defines another preparation and requires a separately specified measure. Neither law nor ergodicity is deduced from deterministic Hamiltonian evolution.