Hamiltonian phase, complete-domain flow and reference measure

Adopt ${\mathrm{A}\mathrm{C}}_{\omega }$ for the density-measure construction (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). A Hamiltonian phase datum consists of a Hausdorff second-countable smooth manifold $M$ of finite even dimension $2n$ without boundary, a symplectic form $\omega$ on $M$, and an autonomous smooth function $H:M\to \mathbb{R}$. Here symplectic means that $\omega$ is a smooth closed two-form and is nondegenerate at every point (Symplectic form and symplectic manifold). The Hamiltonian vector field $X_H$ is the unique smooth field satisfying ${\iota }_{X_H}\omega =dH$ (Hamiltonian vector field and Hamiltonian function, Hamiltonian vector fields exist uniquely for smooth functions). For each $z\in M$, let $I_z\subseteq \mathbb{R}$ be the maximal open interval containing $0$ on which the unique maximal $C^{\mathrm{\infty }}$ integral curve ${\gamma }_z:I_z\to M$ satisfying ${\dot{\gamma }}_z(t)=X_H({\gamma }_z(t))$ and ${\gamma }_z(0)=z$ is defined; its existence and uniqueness follow from the smooth ODE theorem (The fundamental theorem for autonomous smooth ODEs). Set $D=\{(t,z):z\in M,t\in I_z\}$ and define the maximal local flow $\mathrm{\Phi }:D\to M$ by $\mathrm{\Phi }(t,z)={\gamma }_z(t)$. Write ${\mathrm{\Phi }}_t(z)=\mathrm{\Phi }(t,z)$ when $(t,z)\in D$. Uniqueness gives ${\mathrm{\Phi }}_0(z)=z$ and ${\mathrm{\Phi }}_t({\mathrm{\Phi }}_s(z))={\mathrm{\Phi }}_{t+s}(z)$ whenever the terms are defined. The domain $D$ is open, and in local charts $\mathrm{\Phi }$ is smooth in the initial state and $C^1$ in time, as supplied by the stated ODE theorem. Completeness of the flow on all of $M$ is not part of this datum.

A Borel set $R\subseteq M$ is a complete invariant domain for this datum if $\{t\}\times R\subseteq D$ and ${\mathrm{\Phi }}_t(R)=R$ for every $t\in \mathbb{R}$. Thus each restricted map ${\mathrm{\Phi }}_t{\mathrm{\mid }}_R:R\to R$ is defined everywhere on $R$. The definition makes no assertion that $M$ itself is complete or that such an $R$ has finite measure.

The symplectic orientation has volume form $\mathrm{\Omega }={\omega }^n\mathrm{/}n!$ (Symplectic manifolds have a canonical orientation and volume form). For a fixed positive reference action $a_0$, define the positive smooth density ${\eta }_0=\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }\mathrm{/}{a}_0^n$ and let $m_0$ be its intrinsic Borel measure (Density bundle and smooth density fields, Pointwise Borel nonnegative densities, Intrinsic density measure and its chart restriction, Positive smooth densities give Radon volume). The measure $m_0$ may have infinite total mass. Use SI units for a physical interpretation. In a standard mechanical cotangent chart, $q^i$ has unit m, $p_i$ has unit kg m/s, $H$ has unit J, $\omega$ and $a_0$ have unit J s, and ${\eta }_0$ is dimensionless. The scale $a_0$ is a normalization convention and does not assert quantization.

An ensemble on $R$ is a Borel probability measure $\mu$ on $R$, distinct from an individual trajectory. If $\mu$ has density $\rho$ with respect to $m_0{\mathrm{\mid }}_R$, then $\rho \ge 0$ is Borel and ${\int }_R\rho d{m}_0=1$. For $\beta >0$, the canonical probability, when it exists, is the Borel measure ${\mu }_{\beta }(B)=Z_{\beta }^{-1}{\int }_B{e}^{-\beta H}d{m}_0,Z_{\beta }={\int }_R{e}^{-\beta H}d{m}_0,$ for Borel $B\subseteq R$, provided $0<Z_{\beta }<\mathrm{\infty }$. For a temperature-parameterized physical interpretation, take $T>0$ as an absolute temperature in K and $k_B>0$ as a supplied energy-temperature conversion constant in J/K, and set $\beta =(k_{B}T{)}^{-1}$ in J${}^{-1}$.

The flow's actual domain, its complete invariant subsets, the reference measure, and any ensemble probability are separate data. Smoothness alone does not imply completeness, finite phase volume, normalizability of a canonical weight, recurrence, or ergodicity.