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Exact ideal-gas canonical and grand laws

Physical scope

Finite-N monatomic classical ideal gas in a bounded position region, with a countable grand-canonical number law; excludes quantum exchange and empirical validation.

Example

Let $d, N \geq 1$ , position volume $V > 0$ , particle mass $m > 0$ , fixed action reference $a_{0} > 0$ , and $β > 0$ . For $H_{N} = \sum_{j = 1}^{d N} p_{j}^{2} / (2 m)$ with position region of volume $V^{N}$ and reference $d q d p / (N! a_{0}^{d N})$ , put $λ_{T} = a_{0} \sqrt{\frac{β}{2 π m}} and Z_{N} = \frac{(V / λ_{T}^{d})^{N}}{N!} .$ Then $U = d N / (2 β)$ , $Var (H_{N}) = d N / (2 β^{2})$ and $C_{V} = d N k_{B} / 2$ . With chemical potential $μ$ , set $a = e^{β μ} V / λ_{T}^{d}$ . The grand normalizer and particle law are $Ξ = e^{a}, P (N = j) = e^{- a} \frac{a^{j}}{j!}, E [N] = Var (N) = a .$ Conditioned on $N$ , the phase state is the canonical one above. These exact identities concern the adopted finite classical preparation. The component factorization is stated in labelled momentum coordinates on the symmetric lift; each single-coordinate marginal is the law of a uniformly selected degree of freedom, while permutation-invariant sums and averages are the quotient observables.

Facts & Assumptions

Given: The ideal Hamiltonian and phase reference, positive temperature, finite position volume, canonical and grand preparations, and the local moment envelopes needed for response derivatives.

[A1]

The canonical and grand probability laws are the adopted preparations; their response identities apply under the local exponential envelopes (Canonical energy and fluctuation responses, Grand responses in beta and chemical potential).

[A2]

$d, N \geq 1$ , $V, m, a_{0}, β > 0$ , and the position domain has volume $V^{N}$ ; $T$ , $k_{B}$ and $β$ follow SI quantities and unit conventions.

[A3]

On compact positive $β$ intervals, Gaussian polynomial moments are bounded by integrable Gaussian tails; on compact $(β, μ)$ sets, the grand factorial series with any fixed polynomial factor in $N$ converges uniformly.

[F1]

The Gaussian integral $\int_{- \infty}^{\infty} e^{- x^{2}} d x = \sqrt{π}$ proves $\int_{R} e^{- x^{2}} d x = \sqrt{π}$ . For each $a > 0$ , the linear diffeomorphism $y = \sqrt{a} x$ has Jacobian $\sqrt{a}$ , giving the local scaling $\int_{R} e^{- a x^{2}} d x = a^{- 1 / 2} \int_{R} e^{- y^{2}} d y = \sqrt{π / a}$ .

Verification

Given: [A1]–[A3] and the Gaussian and response interfaces [F1].

1.1A2F1algebra

Separate the position and momentum integrals and apply [F1] in each of the $d N$ momentum coordinates. This gives $Z_{N} = \frac{V^{N}}{N! a_{0}^{d N}} {(\frac{2 π m}{β})}^{d N / 2} = \frac{(V / λ_{T}^{d})^{N}}{N!} .$ The position density is uniform on the region and each momentum factor is a centered Gaussian of variance $m / β$ .

2.1A1A2A3F1step 1.1algebra

For $β$ in a compact interval $[β_{0}, β_{1}] \subset (0, \infty)$ , the factors $H_{N}^{r} e^{- β H_{N}}$ for $r \leq 2$ are bounded by a constant times $e^{- β_{0} H_{N} / 2}$ , since $x^{r} e^{- β_{0} x / 2}$ is bounded for $x \geq 0$ ; the latter has a finite Gaussian integral by [F1]. Thus [A3] verifies the canonical response theorem's envelope, which gives $U = - \partial_{β} \log Z_{N} = d N / (2 β)$ , $Var (H_{N}) = \partial_{β}^{2} \log Z_{N} = d N / (2 β^{2})$ , and the chain rule gives $C_{V} = d N k_{B} / 2$ .

2.2A1A2step 1.1algebra

The grand sum has sector weight $e^{β μ j} Z_{j} = a^{j} / j!$ , so its exponential series gives $Ξ = e^{a}$ and $P (N = j) = e^{- a} a^{j} / j!$ . Summing the shifted series gives $E [N] = a$ and $E [N (N - 1)] = a^{2}$ , hence $Var (N) = a$ .

3.1A1A3F1step 1.1step 2.2∎

On a compact $(β, μ)$ set, $a$ is bounded by some $A < \infty$ . For each fixed $r$ , the ratio of successive terms in $\sum_{j \geq 0} (1 + j^{r}) A^{j} / j!$ tends to zero, uniformly once $j$ is large; therefore the series converges uniformly. Combined with the Gaussian bound in step 2.1, this verifies the grand theorem's energy-number envelope. Applying its response identity gives $\partial_{μ} ⟨ N ⟩ = β Var (N) = β a$ . Within each number sector, dividing the sector Boltzmann density by its sector probability cancels the grand factor, leaving precisely the canonical conditional phase law.

Remarks

The action reference keeps $λ_{T}$ a length and $V / λ_{T}^{d}$ dimensionless. No quantum degeneracy criterion or experimental comparison is included.

Exact ideal-gas canonical and grand laws

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources