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Full ideal-gas shell moments and entropy conventions

Physical scope

Finite ideal gas on a compact flat configuration torus at positive regular shell energy; compares two explicitly defined entropy conventions without asserting which one is physically preferred.

Example

Let integers $d, N \geq 1$ satisfy $n = d N \geq 3$ . Take a flat one-particle torus $Q = R^{d} / Λ$ , where $Λ = \sum_{i = 1}^{d} Z b_{i}$ for a basis $b_{1}, \dots, b_{d}$ of $R^{d}$ and $V : = ∣ \det (b_{1}, \dots, b_{d}) ∣ > 0$ , and set $Q_{N} = Q^{N} ≅ R^{n} / Λ^{N}$ , of position volume $V^{N}$ . Use equal masses $m > 0$ , Hamiltonian $H = \sum_{j = 1}^{n} p_{j}^{2} / (2 m)$ and reference $d q d p / (N! a_{0}^{n})$ with fixed action scale $a_{0} > 0$ . For $E > 0$ , write $α = n / 2$ . The density of shell measure is $D (E) = \frac{V^{N} (2 m)^{n / 2} α B_{n}}{N! a_{0}^{n}} E^{α - 1},$ where $B_{n}$ is the volume of the unit ball in $R^{n}$ . Under the full shell preparation, $K = E$ exactly and each quadratic momentum component has mean $E / n$ . Define the cumulative phase volume below energy $E$ by $W (E) : = m_{N}^{p h y s} ({H < E})$ . Choose an energy reference $E_{0} > 0$ in joules and a positive phase-volume reference $W_{0}$ in the same units as $W (E)$ . The reference $d q d p / (N! a_{0}^{n})$ is dimensionless, so $W (E)$ and $W_{0}$ are dimensionless and $E_{0} D (E)$ is dimensionless. The shell entropy $S_{s h} = k_{B} \log (E_{0} D (E))$ and cumulative volume entropy $S_{v o l} = k_{B} \log (W (E) / W_{0})$ have derivatives $\partial_{E} S_{s h} = k_{B} \frac{α - 1}{E}, \partial_{E} S_{v o l} = k_{B} \frac{α}{E} .$ They differ by $k_{B} / E$ at finite $n$ ; neither convention is selected by this calculation.

Facts & Assumptions

Given: The ideal quadratic Hamiltonian, finite flat configuration volume, positive regular shell energy, declared reference and adopted shell law.

[A1]

The reference uses the declared $1 / (N! a_{0}^{n})$ counting convention (Classical particle counting and physical observables), and the full regular-shell law is adopted as in Regular full-shell or finite-band preparation.

[A2]

$n \geq 3$ , $E > 0$ , $m > 0$ , and the position torus has volume $V^{N}$ ; the fixed positive action reference $a_{0}$ has units J s, the energy reference $E_{0}$ has units J, and $W_{0}$ has the same dimensionless phase-volume units as $W (E)$ . Entropy units use SI quantities and unit conventions, and the action unit uses Dimensions and SI unit conventions of the non-relativistic classical mechanics framework. The physical reference is that of [A1]; if shell gradients are used, apply the explicit conjugate-coordinate normalization of Regular full-shell or finite-band preparation. On $Σ_{E}$ , $p \neq 0$ and $\nabla_{p} H = p / m \neq 0$ , so $E$ is a regular value. For each compact energy interval $I_{E} \subset (0, \infty)$ , $H^{- 1} (I_{E}) = Q_{N} \times {p : ∣ p ∣^{2} / (2 m) \in I_{E}}$ is compact because $Q_{N}$ is compact and the momentum set is bounded. Hence the shell is nonempty and the compact regular-band hypotheses hold.

[F1]

For kinetic-plus-potential shells, the position marginal and component moments are those of Kinetic-plus-potential shell marginals and moments.

Verification

Given: [A1]–[A2] and the shell marginal identity [F1].

1.1A1A2F1algebra

In the kinetic shell theorem set $U = 0$ , $C_{m} = (2 m)^{n / 2}$ and $α = n / 2$ . The configuration integral of $1$ is $V^{N}$ , so its exact density formula gives the displayed $D (E)$ . For each $j$ the conditional component mean is $E / n$ , and on the shell $\sum_{j} p_{j}^{2} / (2 m) = E$ pointwise; therefore $K = E$ and its variance is zero.

2.1A2F1step 1.1algebra∎

Integrating $D (s) = C s^{α - 1}$ from $0$ to $E$ gives cumulative phase volume $W (E) = \int_{0}^{E} C s^{α - 1} d s = \frac{C}{α} E^{α},$ where $C = V^{N} (2 m)^{n / 2} α B_{n} / (N! a_{0}^{n})$ . Thus $\log (E_{0} D (E)) = \log (E_{0} C) + (α - 1) \log E$ and $\log (W (E) / W_{0}) = \log (C / (α W_{0})) + α \log E$ . Differentiation gives the two displayed derivatives. Since their difference is $k_{B} / E > 0$ , they are distinct at every finite $E > 0$ .

Remarks

The individual component moments are coordinate statements for the symmetric labelled lift of Classical particle counting and physical observables. A fixed label is not physical on the quotient, but permutation symmetry makes the component law equal to that of a uniformly selected degree of freedom; the total kinetic energy and its average per degree are quotient observables.

Full ideal-gas shell moments and entropy conventions

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources