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Exact finite ideal-gas volume fluctuations

Physical scope

Fixed-N ideal gas in the declared dV/V0 isothermal-isobaric preparation with positive external pressure; excludes alternative piston priors and experimental claims.

Example

Let $N \geq 1$ , $d \geq 1$ , $n = d N$ , $m > 0$ , $a_{0} > 0$ , $V_{0} > 0$ , $T > 0$ and $P_{e x t} > 0$ . Use the unit-Lebesgue-volume shape $Q_{1} \subset R^{d}$ and scaled coordinates $y \in Q_{1}^{N}$ , $p \in R^{n}$ of Isothermal-isobaric joint preparation, with physical positions $q_{i} = V^{1 / d} y_{i}$ . Define the kinetic energy and Hamiltonian by $K (p) : = \sum_{j = 1}^{n} \frac{p_{j}^{2}}{2 m}, H_{N} (V, y, p) : = K (p) .$ In the adopted joint reference $d V / V_{0}$ times the scaled position-momentum reference, the weight is $V^{N} e^{- β (K + P_{e x t} V)}$ . Its normalization is $Δ_{N} = \frac{1}{V_{0} a_{0}^{n}} {(\frac{2 π m}{β})}^{n / 2} \frac{1}{(β P_{e x t})^{N + 1}} .$ The volume mean and variance and mean enthalpy are $⟨ V ⟩ = \frac{N + 1}{β P_{e x t}}, Var (V) = \frac{N + 1}{(β P_{e x t})^{2}}, ⟨ K + P_{e x t} V ⟩ = \frac{n / 2 + N + 1}{β} .$ The extra one in the finite- $N$ volume factor follows from the stated $d V / V_{0}$ prior; it is not a universal piston correction.

Facts & Assumptions

Given: The ideal-gas Hamiltonian, shape-scaled volume domain, adopted $d V / V_{0}$ preparation and positive finite parameters.

[A1]

The volume density is proportional to $V^{N} e^{- β (K + P_{e x t} V)}$ under the adopted preparation (Isothermal-isobaric joint preparation, inherited by Isobaric volume and enthalpy fluctuations).

[A2]

$β > 0$ , $P_{e x t} > 0$ , and the spatial reference shape has unit dimensionless volume; $V_{0}$ is a fixed positive volume scale. Energy and action units follow Dimensions and SI unit conventions of the non-relativistic classical mechanics framework; temperature and $k_{B}$ follow SI quantities and unit conventions. The adopted preparation fixes $V, V_{0}$ in m $^{d}$ and $P_{e x t}$ in J m $^{- d}$ , so $P_{e x t} V$ is an energy.

[F1]

The Gaussian integral $\int_{- \infty}^{\infty} e^{- x^{2}} d x = \sqrt{π}$ gives the unit integral $\int_{R} e^{- x^{2}} d x = \sqrt{π}$ . For $a = β / (2 m) > 0$ , the linear substitution $y = \sqrt{a} p$ gives $\int_{R} e^{- β p^{2} / (2 m)} d p = \sqrt{2 π m / β}$ ; applying it in each of the $n$ momentum coordinates yields $(2 π m / β)^{n / 2}$ .

[F2]

The isobaric response identities apply under their local moment envelopes (Isobaric volume and enthalpy fluctuations).

Verification

Given: [A1]–[A2] and the Gaussian and isobaric-response facts [F1]–[F2].

1.1A1A2F1algebra

Integrating the momentum factors gives [F1]; integrating the unit shape coordinates gives one, while the configuration scaling contributes $V^{N}$ . Hence $Δ_{N} = \frac{1}{N! V_{0} a_{0}^{n}} {(\frac{2 π m}{β})}^{n / 2} \int_{0}^{\infty} V^{N} e^{- s V} d V, s = β P_{e x t} > 0.$ Let $I_{j} (s) = \int_{0}^{\infty} V^{j} e^{- s V} d V$ . Direct integration gives $I_{0} = 1 / s$ ; integration by parts, with $V^{j} e^{- s V} \to 0$ at infinity and vanishing lower boundary for $j \geq 1$ , gives $I_{j} = j I_{j - 1} / s$ . Thus $I_{j} = j! / s^{j + 1}$ , and $I_{N} = N! / s^{N + 1}$ cancels the $N!$ in the reference; the displayed finite positive normalizer follows.

2.1A1A2step 1.1algebra

The normalized volume density is $V^{N} e^{- s V} / I_{N} (s)$ . Applying the same integral formula to $I_{N + 1}$ and $I_{N + 2}$ gives $⟨ V ⟩ = (N + 1) / s$ and $⟨ V^{2} ⟩ = (N + 1) (N + 2) / s^{2}$ . Subtracting the square of the mean gives $Var (V) = (N + 1) / s^{2}$ . Since $s = β P_{e x t}$ , these are the claimed volume moments.

2.2A2F1F2step 1.1algebra

To check [F2], first use fixed sufficient statistics $(K, V)$ with coefficients $(- β, - β P_{e x t})$ . On a compact positive $(β, P_{e x t})$ set, choose lower bounds $β_{0}, P_{0} > 0$ . For $K, V \geq 0$ , absorbing the quadratic polynomial into half of each exponential gives $e^{- β (K + P_{e x t} V)} (1 + K^{2} + V^{2}) \leq C e^{- β_{0} K / 2 - β_{0} P_{0} V / 2} .$ This is integrable against the full reference $ν_{N}$ : the momentum integral is Gaussian by [F1] and the volume integral is $I_{N} (β_{0} P_{0} / 2) < \infty$ by step 1.1. At any fixed pressure $P_{*} > 0$ , the statistics $(B_{*}, V) = (K + P_{*} V, V)$ satisfy $u B_{*} + v V = u K + (u P_{*} + v) V$ . In a small compact neighborhood of $(u, v) = (- β, 0)$ both coefficients are bounded strictly below zero; $1 + B_{*}^{2} + V^{2} \leq C_{*} (1 + K^{2} + V^{2})$ gives the same Gaussian-volume majorant. Thus the exact local exponential envelopes for [F2] hold, including the kinetic-energy factor.

3.1A2F1F2step 2.1step 2.2algebra∎

The momentum factors are independent of $V$ . For $a = β / (2 m) > 0$ , the boundary term $p e^{- a p^{2}}$ vanishes at both infinities, and integration of $(p e^{- a p^{2}})^{'} = e^{- a p^{2}} - 2 a p^{2} e^{- a p^{2}}$ gives $\int p^{2} e^{- a p^{2}} d p = (2 a)^{- 1} \int e^{- a p^{2}} d p$ . Dividing by the positive Gaussian normalizer [F1] and summing the $n$ components gives $⟨ K ∣ V ⟩ = n / (2 β)$ . Step 2.1 gives $P_{e x t} ⟨ V ⟩ = (N + 1) / β$ ; adding the two finite means proves $⟨ K + P_{e x t} V ⟩ = (n / 2 + N + 1) / β$ . The envelope check in step 2.2 also licenses the response identities [F2] for this preparation.

Remarks

If $P_{e x t} = 0$ , the normalization contains $\int_{0}^{\infty} V^{N} d V$ and diverges. If $P_{e x t} < 0$ , its integrand grows exponentially at large $V$ and also diverges. Thus the positive-pressure condition is essential.

Exact finite ideal-gas volume fluctuations

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources