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Exact canonical and grand ideal-gas preparations

Physical scope

Finite-N canonical sectors of a classical ideal gas of one operationally indistinguishable species on a bounded rectangular box, their grand-canonical sum including N=0, and a separately adopted three-dimensional classical H=c|p| branch; the Lebesgue product and polar calculations assume ACω; excludes quantum statistics, actual equilibration, and identifying the massless branch with m->0.

Statement

Assume ${A C}_{ω}$ as defined in The Axiom of Countable Choice ( ${A C}_{ω}$ ), for the Euclidean Lebesgue product and polar-coordinate suppliers used below. Let $\exp (x) : = \sum_{n = 0}^{\infty} x^{n} / n!$ be the real exponential of The real exponential function and the number $e$ by a power series, and let $\log$ be the natural logarithm of The natural logarithm as the inverse of the exponential function, with $\log^{'} (x) = 1 / x$ for $x > 0$ by The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t. Let $d, N \geq 1$ and let $B = \prod_{j = 1}^{d} [0, L_{j}] \subset R^{d}$ , where each side length $L_{j} > 0$ is in metres and $V : = \prod_{j} L_{j}$ is in $m^{d}$ . Let $m > 0$ be the common particle mass in kg, $β > 0$ in J $^{- 1}$ , and let $h_{0} : = a_{0} > 0$ be the fixed action reference in J s from Declared example preparations and thermodynamic interpretation. Take one species of $N$ operationally indistinguishable particles as in Classical particle counting and physical observables. On its auxiliary labelled cover, positions $q_{i} \in B$ are in metres and momenta $p_{i} \in R^{d}$ are in kg m s $^{- 1}$ ; the Hamiltonian is permutation invariant and the reference measure is $d ν_{N} = \frac{d q d p}{N! h_{0}^{d N}} .$ The symmetric labelled law represents the quotient preparation; a one-particle marginal below means the law of a uniformly selected particle. For $N \geq 1$ , define $H_{N} (q, p) = \sum_{i = 1}^{N} \frac{∣ p_{i} ∣^{2}}{2 m}$ and use the canonical preparation of Declared example preparations and thermodynamic interpretation. For the vacuum sector $N = 0$ , define $B^{0} \times R^{d \cdot 0}$ to be a singleton with reference measure $ν_{0} = 1$ , set $H_{0} = 0$ , and use empty-product conventions $V^{0} = 0! = h_{0}^{0} = 1$ ; thus $Z_{0} = 1$ . The mass parameter $m$ applies when particles are present; the vacuum sector has no mass coordinate. For all $N \geq 0$ , set $λ : = h_{0} \sqrt{β / (2 π m)}$ (metres) and $Z_{N} = \int e^{- β H_{N}} d ν_{N} = \frac{V^{N}}{N! λ^{d N}} .$ For $N \geq 1$ , the labelled-cover position coordinates are independent uniform points of $B$ and are independent of the momenta; all momentum components are independent centered Gaussians of variance $m / β$ . The joint law is permutation invariant. The energy satisfies $U = E H_{N} = d N / (2 β)$ and $Var (H_{N}) = d N / (2 β^{2})$ , with variance in J $^{2}$ . With the adopted Boltzmann constant $k_{B} > 0$ in J K $^{- 1}$ and temperature $T : = (k_{B} β)^{- 1} > 0$ in K from Declared example preparations and thermodynamic interpretation, define $F = - β^{- 1} \log Z_{N}$ and scale all box side lengths uniformly when varying $V$ , holding the shape, $N, T, d, m, h_{0}$ fixed. The model pressure $P : = - {(\partial F / \partial V)}_{N, T, d, m, h_{0}, s h a p e}$ has units J m $^{- d}$ (equivalently N m $^{1 - d}$ ; Pa when $d = 3$ ) and obeys $P V = N / β = N k_{B} T$ . The fixed-reference entropy is $S = k_{B} (\log Z_{N} + β U)$ and retains the chosen $N!$ and $h_{0}$ convention.

For $d = 3$ , the speed $v = ∣ p ∣ / m$ of a uniformly selected particle, in m s $^{- 1}$ , has density $f (v) = 4 π {(\frac{m β}{2 π})}^{3 / 2} v^{2} e^{- β m v^{2} / 2}, v \geq 0.$ Its squared-speed expectation, modal speed, and mean speed are $E [v^{2}] = 3 / (m β)$ in m $^{2}$ s $^{- 2}$ , $\sqrt{2 / (m β)}$ in m s $^{- 1}$ , and $E [v] = \sqrt{8 / (π m β)}$ in m s $^{- 1}$ , respectively.

For the separate grand-canonical branch of Declared example preparations and thermodynamic interpretation, let $μ_{c h e m}$ be a chemical potential in joules and $z = \exp (β μ_{c h e m}) > 0$ the dimensionless activity. Specify its full preparation on $Γ_{g} : = ⨆_{j \geq 0} ({j} \times Γ_{j})$ , where $Γ_{j} = B^{j} \times R^{d j}$ for $j \geq 1$ and $Γ_{0}$ is the vacuum singleton. Use the sigma-algebra of sets $A$ whose slices $A_{j} : = {x \in Γ_{j} : (j, x) \in A}$ are Borel, and adopt, whenever $0 < Ξ : = \sum_{j \geq 0} z^{j} Z_{j} < \infty$ , the probability law $P_{g} (A) : = Ξ^{- 1} \sum_{j \geq 0} z^{j} \int_{A_{j}} e^{- β H_{j}} d ν_{j} .$ The particle count is the map $N : Γ_{g} \to {0, 1, \dots}$ , $N (j, x) = j$ . This full-law assumption supplements the sector weights; those weights alone do not determine the within-sector law. Then $Ξ = \sum_{N = 0}^{\infty} z^{N} Z_{N} = \exp (z V / λ^{d}), P (N = j) = \exp (- ν) \frac{ν^{j}}{j!}, ν : = z V / λ^{d},$ so $E N = Var (N) = ν$ and the conditional law given $N$ is the canonical law above.

A distinct three-dimensional classical ultra-relativistic model adopts $H_{u r} (p) = c ∣ p ∣$ for every $p \in R^{3}$ , with $c > 0$ in m s $^{- 1}$ and $p$ in kg m s $^{- 1}$ , in the same box volume $V$ in m $^{3}$ . This extends the nonzero massless momentum formula continuously to $p = 0$ only to define a density on all of $R^{3}$ ; the point has Lebesgue measure zero and no physical zero-momentum massless state is asserted. Its separate canonical partition function is $Z_{N, u r} = \frac{V^{N}}{N! h_{0}^{3 N}} {(\frac{8 π}{(β c)^{3}})}^{N},$ with $U_{u r} = 3 N / β$ J and $Var (H_{u r}) = 3 N / β^{2}$ J $^{2}$ . Define its fixed-volume heat capacity by $C_{V, u r} : = (\partial U_{u r} / \partial T)_{V, N, c, h_{0}} = 3 N k_{B}$ ; it has units J K $^{- 1}$ . Its three-dimensional pressure obeys $P V = N / β$ and has units Pa. This branch is not obtained by setting $m = 0$ in the Gaussian model and makes no photon-number or quantum-statistics claim.

Facts & Assumptions

Given: ${A C}_{ω}$ , $d, N \geq 1$ , the rectangular box $B$ with side lengths in metres and volume $V$ in m $^{d}$ , one operationally indistinguishable species, $m > 0$ kg, $h_{0} = a_{0} > 0$ J s, $β > 0$ J $^{- 1}$ , and the Hamiltonian and symmetric reference measure in the Statement.

[F1]

Declared example preparations and thermodynamic interpretation: for a specified model with finite positive partition function, the canonical density is $Z^{- 1} e^{- β H}$ relative to its fixed reference; grand-sector weights are $z^{N} Z_{N} / Ξ$ . This is an adopted preparation, not an equilibration theorem.

[F2]

Gaussian, gamma and beta product integrals: positive Gaussian coefficients have the stated Gaussian moments and finite products; gamma integrals, equal-scale gamma convolution, and moment differentiation follow from its explicit evaluations and Gamma recurrence.

[F3]

Under ${A C}_{ω}$ , product Lebesgue measure agrees with Euclidean Lebesgue measure on Borel rectangles (On Borel subsets of R^{m+n}, the product lambda_m times lambda_n agrees with lambda_{m+n}).

[F4]

Lebesgue measure of a rectangular box is the product of its side lengths (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included).

[F5]

The Lebesgue factors are sigma-finite; Tonelli factors nonnegative product integrals and Fubini factors absolutely integrable signed product integrals (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, Fubini's theorem for L^1 functions on a sigma-finite product).

[F6]

Polar coordinates in $R^{3}$ give $\int g (p) d λ_{3} (p) = \int_{0}^{\infty} \int_{S^{2}} g (r ω) r^{2} d σ (ω) d r$ for nonnegative Borel $g$ , under ${A C}_{ω}$ ; $σ (S^{2}) = 4 π$ by the unit-ball volume formula (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma, The closed form for the volume of the unit $n$ -ball).

[F7]

The Gaussian integrals and moments in [F2] are improper Riemann integrals. Under ${A C}_{ω}$ , their bounded truncations agree with Lebesgue integrals, and monotone convergence passes nonnegative truncations to the full line (Riemann–Lebesgue comparison for distribution test integrands, Monotone convergence for the integral).

[F8]

Massless test-particle action and momentum scale: the adopted massless test sector has nonzero future-null mechanical momentum $P$ and does not identify it with $m U$ at $m = 0$ .

[F9]

Mechanical momentum and observer energy: for an observer $n$ with $g (n, n) = - 1$ , define spatial momentum $q_{n} (P) = P + g (P, n) n$ and observer energy $E_{n} (P) = - c g (P, n)$ ; in the massless sector $P$ is nonzero future null.

[F10]

Observers, rest spaces and the spatial projection at an event: in a time-oriented Lorentzian background $(M, g, N)$ , $N$ is the smooth future timelike field; at an event $n = N / \sqrt{- g (N, N)}$ is an observer, whose rest space is $n^{⊥}$ and whose spatial projection is $h_{n} (P) = P + g (n, P) n$ .

[F11]

The observer rest-space decomposition and the spatial projection: $g (h_{n} (P), h_{n} (P)) = g (P, P) + g (P, n)^{2}$ , and $g$ is positive definite on the three-dimensional rest space $n^{⊥}$ .

[F12]

Classical particle counting and physical observables: for one operationally indistinguishable species, the symmetric labelled lift uses the factor $1 / N!$ ; label-specific one-particle marginals of a permutation-invariant state are interpreted as those of a uniformly selected particle.

[F13]

The real exponential function and the number $e$ by a power series: $\exp (x)$ is the everywhere-convergent series $\sum_{n \geq 0} x^{n} / n!$ .

[F14]

The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ : $\exp (- x) = 1 / \exp (x)$ and $\exp (x) > 0$ for real $x$ .

[F15]

The natural logarithm as the inverse of the exponential function: $\log$ is the real natural logarithm on $(0, \infty)$ .

[F16]

The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: for $x > 0$ , $\log^{'} (x) = 1 / x$ .

[F17]

Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm gives $\log (x y) = \log x + \log y$ and $\log (x / y) = \log x - \log y$ for positive $x, y$ ; finite iteration gives $\log (x^{k}) = k \log x$ for integers $k \geq 0$ .

[F18]

Under ${A C}_{ω}$ , every point and coordinate hyperplane in $R^{3}$ is Lebesgue null (A box with a degenerate side is Lebesgue null, and so is every coordinate hyperplane in $R^{n}$ ).

[A1]

The canonical probability laws, grand-sector weights and thermodynamic identifications are adopted as in [F1], with the full grand law specified in [A3]; the calculations establish conditional consequences only. The one-species operational indistinguishability and its permutation-invariant labelled lift are model assumptions as in [F12], not consequences of common mass.

[A2]

The ultra-relativistic branch separately adopts the classical Hamiltonian $H_{u r} (p) = c ∣ p ∣$ in one observer's orthonormal spatial momentum coordinates. It is not a massless limit of the massive Hamiltonian and is not a quantum radiation model.

[A3]

In the grand branch adopt the full law $P_{g}$ on the disjoint union and slice sigma-algebra specified in the Statement. In particular, for a Borel $E \subseteq Γ_{j}$ , $P_{g} ({j} \times E) = Ξ^{- 1} z^{j} \int_{E} e^{- β H_{j}} d ν_{j}$ . This is an additional preparation assumption, not a consequence of sector weights alone.

Proof

Given: ${A C}_{ω}$ , $d, N \geq 1$ , the rectangular box $B$ with side lengths in metres and volume $V$ in $m^{d}$ , one operationally indistinguishable species, $m > 0$ kg, $h_{0} = a_{0} > 0$ J s, $β > 0$ J $^{- 1}$ , and the Hamiltonian and symmetric reference measure in the Statement.

1.1givenF1F2F3F4F5F7F12algebra

By [F3], Euclidean Lebesgue measure in the position and momentum coordinates is their product measure; [F5] then factors the nonnegative normalization integral. The box-volume formula [F4] gives $λ_{d N} (B^{N}) = V^{N}$ . Each of the $d N$ scalar momentum factors contributes $\int_{R} e^{- β p^{2} / (2 m)} d p = \sqrt{2 π m / β}$ by [F2] and [F7]. Including the stipulated factor $1 / (N! h_{0}^{d N})$ from [F12] yields $Z_{N} = V^{N} (N! h_{0}^{d N})^{- 1} (2 π m / β)^{d N / 2} = V^{N} / (N! λ^{d N})$ .

1.2givenA2F8F9F10F11algebra

For the separate massless branch, let $N_{t i m e}$ denote the background's smooth future-timelike field $N$ from [F10] and normalize it at the event by $n = N_{t i m e} / \sqrt{- g (N_{t i m e}, N_{t i m e})}$ . [F8] makes $P$ future null, so $g (P, n) < 0$ . By [F11], $∣ q_{n} (P) ∣^{2} = g (q_{n} (P), q_{n} (P)) = g (P, n)^{2}$ , hence $E_{n} (P) = - c g (P, n) = c ∣ q_{n} (P) ∣$ . Identifying $q_{n}$ with its components in an orthonormal basis of the positive rest space gives the adopted $H_{u r} (p) = c ∣ p ∣$ with the stated units.

2.1givenF2F3F5step 1.1

After normalization, the canonical density is the product of the uniform position density on $B^{N}$ and the $d N$ scalar Gaussian momentum densities. For measurable coordinate rectangles, [F3] and [F5] show their probabilities factor; hence the positions and all momentum components are independent, each position is uniform on $B$ , and each momentum component is centered with variance $m / β$ by [F2].

2.2A1A3F1F13F14step 1.1algebra

In the grand preparation [A3] with sector weights from [F1], use the vacuum convention $Z_{0} = 1$ and substitute $Z_{N} = V^{N} / (N! λ^{d N})$ for all $N \geq 0$ into the sector sum. With $ν = z V / λ^{d}$ , [F13] gives $Ξ = \sum_{N \geq 0} ν^{N} / N! = \exp (ν)$ , and [F14] gives the normalized sector weight $ν^{N} / (N! \exp (ν)) = \exp (- ν) ν^{N} / N!$ . Each summand in [A3] is a nonnegative measure on its sector, so countable additivity and this finite positive total mass make $P_{g}$ a probability measure.

2.3A1A2F1F2F3F5F6F18step 1.1algebra

By [F6], the momentum integral for one ultra-relativistic particle is $\int_{R^{3}} e^{- β c ∣ p ∣} d λ_{3} (p) = 4 π \int_{0}^{\infty} r^{2} e^{- β c r} d r = 4 π \frac{Γ (3)}{(β c)^{3}} = \frac{8 π}{(β c)^{3}},$ using [F2]. The point $p = 0$ is Lebesgue null by [F18], so the continuous extension adds no massless zero-momentum state. Product measure and Tonelli [F3, F5] give the $N$ -particle product, and the stated positional/counting reference then gives $Z_{N, u r}$ .

3.1givenF2step 2.1algebra

Each scalar momentum coordinate has second moment $m / β$ and fourth moment $3 m^{2} / β^{2}$ from [F2]. Thus each term $p_{i α}^{2} / (2 m)$ has mean $1 / (2 β)$ and variance $1 / (2 β^{2})$ ; independence from step 2.1 makes means and variances add, giving the stated $U$ and $Var (H_{N})$ .

3.2givenF2F6step 2.1algebra

For $d = 3$ , write $a = β / (2 m)$ . The one-particle momentum density is $(a / π)^{3 / 2} e^{- a ∣ p ∣^{2}}$ . By [F6], for every Borel $A \subseteq [0, \infty)$ the probability that $v = ∣ p ∣ / m$ lies in $A$ is $4 π (a / π)^{3 / 2} \int_{m A} r^{2} e^{- a r^{2}} d r .$ The substitution $r = m v$ therefore gives the speed density $4 π (m β / (2 π))^{3 / 2} v^{2} e^{- β m v^{2} / 2}$ on $v \geq 0$ .

3.3A3F13F14step 1.1step 2.2algebra

Shifting the index in the absolutely convergent series [F13] gives $\sum_{j \geq 0} j ν^{j} / j! = ν \exp (ν)$ and $\sum_{j \geq 0} j (j - 1) ν^{j} / j! = ν^{2} \exp (ν)$ . After normalization by [F14], $E N = ν$ and $E [N (N - 1)] = ν^{2}$ , hence $Var (N) = E [N (N - 1)] + E N - (E N)^{2} = ν$ . For each $j \geq 0$ , step 2.2 gives $P_{g} (N = j) = z^{j} Z_{j} / Ξ > 0$ . For any Borel $E \subseteq Γ_{j}$ , [A3] therefore gives $P_{g} ({j} \times E ∣ N = j) = (Ξ^{- 1} z^{j} \int_{E} e^{- β H_{j}} d ν_{j}) / (Ξ^{- 1} z^{j} Z_{j}) = Z_{j}^{- 1} \int_{E} e^{- β H_{j}} d ν_{j}$ , exactly the canonical law, including the vacuum singleton.

4.1A1F1F15F16F17step 1.1step 3.1algebra

By [F17] and [A1], $F = - β^{- 1} [N \log (V / λ^{d}) - \log N!]$ for fixed $T, N$ and fixed box shape. The log derivative [F16] gives $- \partial_{V} F = N / (β V)$ , with $P$ in J m $^{- d}$ , and therefore $P V = N / β$ . The entropy definition in [F1] gives $S = k_{B} (\log Z_{N} + β U)$ with the same fixed $h_{0}$ and factorial convention.

4.2F2step 3.2algebra

The speed density is proportional to $v^{2} e^{- a v^{2}}$ with $a = β m / 2$ . Its derivative has the sign of $2 v (1 - a v^{2})$ for $v > 0$ , so its unique positive maximum occurs at $v_{m o d e} = a^{- 1 / 2}$ . The second moment is $3 / (2 a) = 3 / (m β)$ by the gamma recurrence in [F2]. The mean is $E v = \frac{4 a^{3 / 2}}{\sqrt{π}} \int_{0}^{\infty} v^{3} e^{- a v^{2}} d v = \frac{4 a^{3 / 2}}{\sqrt{π}} \frac{Γ (2)}{2 a^{2}} = \frac{2}{\sqrt{π a}} = \sqrt{\frac{8}{π β m}} .$ These are model-distribution moments, not values for an individual measured particle.

5.1A1F1F2F5F16step 2.3algebra∎

Under the normalized radial density proportional to $r^{2} e^{- β c r}$ , gamma moments give mean energy $3 / β$ and variance $3 / β^{2}$ per particle. Independence from the product preparation adds these to $U_{u r} = 3 N / β$ and $Var (H_{u r}) = 3 N / β^{2}$ ; differentiating at fixed $V, N, c, h_{0}$ with $β^{- 1} = k_{B} T$ gives $C_{V, u r} = 3 N k_{B}$ J K $^{- 1}$ . Since $Z_{N, u r}$ is proportional to $V^{N}$ , [F16] gives $- \partial_{V} F = N / (β V)$ and $P V = N / β$ , with $P$ in Pa for this three-dimensional branch.

Remarks

The source M18 Ising-chain supplier is not used here. The massless momentum relation is a consequence of the adopted null-sector convention and the stated observer projection; the classical canonical $H = c ∣ p ∣$ preparation is a separate model assumption. No empirical data or precision are asserted.

Exact canonical and grand ideal-gas preparations

Statement

Facts & Assumptions

Proof

Remarks

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