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Gaussian and gamma product tools

1 · Prerequisites

2 · Summary

This page supplies finite Gaussian and gamma product identities, a Gaussian covariance calculation under the stated Countable Choice measure hypothesis, the normalized marginal of a specified spherical-shell preparation under Countable Choice, and fine-entropy conservation under supplied smooth volume-preserving flow data. Physical results then calculate canonical and grand-canonical ideal gases, a finite Ising ring, a one-dimensional hard-rod gas, and a finite spherical-shell gas. Each physical conclusion is conditional on its explicitly adopted preparation, reference measure, units and finite-domain assumptions; the continuous Lebesgue product and Jordan-volume calculations state their Countable Choice conditions. No experiment, equilibration mechanism, or general ensemble-equivalence claim is asserted. Moment differentiation follows the explicitly evaluated moments and Gamma recurrence; finite-ring transfer identities use the published periodic Ising-chain theorem. The continuous models share the explicitly adopted positive action reference, with their original label-counting conventions retained. Draft status and the ordinary independent review obligations remain.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

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Gaussian, gamma and beta product integrals

Statement

For $a > 0$ and an integer $r \geq 0$ , define $I_{r} (a) : = \int_{R} x^{2 r} e^{- a x^{2}} d x$ as the two-sided improper Riemann integral. Then

$I_{r} (a) = \frac{Γ (r + \frac{1}{2})}{a^{r + 1 / 2}}, \int_{R} x^{2 r + 1} e^{- a x^{2}} d x = 0,$

where $Γ (s) : = \int_{0}^{\infty} t^{s - 1} e^{- t} d t$ for $s > 0$ . In particular, $I_{0} (a) = \sqrt{π / a}$ and $I_{1} (a) = \sqrt{π} / (2 a^{3 / 2})$ . For every compact interval $K = [a_{0}, a_{1}] \subset (0, \infty)$ , $I_{r}$ is continuously differentiable on $K$ and

$I_{r}^{'} (a) = - I_{r + 1} (a) .$

For $s, t > 0$ , put $B (s, t) : = \int_{0}^{1} u^{s - 1} (1 - u)^{t - 1} d u$ . Then $B (s, t)$ is finite and

$Γ (s) Γ (t) = Γ (s + t) B (s, t), Γ (s + 1) = s Γ (s), Γ (\frac{1}{2}) = \sqrt{π} .$

Consequently, if independent positive random variables have densities $x^{s - 1} e^{- x} / Γ (s)$ and $x^{t - 1} e^{- x} / Γ (t)$ , their sum has density $x^{s + t - 1} e^{- x} / Γ (s + t)$ .

Facts & Assumptions

Given: $a > 0$ , an integer $r \geq 0$ , and positive real parameters $s, t$ .

[F1]

The improper integral of $e^{- x^{2}}$ over $R$ is finite and positive: the improper Gaussian integral is finite and positive.

[F2]

The square of the one-dimensional Gaussian integral is the plane Gaussian integral: the square of the one-dimensional Gaussian integral equals the nonnegative plane Gaussian integral.

[F3]

The plane Gaussian integral equals $π$ by polar coordinates: the nonnegative plane Gaussian integral equals $π$ , with the polar substitutions justified on compact half-annuli away from the origin.

[F4]

Change of variables for an injective $C^{1}$ map on a compact Jordan set: an injective $C^{1}$ map with invertible derivative on an open neighborhood of a compact Jordan set transforms a bounded integrable function by the absolute Jacobian determinant.

[F5]

Continuity and derivatives of positive-base real powers: for any fixed real exponent $γ$ , the positive-base power $a^{γ}$ is differentiable on $(0, \infty)$ , with derivative $γ a^{γ - 1}$ ; these positive-base powers are continuous.

[F6]

Measures agreeing on a generating pi-system are equal under an increasing finite-measure exhaustion from that pi-system: two measures agreeing on a generating pi-system are equal if that system contains an increasing common finite-measure exhaustion of the whole space.

Proof

1.1givenalgebra

For $s > 0$ , the gamma integrand on $(0, 1)$ is bounded by $t^{s - 1}$ , whose integral is $1 / s$ . For $t \geq 1$ , $t^{s - 1} e^{- t / 2}$ is bounded, so $t^{s - 1} e^{- t} \leq C_{s} e^{- t / 2}$ and the tail converges. On a compact positive $s$ -interval, the lower endpoint bounds the near-zero power and the upper endpoint bounds the exponential tail uniformly. Hence $Γ (s)$ is finite and has controlled endpoint tails.

1.2F2F3algebra

Let $I = \int_{R} e^{- x^{2}} d x$ . By [F2], $I^{2}$ equals the nonnegative plane Gaussian integral, and [F3] evaluates that integral as $π$ . Hence $I^{2} = π$ .

2.1F1step 1.2algebra

By [F1], $I$ is finite and positive. The positive square root in step 1.2 is therefore $I = \sqrt{π}$ .

2.2step 1.1algebra

Integration by parts on $[ε, R]$ gives $\int_{ε}^{R} t^{s} e^{- t} d t = [- t^{s} e^{- t}]_{ε}^{R} + s \int_{ε}^{R} t^{s - 1} e^{- t} d t$ . The boundary terms vanish by step 1.1, so $Γ (s + 1) = s Γ (s)$ ; direct integration gives $Γ (1) = 1$ .

2.3step 1.1algebra

For $s, t > 0$ , the beta integrand is integrable near $0$ by comparison with $u^{s - 1}$ and near $1$ by comparison with $(1 - u)^{t - 1}$ . On each compact rectangle $[1 / j, j]^{2}$ the continuous product integrand $x^{s - 1} y^{t - 1} e^{- (x + y)}$ has integral equal to the product of its one-dimensional integrals, as follows by factoring the finite rectangular Riemann sums. These nested rectangles exhaust the positive quadrant and their integrals tend to $Γ (s) Γ (t)$ by step 1.1.

3.1step 1.2step 2.1algebra

Substitution $t = x^{2}$ on compact half-intervals and passage to the convergent improper limits gives $Γ (\frac{1}{2}) = 2 \int_{0}^{\infty} e^{- x^{2}} d x = I = \sqrt{π}$ by steps 1.2 and 2.1.

3.2step 1.1step 2.3F4algebra

On $A_{j} = [1 / j, j] \times [1 / j, 1 - 1 / j]$ in coordinates $(r, u)$ , the map $Φ (r, u) = (r u, r (1 - u))$ is injective, has determinant $- r \neq 0$ , and is a $C^{1}$ diffeomorphism on an open neighborhood. Its image is a compact Jordan set. The images exhaust the positive quadrant cofinally with the rectangles in step 2.3: each compact rectangle stays a positive distance from both axes and has bounded $x + y$ , while each image is compact and stays away from the axes. Applying [F4] on $A_{j}$ and taking the increasing limits gives $Γ (s) Γ (t) = \int_{0}^{\infty} r^{s + t - 1} e^{- r} d r B (s, t) = Γ (s + t) B (s, t)$ . The limits are finite by step 1.1 and the endpoint bounds in step 2.3.

4.1givenF1F4step 1.1step 1.2step 2.1step 2.2step 3.1algebra

For integer $r \geq 0$ , $x^{2 r} e^{- a x^{2} / 2}$ is bounded on $R$ , so $x^{2 r} e^{- a x^{2}} \leq C e^{- a x^{2} / 2}$ and [F1] gives absolute convergence; odd moments converge absolutely and vanish by reflection. On compact intervals the substitution $y = \sqrt{a} x$ is covered by [F4], and its improper limit gives $I_{0} (a) = a^{- 1 / 2} I_{0} (1) = \sqrt{π / a}$ by steps 1.2 and 2.1. The substitution $t = a x^{2}$ on both half-lines gives $I_{r} (a) = a^{- r - 1 / 2} Γ (r + \frac{1}{2})$ ; finiteness follows from step 1.1, the half-integer starting value from step 3.1, and recurrence from step 2.2.

4.2givenstep 1.1step 3.2F4algebra

Let $C = [c, d] \subset (0, \infty)$ and set $Q_{j} = [1 / j, j]^{2}$ . The event $A_{j} = Q_{j} \cap {(x, y) : x + y \in C}$ is Jordan measurable. On each grid rectangle $I \times J \subset Q_{j}$ , independence gives probability $P (X \in I) P (Y \in J) = (\int_{I} f_{s}) (\int_{J} f_{t})$ ; because the marginal densities are continuous on $Q_{j}$ , the finite rectangular Riemann sums factor and converge to the integral of $f_{s} (x) f_{t} (y)$ on that rectangle. The inner and outer grid unions for $A_{j}$ differ only in cells meeting its finitely many line-segment boundary pieces; their total area tends to zero, and $f_{s} f_{t}$ is bounded on $Q_{j}$ . Thus $P ((X, Y) \in A_{j}) = \int_{A_{j}} f_{s} f_{t} d x d y$ . As $Q_{j}$ increases to the positive quadrant, countable additivity gives probabilities increasing to $P (X + Y \in C)$ , and the nonnegative improper integrals increase to the integral over $x, y > 0$ . The images of $C \times [1 / j, 1 - 1 / j]$ under $(r, u) \mapsto (r u, r (1 - u))$ increase to $A_{C} = {x, y > 0 : x + y \in C}$ . On each such compact set the map is injective with absolute Jacobian $r$ , so [F4] transforms the integral to $r^{s + t - 1} e^{- r} u^{s - 1} (1 - u)^{t - 1} / (Γ (s) Γ (t))$ . Its rectangular Riemann sums factor; as $j \to \infty$ the $u$ -integral tends to $B (s, t)$ . By step 3.2, $P (X + Y \in C) = \int_{C} r^{s + t - 1} e^{- r} / Γ (s + t) d r$ . A singleton C has probability zero by the same zero-area grid-boundary argument. The density integrates to one by step 1.1.

5.1F5step 2.2step 4.1algebra

Fix the integer $r \geq 0$ . Step 4.1 evaluated $I_{r} (a) = Γ (r + \frac{1}{2}) a^{- r - 1 / 2}$ for every $a > 0$ using only convergent improper substitutions, and step 2.2 proved the Gamma recurrence by integration by parts; neither argument used parameter differentiation. By [F5], differentiation of this established explicit function gives $I_{r}^{'} (a) = - (r + \frac{1}{2}) Γ (r + \frac{1}{2}) a^{- r - 3 / 2} = - Γ (r + \frac{3}{2}) a^{- r - 3 / 2} = - I_{r + 1} (a) .$ The same positive-base power formula makes this derivative continuous on $(0, \infty)$ , hence on a neighborhood of every compact $K \subset (0, \infty)$ . This proves the full parameter clause without interchanging a derivative and an improper integral.

6.1F6step 4.2construct∎

Let $μ$ be the law of $X + Y$ on $(0, \infty)$ and let $ν$ be the probability measure with the density in step 4.2. The collection $P = {\emptyset} \cup {[a, b] : 0 < a \leq b < \infty}$ is closed under intersections, allowing a singleton or the empty set. It generates the Borel sigma-algebra of the positive half-line: its members are closed there, and every relatively open interval is a countable union of closed subintervals with positive endpoints; unbounded intervals are obtained by an additional increasing union. Step 4.2 gives equality of $μ$ and $ν$ on every member of $P$ , including the singleton case. The sets $P_{j} = [1 / j, j]$ for integers $j \geq 1$ increase to $(0, \infty)$ , lie in $P$ , and have common measure at most $1$ . Thus [F6] proves $μ = ν$ on all Borel sets, which establishes the claimed gamma convolution law.

Remarks

The parameter-derivative clause follows from the independently evaluated moments and Gamma recurrence. No research-module token is used as a logical supplier, and these integral identities make no empirical claim. No claim is made for $a \leq 0$ , $s \leq 0$ or $t \leq 0$ .

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Jacobian identity for a specified smooth diffeomorphic flow

Statement

Let $n \geq 1$ , let $I \subseteq R$ be an open interval containing $0$ , and let $U \subseteq R^{n}$ be open. Suppose $χ : I \times U \to R^{n}$ is jointly $C^{2}$ , each $χ_{t} (a) : = χ (t, a)$ is a $C^{2}$ diffeomorphism from $U$ onto an open set $U_{t}$ , and $χ_{0} (a) = a$ . Let $D : = {(t, x) : t \in I, x \in U_{t}}$ be open and let $b : D \to R^{n}$ be $C^{1}$ with $\partial_{t} χ (t, a) = b (t, χ (t, a))$ . Define $F (t, a) : = D_{a} χ (t, a)$ , $J (t, a) : = \det F (t, a)$ , and $div b : = \sum_{i = 1}^{n} \partial_{x_{i}} b_{i}$ . Then

$\partial_{t} F = (D_{x} b) (t, χ) F, \partial_{t} J = J (div b) (t, χ), J (0, a) = 1.$

In particular $J (t, a) > 0$ for all $(t, a) \in I \times U$ , and if $div b = 0$ on $D$ , then $J (t, a) = 1$ . The same identities hold on the flat torus $T^{n} : = R^{n} / Λ$ , where $Λ = Z ℓ_{1} + \dots + Z ℓ_{n}$ for a linearly independent basis $ℓ_{1}, \dots, ℓ_{n}$ and the smooth structure and Euclidean metric descend from $R^{n}$ . In that clause, let $χ : I \times T^{n} \to T^{n}$ be jointly $C^{2}$ , with each $χ_{t}$ a diffeomorphism and $χ_{0} = id$ . Let $π : T T^{n} \to T^{n}$ be the tangent-bundle projection, and let $b : I \times T^{n} \to T T^{n}$ have $C^{1}$ chart coefficients and satisfy $π (b (t, a)) = a$ and $\partial_{t} χ (t, a) = b (t, χ (t, a))$ . In every translation chart write $b = \sum_{i} b^{i} \partial_{x^{i}}$ and define $div b : = \sum_{i} \partial_{x^{i}} b^{i}$ . Define $F (t, a)$ to be the matrix of $D_{a} χ_{t} : T_{a} T^{n} \to T_{χ_{t} (a)} T^{n}$ in the translation frames and $J (t, a) : = \det F (t, a)$ . Translation-chart changes have derivative $I$ , so these matrices, determinants and divergences agree on overlaps. These are identities for supplied flow data; no existence or global-flow theorem is asserted.

Facts & Assumptions

Given: The interval $I$ , open set $U$ , jointly $C^{2}$ family $χ$ , diffeomorphism and initial-identity hypotheses, and $C^{1}$ velocity field $b$ in the Statement.

[F1]

Clairaut--Schwarz theorem for continuous second partial derivatives: the two mixed second partial derivatives of a $C^{2}$ scalar function on an open Euclidean set agree.

[F2]

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ : for totally differentiable maps, the derivative of a composite is the composite of the derivatives.

[F3]

The derivative of the determinant along an invertible C¹ matrix curve: $(\det A)^{'} = \det A tr (A^{- 1} A^{'})$ : if $A (t)$ is a differentiable invertible real square-matrix curve, then $(\det A)^{'} = \det A tr (A^{- 1} A^{'})$ .

[F4]

For $A \in M_{m \times n} (F)$ and $B \in M_{n \times m} (F)$ , $tr (A B) = tr (B A)$ : for matching finite real square matrices, $tr (A B) = tr (B A)$ .

[F5]

The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ : a function continuous on a closed interval and differentiable on its interior has an interior point where its derivative equals the interval's average slope.

[F6]

Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a, b]$ takes every value between $f (a)$ and $f (b)$ : a continuous real function on a closed interval takes every value between its endpoint values.

Proof

1.1givenF1F2algebra

For each component $χ_{i}$ , joint $C^{2}$ regularity and [F1] give $\partial_{t} \partial_{a_{j}} χ_{i} = \partial_{a_{j}} \partial_{t} χ_{i}$ . Differentiating the given velocity equation in $a$ and applying [F2] yields $\partial_{t} F = L F$ , where $L (t, a) : = (D_{x} b) (t, χ (t, a))$ .

2.1givenF2F3step 1.1

Each $F (t, a)$ is invertible: differentiating $χ_{t}^{- 1} \circ χ_{t} = id$ by [F2] gives $D χ_{t}^{- 1} (χ_{t} (a)) F (t, a) = I$ . Thus [F3] applies to the $C^{1}$ matrix curve $t \mapsto F (t, a)$ and gives $\partial_{t} J = J tr (F^{- 1} L F)$ .

3.1F4step 1.1step 2.1algebra

By [F4], $tr (F^{- 1} L F) = tr (F F^{- 1} L) = tr L = \sum_{i} \partial_{x_{i}} b_{i} (t, χ (t, a))$ . Also $F (0, a) = I$ , so $J (0, a) = 1$ .

4.1givenF6step 2.1step 3.1

The determinant $J (t, a)$ is continuous and nonzero on $I$ by step 2.1. If it were negative for some $t$ , [F6] on the closed segment with endpoints $0$ and $t$ would give a point where $J = 0$ , contradicting invertibility. Hence $J > 0$ .

4.2givenF5step 3.1algebra

If $div b = 0$ , step 3.1 gives $\partial_{t} J = 0$ . For each fixed $a$ , the case $t = 0$ is already $J (0, a) = 1$ . For $t \neq 0$ , apply [F5] to $J (\cdot, a)$ on $[\min (0, t), \max (0, t)]$ , whose endpoints are distinct; since its derivative is zero, its endpoint difference is zero, so $J (t, a) = J (0, a) = 1$ .

5.1givenstep 1.1step 2.1step 3.1step 4.1step 4.2∎

For the torus clause, fix $a \in T^{n}$ and $t_{0} \in I$ . By continuity, choose a time neighborhood of $t_{0}$ on which $a$ and $χ_{t} (a)$ stay in translation charts. Their transition maps are translations, so the coordinate matrices of $D_{a} χ_{t}$ , their determinants, and the coordinate divergence of $b$ agree on overlaps. In these charts the Euclidean calculations in steps 1.1–3.1 prove both differential identities, and $χ_{0} = id$ gives $J (0, a) = 1$ . Since $t_{0}$ was arbitrary, the identities hold throughout $I$ ; applying steps 4.1 and 4.2 on the closed segment between $0$ and any $t \in I$ gives positivity and the divergence-free conclusion.

Remarks

The hypotheses describe an already supplied family of diffeomorphisms and its velocity field. This lemma does not prove that an ODE generates such a family, that a trajectory is global, or that any physical fluid or Hamiltonian model satisfies the hypotheses.

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Fine entropy under a specified incompressible periodic flow

Statement

Use $\log$ for the natural logarithm of The natural logarithm as the inverse of the exponential function. Let $T^{n} : = R^{n} / Λ$ be a flat translation torus, where $Λ = Z ℓ_{1} + \dots + Z ℓ_{n}$ for a linearly independent basis $ℓ_{1}, \dots, ℓ_{n}$ , with the smooth structure and Euclidean metric descended from $R^{n}$ . Let $μ = c d a$ be its fixed translation-invariant volume measure in dimensionless translation coordinates, with $c > 0$ . Let $I \subseteq R$ be an open interval containing $0$ . Suppose $χ : I \times T^{n} \to T^{n}$ is jointly $C^{2}$ , each $χ_{t}$ is a diffeomorphism, and $χ_{0} = id$ . Let $π : T T^{n} \to T^{n}$ be the tangent-bundle projection and let $b : I \times T^{n} \to T T^{n}$ have $C^{1}$ chart coefficients with $π (b (t, a)) = a$ and $\partial_{t} χ (t, a) = b (t, χ (t, a))$ . In translation charts write $b = \sum_{i} b^{i} \partial_{x^{i}}$ and define $div b = \sum_{i} \partial_{x^{i}} b^{i}$ . Assume $div b = 0$ in these charts. Let $f_{0} \in C^{1} (T^{n})$ satisfy $0 < m \leq f_{0} \leq M < \infty$ and $\int f_{0} d μ = 1$ , and define $f_{t} : = f_{0} \circ χ_{t}^{- 1}$ . Then $f_{t}$ is a probability density for every $t \in I$ , the integral $H (f_{t}) : = - \int_{T^{n}} f_{t} \log f_{t} d μ$ is finite, and $H (f_{t}) = H (f_{0})$ . Here $H$ is the dimensionless fine entropy of this density; multiplication by a physical constant such as $k_{B}$ is a separate interpretation.

Facts & Assumptions

Given: The flat torus, fixed measure $μ = c d a$ , jointly $C^{2}$ diffeomorphisms $χ_{t}$ , divergence-free velocity, and positive $C^{1}$ probability density $f_{0}$ bounded above and below as in the Statement.

[F1]

Jacobian identity for a specified smooth diffeomorphic flow: on the flat translation torus, for a jointly $C^{2}$ diffeomorphic family $χ_{t}$ with $χ_{0} = id$ and a $C^{1}$ time-dependent vector field $b$ satisfying $\partial_{t} χ (t, a) = b (t, χ (t, a))$ , the local-chart Jacobian satisfies $\partial_{t} J = J div b$ , is positive, and equals $1$ when the chart divergence $\sum_{i} \partial_{x^{i}} b^{i}$ vanishes.

[F2]

Change of variables for an injective $C^{1}$ map on a compact Jordan set: an injective $C^{1}$ map with invertible derivative on an open neighborhood of a compact Jordan set obeys the Riemann change-of-variables formula for bounded integrable functions.

[F3]

The natural logarithm as the inverse of the exponential function defines $\log$ on $(0, \infty)$ as the inverse of the real exponential; Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm makes it continuous there. In particular it is bounded and continuous on every compact interval $[m, M] \subset (0, \infty)$ .

Proof

1.1givenF1

Apply [F1] to the given family. Its hypotheses are exactly the supplied $C^{2}$ diffeomorphisms, identity initial map and $C^{1}$ divergence-free velocity, so every local translation-chart Jacobian is $J_{t} = 1$ .

2.1givenF1F2step 1.1

For fixed $t$ , around each point choose a source chart and a smaller coordinate neighborhood whose closure maps under $χ_{t}$ into a target chart. Continuity and compactness give finitely many closed coordinate boxes $Q_{j}$ with interiors covering the torus and with each $χ_{t} (Q_{j})$ inside a target chart. Put $E_{1} = Q_{1}$ and $E_{j} = Q_{j} ∖ ⋃_{i < j} int (Q_{i})$ ; each $E_{j}$ is compact, the pieces cover the torus, and any overlap lies on one of the box faces. Each $E_{j}$ is Jordan because its boundary lies in a finite union of coordinate faces, which are compact $C^{1}$ hypersurface patches. A $C^{1}$ graph patch has bounded slope on a compact subpatch, so its mesh- $h$ box cover has total $n$ -volume $O (h)$ and hence content zero. On compact chart neighborhoods $χ_{t}$ has bounded derivative, so it maps these zero-content boundaries to zero-content boundaries. Each compact piece and its image therefore have Jordan boundaries and lie in source and target charts; [F2] applies there. Summing over the pieces, overlaps do not affect the integrals, translation chart changes preserve $d a$ , and the constant $c$ is the same on both sides. Thus for every continuous $φ$ on the torus, $\int φ \circ χ_{t} d μ = \int φ J_{t} d μ = \int φ d μ$ .

3.1givenstep 2.1

The function $f_{t} = f_{0} \circ χ_{t}^{- 1}$ is positive and continuous. The substitution of step 2.1 with $φ = f_{t}$ gives $\int f_{t} d μ = \int f_{0} d μ = 1$ , so $f_{t}$ is normalized.

4.1givenF3step 2.1algebra∎

The bounds $m \leq f_{0} \leq M$ also hold for $f_{t}$ ; by [F3], $\log$ is continuous on $[m, M]$ , so $f_{t} \log f_{t}$ is bounded and continuous on the compact torus and its integral is finite. Applying step 2.1 to $φ = f_{t} \log f_{t}$ gives $\int f_{t} \log f_{t} d μ = \int (f_{t} \log f_{t}) \circ χ_{t} d μ = \int f_{0} \log f_{0} d μ$ , since $f_{t} \circ χ_{t} = f_{0}$ . Negating proves $H (f_{t}) = H (f_{0})$ .

Remarks

This is a statement about supplied smooth mathematical flow data. It does not assert existence of a flow for a vector field, Hamiltonian structure, relaxation, mixing, ergodicity, monotonicity of coarse entropy, or a physical entropy interpretation.

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Gaussian covariance for positive quadratic forms and a boundary counterexample

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ as defined in The Axiom of Countable Choice ( ${A C}_{ω}$ ). Let $n \geq 1$ , let $K \in R^{n \times n}$ be symmetric positive definite, and let $β > 0$ . For $x \in R^{n}$ put $H (x) = \frac{1}{2} x^{T} K x$ , and use the Euclidean Lebesgue measure $λ_{n}$ to define the density $Z^{- 1} e^{- β H (x)}$ . Then

$Z = \int_{R^{n}} e^{- β H (x)} d x = {(\frac{2 π}{β})}^{n / 2} (\det K)^{- 1 / 2}, E [x] = 0, E [x x^{T}] = β^{- 1} K^{- 1}, E [H] = \frac{n}{2 β} .$

Positive definiteness and $β > 0$ are essential for this normalized full-space Gaussian. The unrestricted identity $E [x H^{'} (x)] = β^{- 1}$ is false without a boundary hypothesis: on $[- L, L]$ with $L > 0$ and $H \equiv 0$ , the normalized expectation on the left is $0$ .

Facts & Assumptions

Given: ${A C}_{ω}$ , $n \geq 1$ , symmetric positive-definite $K$ , $β > 0$ , and $H (x) = \frac{1}{2} x^{T} K x$ on $R^{n}$ .

[F1]

Gaussian, gamma and beta product integrals: for $a > 0$ , $\int_{R} e^{- a x^{2}} d x = \sqrt{π / a}$ , odd Gaussian moments vanish, and the second moment is $\sqrt{π} / (2 a^{3 / 2})$ .

[F2]

Under ${A C}_{ω}$ , a $C^{1}$ diffeomorphism transforms nonnegative Lebesgue integrals by its absolute Jacobian (A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions).

[F3]

Under ${A C}_{ω}$ , Euclidean Lebesgue measure is the finite product of one-dimensional Lebesgue measures on Borel rectangles, and each box has measure equal to the product of its side lengths (On Borel subsets of R^{m+n}, the product lambda_m times lambda_n agrees with lambda_{m+n}, 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); each factor is sigma-finite. Tonelli factors nonnegative products, and Fubini factors absolutely integrable signed products (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, Fubini's theorem for L^1 functions on a sigma-finite product).

[F4]

The Gaussian integrals and moments in [F1] are improper Riemann integrals. On bounded intervals they agree with Lebesgue integrals under ${A C}_{ω}$ , and monotone convergence passes their nonnegative truncations to $R$ (Riemann–Lebesgue comparison for distribution test integrands, Monotone convergence for the integral).

Proof

1.1givenconstructalgebra

We construct $R$ with $K = R^{T} R$ by induction on $n$ . For $n = 1$ , take $R = \sqrt{K_{11}} > 0$ . For $n > 1$ , write $K = (\begin{array}{cc} a & b^{T} \\ b & C \end{array})$ ; positive definiteness gives $a > 0$ and $S : = C - b b^{T} / a$ positive definite, since $y^{T} S y = (- b^{T} y / a, y)^{T} K (- b^{T} y / a, y) > 0$ for $y \neq 0$ . By induction $S = R_{1}^{T} R_{1}$ ; then $R = (\begin{array}{cc} \sqrt{a} & b^{T} / \sqrt{a} \\ 0 & R_{1} \end{array})$ satisfies $R^{T} R = K$ . Its diagonal entries are positive, so $\det R > 0$ ; taking determinants gives $\det K = (\det R)^{2}$ and therefore $\det R = \sqrt{\det K}$ .

2.1givenF1F2F3F4step 1.1algebra

Put $T (x) = R x$ . This is a $C^{1}$ diffeomorphism of $R^{n}$ with $∣ \det D T ∣ = \det R$ . Since $H (x) = ∣ R x ∣^{2} / 2$ , [F2] gives $Z = (\det R)^{- 1} \int_{R^{n}} e^{- β ∣ y ∣^{2} / 2} d λ_{n} (y) .$ By [F3], $λ_{n}$ is the finite product of $n$ sigma-finite one-dimensional Lebesgue measures; Tonelli and the factorization $e^{- β ∣ y ∣^{2} / 2} = \prod_{i} e^{- β y_{i}^{2} / 2}$ give the integral as $I_{0} (β / 2)^{n}$ . By [F1] and [F4], $I_{0} (β / 2) = \sqrt{2 π / β}$ . Hence $Z = (2 π / β)^{n / 2} (\det K)^{- 1 / 2},$ which is finite and positive.

3.1F1F3step 2.1algebra

Let $Y$ have the normalized density proportional to $e^{- β ∣ y ∣^{2} / 2}$ with respect to $λ_{n}$ . For each $i$ , [F3] and Tonelli factor the nonnegative second-moment integral into $I_{1} (β / 2) I_{0} (β / 2)^{n - 1}$ , while the normalizer is $I_{0} (β / 2)^{n}$ ; [F1] gives $E [Y_{i}^{2}] = I_{1} (β / 2) / I_{0} (β / 2) = 1 / β$ . Also $\int ∣ y_{i} ∣ e^{- β ∣ y ∣^{2} / 2} d λ_{n} < \infty$ , since $∣ y_{i} ∣ \leq 1 + y_{i}^{2}$ and the same Tonelli factorization makes both integrals finite. For $i \neq j$ , $∣ y_{i} y_{j} ∣ \leq (y_{i}^{2} + y_{j}^{2}) / 2$ , so the cross moments are absolutely integrable. Fubini [F3] factors these signed moments; the one-dimensional first and odd moments vanish by [F1]. Thus $E [Y] = 0$ and $E [Y Y^{T}] = β^{- 1} I$ .

4.1F2step 1.1step 3.1algebra

By [F2], the normalized law of $X$ is the pullback of the law of $Y$ under $x = R^{- 1} y$ . The absolute integrability established in step 3.1 permits the same change of variables for signed first and second moments by applying [F2] to their positive and negative parts. Hence $E [X] = 0$ and $E [X X^{T}] = R^{- 1} (β^{- 1} I) R^{- T} = β^{- 1} (R^{T} R)^{- 1} = β^{- 1} K^{- 1} .$ Since $H (X) = ∣ Y ∣^{2} / 2$ , summing the diagonal second moments gives $E [H] = n / (2 β)$ .

5.1givenF3algebra∎

For the stated boundary counterexample, the density on $[- L, L]$ is the constant $(2 L)^{- 1}$ because $H = 0$ and the normalizing integral is $2 L$ by [F3]. Since $H^{'} (x) = 0$ everywhere, $E [x H^{'} (x)] = 0 \neq β^{- 1}$ for $β > 0$ .

Remarks

If $K$ has a nonzero kernel, the integrand is constant along an unbounded kernel direction and $Z = \infty$ ; if $β \leq 0$ , the full-space integral also diverges. The theorem does not assert a named-degree-of-freedom equipartition rule on arbitrary domains or for discrete variables.

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Exact spherical-shell coordinate marginal and fixed-block Gaussian limit

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ as defined in The Axiom of Countable Choice ( ${A C}_{ω}$ ). Let $D \geq 2$ , $1 \leq k < D$ , and $R > 0$ . Let $Z_{1}, \dots, Z_{D}$ be independent real random variables with common Lebesgue probability density $ϕ (x) : = (2 π)^{- 1 / 2} e^{- x^{2} / 2}$ ; its integral is one by the Gaussian integral in Gaussian, gamma and beta product integrals, and this is the standard-normal coordinate law. Set $Z = (Z_{1}, \dots, Z_{D})$ and define $P : = R Z / ∣ Z ∣$ when $Z \neq 0$ , and $P : = 0$ when $Z = 0$ . The joint density of $Z$ makes $P (Z = 0) = 0$ . The distribution of $P$ is rotationally invariant, meaning that for every orthogonal map $Q \in O (D)$ and Borel set $A \subseteq R^{D}$ , $P (Q P \in A) = P (P \in A)$ . It is concentrated on the sphere in the precise sense that $P (∣ P ∣ = R) = 1$ . Its first $k$ coordinates have Lebesgue density

$f_{D, k} (y) = \frac{Γ (D / 2)}{Γ ((D - k) / 2) (π R^{2})^{k / 2}} {(1 - \frac{∣ y ∣^{2}}{R^{2}})}^{(D - k - 2) / 2} (∣ y ∣ < R),$

and zero for $∣ y ∣ \geq R$ . This density is integrable even when it diverges at the boundary. Also $E [P_{i}] = 0$ , $E [P_{i} P_{j}] = 0$ for $i \neq j$ , and $E [P_{i}^{2}] = R^{2} / D$ .

For fixed $k$ and $σ > 0$ , set $R_{D} = \sqrt{D} σ$ and let $D \to \infty$ through integers $D > k$ . If $g : R^{k} \to R$ is bounded and Lipschitz in the Euclidean norm, meaning that finite constants $B_{g}, L_{g}$ satisfy $∣ g (x) ∣ \leq B_{g}$ and $∣ g (x) - g (y) ∣ \leq L_{g} ∣ x - y ∣$ for all $x, y \in R^{k}$ , then

$E g ((P_{1}, \dots, P_{k})) ⟶ E g (σ (Z_{1}, \dots, Z_{k})) .$

This is a fixed-block bounded-Lipschitz test limit; it makes no claim uniform in $k$ or for unbounded test functions. If $k = D$ , the projection is singular on the sphere and the displayed Lebesgue density is not asserted.

Facts & Assumptions

Given: The Axiom of Countable Choice ${A C}_{ω}$ , integers $D, k$ with $D \geq 2$ and $1 \leq k < D$ , radius $R > 0$ , and the product probability law of $D$ independent coordinates with density $ϕ (x) = (2 π)^{- 1 / 2} e^{- x^{2} / 2}$ .

[F1]

Gaussian, gamma and beta product integrals: $Γ (a)$ is finite for $a > 0$ , $Γ (1 / 2) = \sqrt{π}$ , positive gamma densities with the same scale convolve by adding their shape parameters, and the Gaussian second and fourth moments are $1$ and $3$ for a standard normal coordinate.

[F2]

A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions: assuming ${A C}_{ω}$ , for a $C^{1}$ diffeomorphism $T : U \to V$ between open subsets of Euclidean space and nonnegative Lebesgue-measurable $f$ on $V$ , the Lebesgue integral transforms by $∣ \det D T ∣$ .

[F3]

On Borel subsets of R^{m+n}, the product lambda_m times lambda_n agrees with lambda_{m+n}: assuming ${A C}_{ω}$ , product Lebesgue measure and Euclidean Lebesgue measure agree on Borel subsets of $R^{m + n}$ for positive integers $m, n$ .

[F4]

Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: for a nonnegative product-measurable function on a product of sigma-finite measure spaces, either order of integration agrees with the product integral.

[F5]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ says every sequence of nonempty sets has a choice function; it is the explicit set-theoretic hypothesis used by [F2].

Proof

1.1givenF2F3F5algebra

Under Countable Choice [F5], the Borel product-measure identity [F3] and independence give the joint Lebesgue density $ϕ_{D} (z) = (2 π)^{- D / 2} e^{- ∣ z ∣^{2} / 2}$ . A singleton has Lebesgue measure zero, so $Z \neq 0$ almost surely. For any orthogonal $Q$ and Borel $A \subseteq R^{D}$ , the map $Q^{- 1} : R^{D} \to R^{D}$ is a $C^{1}$ diffeomorphism with $∣ \det Q^{- 1} ∣ = 1$ ; applying [F2] to $1_{A} ϕ_{D}$ and using $∣ Q^{- 1} z ∣ = ∣ z ∣$ gives $P (Q Z \in A) = P (Z \in A)$ . Since $Q (R Z / ∣ Z ∣) = R Q Z / ∣ Q Z ∣$ off the null event, the law of $P$ is invariant under every orthogonal map, and $∣ P ∣ = R$ almost surely.

1.2F1F2algebra

Write $Z = (A, B)$ with $A \in R^{k}$ and $B \in R^{D - k}$ . The vectors are independent. On each open half-line $(0, \infty)$ and $(- \infty, 0)$ , the map $z \mapsto z^{2} / 2$ is a $C^{1}$ diffeomorphism onto $(0, \infty)$ ; applying [F2] to the Gaussian density times the indicator of any Borel target set on each branch shows that each $Z_{j}^{2} / 2$ has density $w^{- 1 / 2} e^{- w} / Γ (1 / 2)$ . Repeated gamma convolution from [F1] then shows $T : = ∣ B ∣^{2}$ has density $t^{b - 1} e^{- t / 2} / (2^{b} Γ (b))$ for $t > 0$ , where $b = (D - k) / 2$ , independently of $A$ .

1.3givenconstructalgebra

For fixed $t > 0$ , the first $k$ coordinates $y = R A / \sqrt{∣ A ∣^{2} + t}$ range over $∣ y ∣ < R$ . The inverse is $A = \sqrt{t} y / \sqrt{R^{2} - ∣ y ∣^{2}}$ . Its derivative has radial eigenvalue $\sqrt{t} R^{2} / (R^{2} - ∣ y ∣^{2})^{3 / 2}$ and $k - 1$ tangential eigenvalues $\sqrt{t} / (R^{2} - ∣ y ∣^{2})^{1 / 2}$ , hence Jacobian $t^{k / 2} R^{2} / (R^{2} - ∣ y ∣^{2})^{k / 2 + 1}$ .

2.1F1F2F3F4step 1.2step 1.3algebra

The map $Ψ (y, t) : = (A (y, t), t)$ from $U : = {(y, t) : ∣ y ∣ < R, t > 0}$ to $V : = R^{k} \times (0, \infty)$ is a $C^{1}$ diffeomorphism, with inverse $(a, t) \mapsto (R a / \sqrt{∣ a ∣^{2} + t}, t)$ . Apply [F2] to the nonnegative joint density of $(A, T)$ and this diffeomorphism. The product Lebesgue measure on $R^{k} \times (0, \infty)$ agrees with Euclidean Lebesgue measure on Borel sets by [F3], and Tonelli's theorem [F4] gives the marginal density by integration in $t$ . Substituting the Gaussian density of $A$ , the gamma density of $T$ , and the Jacobian from step 1.3 yields $f_{D, k} (y) = \frac{R^{2}}{(2 π)^{k / 2} 2^{b} Γ (b) (R^{2} - ∣ y ∣^{2})^{k / 2 + 1}} \int_{0}^{\infty} t^{D / 2 - 1} \exp ⁣ (- \frac{t R^{2}}{2 (R^{2} - ∣ y ∣^{2})}) d t$ . The substitution $u = t R^{2} / [2 (R^{2} - ∣ y ∣^{2})]$ evaluates the integral as $Γ (D / 2) [2 (R^{2} - ∣ y ∣^{2}) / R^{2}]^{D / 2}$ . Since $D / 2 = k / 2 + b$ , simplification gives the displayed density. Its boundary singularity, when present, has exponent $b - 1 > - 1$ and is integrable in the radial variable; the pushforward construction shows total mass one. Because $T = ∣ B ∣^{2} > 0$ almost surely, the marginal has no boundary atom at $∣ y ∣ = R$ .

2.2step 1.1givenalgebra

Sign changes of one Gaussian coordinate show $E [P_{i}] = 0$ and $E [P_{i} P_{j}] = 0$ for $i \neq j$ . Permutation invariance makes all $E [P_{i}^{2}]$ equal, while $\sum_{i = 1}^{D} P_{i}^{2} = R^{2}$ almost surely; summing expectations gives $E [P_{i}^{2}] = R^{2} / D$ .

3.1F1step 1.1algebra∎

For each finite $D$ , take the product Gaussian vector $Z = (Z_{1}, \dots, Z_{D})$ in that dimension and set $S_{D} : = D^{- 1} \sum_{i = 1}^{D} Z_{i}^{2}$ . By [F1] and independence, $E S_{D} = 1$ , $E [(S_{D} - 1)^{2}] = 2 / D$ , and $E ∣ Z_{1 : k} ∣^{2} = k$ . Fix $0 < δ < 1 / 2$ and $M > 0$ . On $E_{D} = {∣ Z_{1 : k} ∣ \leq M, ∣ S_{D} - 1 ∣ \leq δ}$ , $S_{D} \in [1 / 2, 3 / 2]$ and the identity $∣ u^{- 1 / 2} - 1 ∣ = ∣ u - 1 ∣ / [\sqrt{u} (1 + \sqrt{u})]$ gives $∣ S_{D}^{- 1 / 2} - 1 ∣ \leq \sqrt{2} δ$ ; hence $∣ σ Z_{1 : k} / \sqrt{S_{D}} - σ Z_{1 : k} ∣ \leq \sqrt{2} σ M δ$ . If $g$ is bounded Lipschitz with bound $B_{g}$ and Lipschitz constant $L_{g}$ , then $M^{2} 1_{{∣ Z_{1 : k} ∣ > M}} \leq ∣ Z_{1 : k} ∣^{2}$ and $δ^{2} 1_{{∣ S_{D} - 1 ∣ > δ}} \leq (S_{D} - 1)^{2}$ imply $P (E_{D}^{c}) \leq k / M^{2} + 2 / (D δ^{2})$ . Therefore $∣ E g (P_{1 : k}) - E g (σ Z_{1 : k}) ∣ \leq L_{g} \sqrt{2} σ M δ + 2 B_{g} (\frac{k}{M^{2}} + \frac{2}{D δ^{2}})$ . Choose $M$ large, then $δ$ small, then $D$ large. The right side becomes arbitrarily small, proving the stated test convergence without assuming independence between $Z_{1 : k}$ and $S_{D}$ .

Remarks

This is a specified rotational Gaussian-direction preparation. It is not a general coarea formula, a proof that every energy-shell measure has this marginal, or a dynamical equilibration result.

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Declared example preparations and thermodynamic interpretation

Physical scope

Separate finite classical canonical, grand-canonical, spherical-shell and product preparations in the named model examples; no claim of equilibration, empirical truth, or statistical inference from observations.

Postulate

Here $\log$ denotes the natural logarithm defined in The natural logarithm as the inverse of the exponential function.

For each named finite classical model in this page, adopt the preparation explicitly stated in its theorem. In a canonical branch, take a measurable phase space with a specified reference measure $ν$ , a measurable Hamiltonian $H$ , an inverse energy parameter $β > 0$ , and a finite positive partition function $Z = \int e^{- β H} d ν$ ; the adopted probability measure is $d P = Z^{- 1} e^{- β H} d ν$ . In a grand-canonical branch, take disjoint particle-number sectors with sector partition functions $Z_{N}$ and a dimensionless activity $z > 0$ for which $Ξ = \sum_{N \geq 0} z^{N} Z_{N}$ is finite; sector $N$ has probability weight $z^{N} Z_{N} / Ξ$ . The spherical-shell and product preparations are separate explicitly specified probability laws in their consuming theorem statements. These are alternative model assumptions, not consequences of a Hamiltonian flow and not assertions that a laboratory system realizes them.

For a canonical model whose reference measure and Hamiltonian are fixed, adopt the statistical thermodynamic definitions $U = E_{P} [H]$ , $F = - β^{- 1} \log Z$ , and $S = k_{B} (\log Z + β U)$ when the displayed integrals are finite. For a volume-dependent model, define the model pressure as $- \partial_{V} F$ only along the declared fixed-shape volume family. Here $k_{B} > 0$ has SI units J K $^{- 1}$ , temperature $T > 0$ has units K, and $β = (k_{B} T)^{- 1}$ has units J $^{- 1}$ . The fixed reference measure is part of the preparation; changing it changes the numerical partition function and entropy convention.

For the continuous particle examples, adopt the operational species and counting convention of Classical particle counting and physical observables. Use one fixed primitive action reference $a_{0} > 0$ in J s, and write $h_{0} : = a_{0}$ where the consuming model uses the notation $h_{0}$ . In a Cartesian physical phase chart $d q d p$ has action dimension $(J s)^{d N}$ , so the reference $d q d p / (\prod_{s} N_{s}! a_{0}^{d N})$ is dimensionless. Each model retains its declared operational labels: an omitted factorial for resolved harmonic coordinates is not replaced by a particle-exchange quotient. The finite spin examples use their stated discrete counting measure, without a continuous phase-cell factor. Choosing a numerical reference action, or identifying it with Planck's constant, is an additional convention; no such value or identification is imposed here. The Gaussian and gamma identities in Gaussian, gamma and beta product integrals establish the mathematical normalizations, not the physical preparation or empirical equilibration.

Remarks

A finite partition function and normalized density are mathematical conditions of each canonical consumer. No negative-temperature unbounded gas, ergodic hypothesis, observed outcome, or universal ensemble-selection principle is included.

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Confined coupled quadratic canonical systems

Physical scope

Finite classical continuous canonical model with positive diagonal mass matrix, strictly positive stiffness matrix, fixed positive action reference and beta>0; the Lebesgue product calculations assume ACω; no unpinned zero mode or empirical equilibration claim.

Statement

Assume ${A C}_{ω}$ as defined in The Axiom of Countable Choice ( ${A C}_{ω}$ ) for the Euclidean Lebesgue product measure used below. Let $n \geq 1$ , let $M = diag (m_{1}, \dots, m_{n})$ with $m_{i} > 0$ in kg, and let $K$ be a real symmetric positive-definite stiffness matrix with entries in J m $^{- 2}$ . For displacement $q \in R^{n}$ in metres and momentum $p \in R^{n}$ in kg m s $^{- 1}$ , define $H (q, p) = \frac{1}{2} p^{T} M^{- 1} p + \frac{1}{2} q^{T} K q .$ Adopt the canonical probability density proportional to $e^{- β H}$ relative to $d q d p / h_{0}^{n}$ , where $β > 0$ has units J $^{- 1}$ and the fixed action reference $h_{0} > 0$ has units J s. Under the preparation of Declared example preparations and thermodynamic interpretation, the dimensionless partition function and moments are $Z = {(\frac{2 π}{β h_{0}})}^{n} \sqrt{\frac{\det M}{\det K}}, E [q] = E [p] = 0, E [q q^{T}] = β^{- 1} K^{- 1}, E [p p^{T}] = β^{- 1} M, E [q p^{T}] = 0.$ The model mean energy is $U = E [H] = n / β$ . With $k_{B} > 0$ in J K $^{- 1}$ and $T > 0$ in K from Declared example preparations and thermodynamic interpretation, so that $β = (k_{B} T)^{- 1}$ , define the fixed-parameter heat-capacity response by $C_{h a r m} : = {(\frac{\partial U}{\partial T})}_{M, K, h_{0}} = n k_{B} .$ It has units J K $^{- 1}$ . This model has no volume variable, so $C_{h a r m}$ is not a fixed-volume derivative.

Facts & Assumptions

Given: ${A C}_{ω}$ , $n \geq 1$ , $M$ positive diagonal, $K$ symmetric positive definite, $β > 0$ , fixed $h_{0} > 0$ , and the displayed continuous-coordinate canonical model.

[F1]

Declared example preparations and thermodynamic interpretation: in each named model with finite positive $Z$ , the canonical probability is $Z^{- 1} e^{- β H}$ relative to the specified reference; $U = E H$ and the fixed-reference thermodynamic interpretation is adopted separately from the probability preparation.

[F2]

Gaussian covariance for positive quadratic forms and a boundary counterexample: assuming ${A C}_{ω}$ , for a symmetric positive-definite quadratic form on $R^{n}$ , the normalized Gaussian integral is $(2 π / β)^{n / 2} (\det K)^{- 1 / 2}$ , with mean zero and covariance $β^{- 1} K^{- 1}$ .

[F3]

Under ${A C}_{ω}$ , Euclidean phase-space Lebesgue measure is the product of the $q$ and $p$ Lebesgue measures; Tonelli factors nonnegative densities and Fubini factors their integrable moments (On Borel subsets of R^{m+n}, the product lambda_m times lambda_n agrees with lambda_{m+n}, Tonelli's theorem for nonnegative measurable functions on a sigma-finite product, Fubini's theorem for L^1 functions on a sigma-finite product).

[A1]

The canonical density is a model preparation, not a claim that an isolated trajectory equilibrates or that a laboratory realizes this law.

Proof

1.1givenF1F2F3algebra

By [F1], the phase-space weight factors as $e^{- β p^{T} M^{- 1} p / 2} e^{- β q^{T} K q / 2}$ . The product-measure identity and Tonelli [F3] justify multiplying the two nonnegative Lebesgue integrals. Applying [F2] to the $q$ factor gives normalization $(2 π / β)^{n / 2} (\det K)^{- 1 / 2}$ and covariance $β^{- 1} K^{- 1}$ .

1.2givenF2algebra

The momentum quadratic form has symmetric positive-definite matrix $M^{- 1}$ . Applying [F2] with $K = M^{- 1}$ gives its integral $(2 π / β)^{n / 2} \sqrt{\det M}$ and covariance of $p$ equal to $β^{- 1} (M^{- 1})^{- 1} = β^{- 1} M$ .

2.1A1F3step 1.1step 1.2algebra

Multiplying the two integrals from steps 1.1–1.2 and dividing by the reference factor $h_{0}^{n}$ gives $Z = (2 π / (β h_{0}))^{n} \sqrt{\det M / \det K}$ . The phase-space Lebesgue measure is the product of its $q$ and $p$ factors by [F3]; the normalized canonical density therefore factors into independent $q$ and $p$ laws. Each factor is even, so its mean is zero, and its covariance is the one computed in steps 1.1–1.2.

3.1A1F1step 1.1step 1.2algebra∎

By [F1], taking the expectation of $H$ and summing the diagonal traces of the two covariance identities gives $U = \frac{1}{2} E [p^{T} M^{- 1} p] + \frac{1}{2} E [q^{T} K q] = \frac{n}{2 β} + \frac{n}{2 β} = \frac{n}{β} .$ The adopted relation $β^{- 1} = k_{B} T$ makes $U = n k_{B} T$ . Holding the model parameters $M, K, h_{0}$ fixed therefore gives $C_{h a r m} = (\partial U / \partial T)_{M, K, h_{0}} = n k_{B}$ , with units J K $^{- 1}$ .

Remarks

Positive definiteness is a confinement condition. If $K$ has a translation zero mode, the full-space configurational integral diverges and this normalized ensemble does not exist without an additional constraint or confinement.

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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.

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Finite periodic Ising-ring correlations and exact decay

Physical scope

Finite periodic one-dimensional nearest-neighbor Ising ring with N>=2, exactly N indexed bonds, J>=0, zero field, and finite beta>0; no general long-range or higher-dimensional claim.

Statement

Use the real exponential $\exp (x) = \sum_{n \geq 0} x^{n} / n!$ of The real exponential function and the number $e$ by a power series and the hyperbolic functions $\cosh x = (\exp x + \exp (- x)) / 2$ , $\sinh x = (\exp x - \exp (- x)) / 2$ . Define $\tanh x : = \sinh x / \cosh x$ ; the denominator is positive for every real $x$ . Let $N \geq 2$ , let $Ω_{N} = {- 1, 1}^{N}$ with cyclic indexing $s_{N + 1} = s_{1}$ , and count exactly the $N$ indexed periodic bonds (so $N = 2$ has two bonds). For coupling $J \geq 0$ in joules, zero field, and $β > 0$ in J $^{- 1}$ , adopt the finite canonical law $P_{N} (s) = Z_{N}^{- 1} \exp ⁣ (β J \sum_{i = 1}^{N} s_{i} s_{i + 1}) .$ Set $t = \tanh (β J) \in [0, 1)$ . For $1 \leq r \leq N - 1$ , $E_{N} [s_{i}] = 0, E_{N} [s_{i} s_{i + r}] = \frac{t^{r} + t^{N - r}}{1 + t^{N}} .$ For each fixed positive integer separation $r$ and $N \to \infty$ , the correlation tends to $t^{r}$ , with absolute error at most $t^{N - r} + t^{N + r}$ . When $J > 0$ , define the limiting correlation length $ξ > 0$ in lattice spacings as the unique number satisfying $t^{r} = e^{- r / ξ}$ for every positive integer separation $r$ ; equivalently, $ξ = - 1 / \log t$ . The zero-field susceptibility per spin, defined by $χ_{N} : = N^{- 1} \partial_{h} E_{N, h} [\sum_{i} s_{i}] ∣_{h = 0}$ for the field-coupled law with energy $- ⁣ J \sum_{i} s_{i} s_{i + 1} - h \sum_{i} s_{i}$ , where the field parameter $h \in R$ is in joules, has limit $χ_{\infty} = β \frac{1 + t}{1 - t} .$ Both $χ_{N}$ and $χ_{\infty}$ have units J $^{- 1}$ per spin (the spin variables are dimensionless). Define the finite-ring model free energy by $F_{N} : = - β^{- 1} \log Z_{N}$ and its limiting free energy per spin, when the limit exists, by $f_{\infty} : = \lim_{N \to \infty} F_{N} / N$ . Then $f_{\infty} = - β^{- 1} \log (2 \cosh (β J))$ , analytic at every finite positive temperature. No statement is made for arbitrary one-dimensional interactions.

Facts & Assumptions

Given: The finite spin space $Ω_{N}$ , the stated bond convention, $N \geq 2$ , $J \geq 0$ , and finite $β > 0$ .

[F1]

Declared example preparations and thermodynamic interpretation: each finite model uses its explicitly adopted canonical probability on the stated reference/counting space; the preparation is a model assumption.

[F2]

One-dimensional periodic Ising-chain limit: for the periodic nearest-neighbor ring with $N \geq 2$ , finite real $J, h$ and $β > 0$ , the same $\cosh$ and $\sinh$ definitions give $T_{s, t} = \exp (β J s t + β h (s + t) / 2)$ and $Z_{N} = tr (T^{N})$ . Its explicit eigenvalues satisfy $λ_{+} > ∣ λ_{-} ∣$ , and the logarithmic partition limit and each fixed finite order of parameter derivatives converge uniformly on compact subsets of $(0, \infty) \times R^{2}$ . This is the current published mathematical supplier for the exact indexed-bond model used here.

[F3]

The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ : $\exp (x) > 0$ for every real $x$ , hence the defined $\cosh x$ is positive and $\tanh x$ is well-defined.

[F4]

The exponential function is strictly increasing: $\exp$ is continuous and $x \leq y$ implies $\exp (x) \leq \exp (y)$ .

[F5]

A real-analytic function on an open subset of $R$ is locally represented by a convergent real power series: a real-valued function on an open interval is real analytic when each point has a neighborhood on which the function equals a convergent real power series.

[F6]

The sum of a real power series is real analytic throughout the open interval determined by its radius: the sum of a real power series is real analytic on the interior of its radius of convergence; the exponential series has infinite radius.

[F7]

Real-analytic functions are closed under sums, products and compositions, and under quotients where the denominator is nonzero: sums, products, quotients with nonzero denominator, and compositions of real-analytic functions on their open domains are real analytic.

[F8]

[The power series for log(1+x) on (-1,1], including the Abel endpoint](/item/thm-log-one-plus-x-power-series): for $∣ u ∣ < 1$ , $\log (1 + u) = \sum_{n = 1}^{\infty} (- 1)^{n + 1} u^{n} / n$ , so $u \mapsto \log (1 + u)$ is real analytic near $u = 0$ .

[F9]

Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm: for positive $x, y$ , $\log (x y) = \log x + \log y$ .

[A1]

For the response clause, the field-coupled finite law is $P_{N, h} (s) \propto \exp (β J \sum_{i} s_{i} s_{i + 1} + β h \sum_{i} s_{i})$ for real field parameter $h$ in joules; this is a finite model family, not an empirical magnetic law.

Proof

1.1givenF2F4algebra

In the zero-field case the transfer matrix in [F2] is $T_{s, t} = e^{K s t}$ with $K = β J$ , so $T = (\begin{array}{cc} e^{K} & e^{- K} \\ e^{- K} & e^{K} \end{array})$ . Direct multiplication gives orthonormal eigenvectors $u_{+} = (1, 1) / \sqrt{2}$ , $u_{-} = (1, - 1) / \sqrt{2}$ with eigenvalues $λ_{+} = 2 \cosh K$ and $λ_{-} = 2 \sinh K$ . Hence $Z_{N} = tr (T^{N}) = λ_{+}^{N} + λ_{-}^{N} = λ_{+}^{N} (1 + t^{N})$ , where $t = λ_{-} / λ_{+} = \tanh K$ . Since $K \geq 0$ , [F4] gives $e^{K} \geq e^{- K}$ , so $λ_{-} = e^{K} - e^{- K} \geq 0$ ; [F2] gives $λ_{+} > ∣ λ_{-} ∣$ . Thus $t \in [0, 1)$ .

2.1A1F1F2step 1.1algebra

By the finite preparation [F1], the partition sum is the normalization for the adopted law. Put $D_{0} = diag (1, - 1)$ ; then $D_{0} u_{+} = u_{-}$ and $D_{0} u_{-} = u_{+}$ . Expanding the finite canonical sum over the $N$ cyclic indices with $s_{i} s_{i + r}$ inserted gives numerator $tr (D_{0} T^{r} D_{0} T^{N - r})$ . Evaluating this trace in the eigenbasis yields $λ_{-}^{r} λ_{+}^{N - r} + λ_{+}^{r} λ_{-}^{N - r}$ . Dividing by $Z_{N}$ from step 1.1 gives the stated correlation formula.

3.1step 2.1algebra

The map $s \mapsto - s$ preserves every bond product and reverses $\sum_{i} s_{i}$ , so at zero field $E_{N} [s_{i}] = 0$ . For fixed $r$ , subtracting $t^{r}$ from the formula of step 2.1 gives $(t^{N - r} - t^{N + r}) / (1 + t^{N})$ ; its absolute value is at most $t^{N - r} + t^{N + r}$ because $0 \leq t < 1$ . Thus the fixed-distance limit is $t^{r}$ . When $0 < t < 1$ , writing $t^{r} = e^{- r / ξ}$ gives $ξ = - 1 / \log t$ ; for $J = 0$ , $t = 0$ and all nonzero-separation correlations vanish.

3.2A1step 2.1algebra

For the field-coupled finite law [A1], differentiating its finite numerator and denominator sums gives $\partial_{h} E_{N, h} [M] = β (E_{N, h} [M^{2}] - E_{N, h} [M]^{2})$ , where $M = \sum_{i} s_{i}$ . At $h = 0$ , spin-flip symmetry gives $E M = 0$ and translation invariance gives $\frac{1}{N} E [M^{2}] = 1 + \sum_{r = 1}^{N - 1} E [s_{1} s_{1 + r}] = 1 + \frac{2 \sum_{r = 1}^{N - 1} t^{r}}{1 + t^{N}} .$ The finite geometric sum tends to $2 t / (1 - t)$ as $N \to \infty$ , so $χ_{N} \to β (1 + t) / (1 - t)$ . This also gives $χ_{N} = β$ when $J = 0$ .

4.1step 1.1step 3.1algebra

From step 1.1, $N^{- 1} \log Z_{N} = \log (2 \cosh (β J)) + N^{- 1} \log (1 + t^{N})$ . Since $1 \leq 1 + t^{N} \leq 2$ , the second term tends to zero, so $f_{\infty} = - β^{- 1} \log (2 \cosh (β J))$ . The correlation length diverges only in the excluded zero-temperature limit $β J \to \infty$ .

5.1F3F4F5F6F7F8F9step 4.1∎

Fix any $β_{0} > 0$ and put $A (β) : = 2 \cosh (β J)$ and $u (β) : = A (β) / A (β_{0}) - 1$ . By [F5, F6, F7], $A$ and $u$ are real analytic near $β_{0}$ ; $A (β_{0}) > 0$ by [F3] and $u (β_{0}) = 0$ . Continuity from [F4] allows a smaller neighborhood where $∣ u (β) ∣ < 1$ . By [F8], [F6], and the product law [F9], $\log A (β) = \log A (β_{0}) + \log (1 + u (β))$ is real analytic there, and [F7] makes $f_{\infty} (β) = - β^{- 1} \log A (β)$ real analytic since $β \neq 0$ . The relation $β = (k_{B} T)^{- 1}$ is real analytic for $T > 0$ , so the limit is analytic at every finite positive temperature as well.

Remarks

The transfer-matrix premise uses the published exact periodic-chain theorem. The local spin-insertion and finite-response arguments prove the additional correlation and susceptibility formulas. No Gaussian-tail research token is used as a supplier.

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Exact Tonks hard-rod pressure and convergent virial series

Physical scope

Finite-N one-dimensional classical hard rods of common length a>=0 with L>Na and quadratic momenta; the Euclidean Lebesgue product and Jordan-volume calculations assume ACω; exact thermodynamic and virial statements only for this Tonks model, not an all-fluid cluster expansion.

Statement

Assume ${A C}_{ω}$ as defined in The Axiom of Countable Choice ( ${A C}_{ω}$ ) for the Euclidean Lebesgue product and Jordan-volume identifications below. Use the natural logarithm $\log$ of The natural logarithm as the inverse of the exponential function. Let $N \geq 1$ , let $a \geq 0$ be a hard-rod length in metres, and let $L > N a$ be the one-dimensional box length in metres. Rod centers are $x_{i} \in [a / 2, L - a / 2]$ with $∣ x_{i} - x_{j} ∣ \geq a$ for $i \neq j$ . Let each rod have mass $m > 0$ kg, momentum $p_{i} \in R$ in kg m s $^{- 1}$ , and Hamiltonian $H = \sum_{i = 1}^{N} p_{i}^{2} / (2 m)$ on the allowed configuration set, with infinite energy outside it. Adopt the reference $d x d p / (N! h_{0}^{N})$ , where $h_{0} > 0$ is a fixed action in J s, and canonical inverse temperature $β > 0$ in J $^{- 1}$ . Put $λ = h_{0} \sqrt{β / (2 π m)}$ . Then $Z_{N} = \frac{(L - N a)^{N}}{N! λ^{N}}, U = E H = \frac{N}{2 β}, p_{1 D} : = - {(\frac{\partial F}{\partial L})}_{N, a, m, β, h_{0}} = \frac{N}{β (L - N a)}, F = - β^{- 1} \log Z_{N} .$ Here $p_{1 D}$ is the one-dimensional force conjugate to length and has units N; it is not a three-dimensional pressure. With number density $ρ = N / L$ , the exact equation is $β p_{1 D} = \frac{ρ}{1 - a ρ} = \sum_{j = 1}^{\infty} a^{j - 1} ρ^{j}, a ρ < 1.$ After the first $M \geq 1$ terms, the exact nonnegative remainder (strictly positive when $a > 0$ ) is $ρ (a ρ)^{M} / (1 - a ρ)$ , bounded by $ρ r^{M} / (1 - r)$ whenever $a ρ \leq r < 1$ . At fixed $ρ > 0$ with $a ρ < 1$ , as $N, L \to \infty$ with $N / L = ρ$ , $\lim \frac{β F}{N} = \log ⁣ (\frac{ρ λ}{1 - a ρ}) - 1,$ with finite-size excess between $0$ and $(1 + \log N) / N$ .

Facts & Assumptions

Given: ${A C}_{ω}$ , $N \geq 1$ , $a \geq 0$ , $L > N a$ , $m > 0$ , $β > 0$ , and the hard-core position set and Hamiltonian in the Statement.

[F1]

Declared example preparations and thermodynamic interpretation: for each specified model with finite positive partition function, the canonical density is $Z^{- 1} e^{- β H}$ relative to its fixed reference, and $F = - β^{- 1} \log Z$ is the adopted free-energy definition.

[F2]

Gaussian, gamma and beta product integrals: for $m, β > 0$ , $\int_{R} e^{- β p^{2} / (2 m)} d p = \sqrt{2 π m / β}$ and the normalized Gaussian has second moment $m / β$ ; the positive gamma integral is finite.

[F3]

Change of variables for an injective $C^{1}$ map on a compact Jordan set: an injective $C^{1}$ map with nonzero Jacobian on an open neighborhood of a compact Jordan set obeys the compact Jordan change-of-variables formula.

[F4]

The natural logarithm as the inverse of the exponential function defines $\log$ on positive reals; Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm makes it continuous and increasing with the product and quotient laws, and The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t gives $\log^{'} (x) = 1 / x$ and $\log x = \int_{1}^{x} d t / t$ . Finite iteration of the product law gives the integer-power identity.

[F5]

Under ${A C}_{ω}$ , Euclidean Lebesgue measure is the product of its coordinate factors, boxes have the product of their side lengths as measure, and bounded Jordan sets are Lebesgue measurable with measure equal to their Jordan content (On Borel subsets of R^{m+n}, the product lambda_m times lambda_n agrees with lambda_{m+n}, 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, Lebesgue outer measure is at most Jordan outer content, and a bounded Jordan measurable set is Lebesgue measurable with Lebesgue measure equal to its Jordan content).

[F6]

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

[F7]

A bounded set is Jordan measurable when its boundary has content zero, and, for $m \geq 1$ , a continuous graph over a closed nondegenerate rectangle in $R^{m}$ has content zero in $R^{m + 1}$ (A bounded set in $R^{m}$ is Jordan measurable iff its boundary is null, equivalently of content zero, The graph of a continuous function on a closed nondegenerate rectangle in $R^{m}$ has content zero in $R^{m + 1}$ ).

[F8]

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

[F9]

A closed bounded subset of $R^{N}$ is compact, in particular each closed box and the ordered simplex below (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

[A1]

The finite hard-core configuration rule and indistinguishable-classical-particle reference convention are the adopted Tonks model assumptions; no interaction beyond hard exclusion is included.

Proof

1.1givenF2F5F6F8algebra

The single-rod momentum Lebesgue integral is $\int_{R} e^{- β p^{2} / (2 m)} d p = \sqrt{2 π m / β}$ by [F2] and [F8]. By [F5] and Tonelli [F6], the $N$ momentum factors integrate to the product of these integrals. Dividing by $h_{0}^{N}$ gives $λ^{- N}$ .

1.2givenF2F8algebra

By [F2] and the bounded Riemann-to-Lebesgue agreement in [F8], each normalized scalar momentum Gaussian has second moment $m / β$ . Thus the mean kinetic energy of each rod is $1 / (2 β)$ and the total mean is $U = N / (2 β)$ .

1.3givenA1F3F5F7F9algebra

Put $ℓ = L - N a > 0$ . For $N = 1$ , the allowed position set is $[a / 2, L - a / 2]$ , whose Lebesgue measure is $L - a = ℓ$ directly by the box formula [F5]; no graph theorem is needed. For $N \geq 2$ , the allowed labelled position set is a finite union of closed bounded ordering polytopes in $[a / 2, L - a / 2]^{N}$ , hence compact by [F9]. Its boundary is contained in finitely many box faces and contact hyperplanes $x_{i} - x_{j} = \pm a$ , each contained in a graph of an affine function over a sufficiently large closed nondegenerate $(N - 1)$ -dimensional coordinate rectangle, so [F7] makes it Jordan measurable. Fix one label ordering $π$ and sort its centers as $x_{π (1)} \leq \dots \leq x_{π (N)}$ . On that sector set $y_{i} = x_{π (i)} - a / 2 - (i - 1) a$ ; the hard-core inequalities become the closed bounded, hence compact, ordered simplex $S_{ℓ} : = {0 \leq y_{1} \leq \dots \leq y_{N} \leq ℓ}$ by [F9]. Its boundary is contained in the box faces and finitely many tie hyperplanes, each contained in such an affine graph over a nondegenerate $(N - 1)$ -dimensional rectangle; [F7] therefore makes $S_{ℓ}$ Jordan measurable. The affine permutation map has absolute determinant $1$ , so [F3] preserves Jordan volume. The $N!$ coordinate permutations of $S_{ℓ}$ partition $[0, ℓ]^{N}$ up to the tie hyperplanes, which have content zero by [F7]; their equal volumes sum to $ℓ^{N}$ , hence each simplex has volume $ℓ^{N} / N!$ . The $N!$ labelled order sectors cover the allowed set and overlap only on the stated content-zero contact hyperplanes. Therefore its Jordan content is $ℓ^{N}$ , and [F5] gives the full labelled position Lebesgue measure $ℓ^{N}$ .

2.1F1F5F6step 1.1step 1.3algebra

The reference measure divides the position-momentum integral by $N! h_{0}^{N}$ . Product Lebesgue measure and Tonelli [F5, F6] separate the position measure $ℓ^{N}$ from step 1.3 and the momentum integral from step 1.1. Thus $Z_{N} = ℓ^{N} / (N! λ^{N})$ .

3.1F1F4step 2.1algebra

By the product, quotient and integer-power laws in [F4], $F = - β^{- 1} [N \log ((L - N a) / λ) - \log N!]$ , using the logarithm derivative in [F4] at fixed $N, a, β$ gives $p_{1 D} = N / [β (L - N a)]$ . Substituting $ρ = N / L$ yields $β p_{1 D} = ρ / (1 - a ρ)$ .

4.1step 3.1algebra

At $a = 0$ , the virial series has first term $ρ$ and all higher terms zero, so its remainder after any $M \geq 1$ terms is zero. For $0 \leq a ρ < 1$ , the finite geometric identity $\sum_{j = 1}^{M} (a ρ)^{j - 1} = (1 - (a ρ)^{M}) / (1 - a ρ)$ gives the stated virial partial sum and exact remainder after multiplication by $ρ$ . If $a ρ \leq r < 1$ , then $(a ρ)^{M} / (1 - a ρ) \leq r^{M} / (1 - r)$ , proving the uniform bound.

5.1F4step 2.1algebra∎

At fixed $ρ > 0$ , $L - N a = N (1 / ρ - a) > 0$ , so the laws in [F4] give $\frac{β F}{N} = - \log ((L - N a) / λ) + \frac{\log N!}{N} .$ For each integer $k = 1, \dots, N - 1$ , monotonicity of $\log$ bounds its integral on $[k, k + 1]$ between $\log k$ and $\log (k + 1)$ . Summing gives $\int_{1}^{N} \log x d x \leq \log N! \leq \int_{1}^{N} \log x d x + \log N .$ For $N = 1$ the sums and integral are empty and zero. Since [F4] gives $(x \log x - x)^{'} = \log x$ , the integral is $N \log N - N + 1$ . Hence $\log N! = N \log N - N + ϵ_{N}$ with $1 \leq ϵ_{N} \leq 1 + \log N$ . Substitution and [F4] cancel $\log N$ , yielding $\frac{β F}{N} = \log ⁣ (\frac{ρ λ}{1 - a ρ}) - 1 + \frac{ϵ_{N}}{N} .$ The excess is nonnegative and at most $(1 + \log N) / N$ . Finally, $\log N = \int_{1}^{N} d x / x \leq \int_{1}^{N} d x / \sqrt{x} = 2 (\sqrt{N} - 1)$ , so that upper bound tends to zero and proves the stated thermodynamic limit.

Remarks

The infinite virial series is the geometric expansion of this exact one-dimensional hard-rod equation of state. It is not a formal or convergent cluster expansion for arbitrary fluids, attractions, dimensions, or interactions.

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Finite spherical-shell gas and its exact marginal

Physical scope

Finite-N classical ideal kinetic energy for one operationally indistinguishable species, conditional on Countable Choice for the exact shell marginal, with a specifically adopted rotational Gaussian-direction preparation; no dynamical ergodicity or general coarea claim.

Statement

Assume ${A C}_{ω}$ as defined in The Axiom of Countable Choice ( ${A C}_{ω}$ ), the mathematical hypothesis required by the spherical-marginal supplier. Let $N, d \geq 1$ , $D = d N \geq 2$ , and let $B = \prod_{j = 1}^{d} [0, L_{j}] \subset R^{d}$ have side lengths $L_{j} > 0$ in metres and volume $V = \prod_{j} L_{j}$ in $m^{d}$ . Let $m > 0$ kg and $E > 0$ joules, and set $R = \sqrt{2 m E}$ in kg m s $^{- 1}$ . Take one operationally indistinguishable species of $N$ particles as specified by Classical particle counting and physical observables. On the auxiliary labelled cover, take independent uniform positions $q_{i} \in B$ with density $V^{- 1}$ and, independently, the momentum law $P = R Z / ∣ Z ∣,$ where $Z \in R^{D}$ has the independent standard-normal coordinate law defined in Exact spherical-shell coordinate marginal and fixed-block Gaussian limit and $P = 0$ is assigned on the null event $Z = 0$ . This product law is invariant under particle permutations, so it defines the stated model preparation on the quotient; a one-particle marginal means the law of a uniformly selected particle. This is a specified spherical-shell preparation: $H (P) = ∣ P ∣^{2} / (2 m) = E$ almost surely, with no assumption about a trajectory or equilibration. For any momentum block of $k$ coordinates with $1 \leq k < D$ , its density is $f_{D, k}$ from Exact spherical-shell coordinate marginal and fixed-block Gaussian limit. In particular the total-energy variance is zero J $^{2}$ .

Compare the separate canonical ideal-gas preparation of Exact canonical and grand ideal-gas preparations with the same $N, d, m, B$ and one-species convention, at $β = D / (2 E) > 0$ . Its mean total energy is also $E$ , but its variance is $D / (2 β^{2}) > 0$ J $^{2}$ . If $N > 1$ , the uniformly selected particle's $d$ -component shell marginal has compact support because $d < D$ , whereas its canonical Gaussian momentum has positive probability outside every finite-radius ball. The two finite preparations therefore differ despite their matched mean energy. For $N = 1$ , the full momentum vector is singular on its sphere and the density formula for $k = D$ is not asserted.

Facts & Assumptions

Given: ${A C}_{ω}$ , the rectangular box with side lengths in metres and volume in m $^{d}$ , the one-species operational indistinguishability assumption, particle number and dimension, positive mass and energy, shell radius $R$ , and the explicitly adopted independent position/spherical-momentum preparation in the Statement.

[F1]

Declared example preparations and thermodynamic interpretation: each shell or product law is an explicitly adopted probability preparation, separate from a canonical law and not inferred from Hamiltonian dynamics.

[F2]

Exact spherical-shell coordinate marginal and fixed-block Gaussian limit: assuming ${A C}_{ω}$ , for $D \geq 2$ , $1 \leq k < D$ , and $R > 0$ , the independent standard-normal coordinate law has an orthogonally invariant direction; $R Z / ∣ Z ∣$ is concentrated on $∣ P ∣ = R$ , its $k$ -coordinate marginal has the stated exact density, and its coordinate second moment is $R^{2} / D$ .

[F3]

Exact canonical and grand ideal-gas preparations: for the massive $d$ -dimensional canonical ideal gas with $β > 0$ , total energy has mean $d N / (2 β)$ and variance $d N / (2 β^{2})$ , and each component of the uniformly selected particle's momentum is Gaussian with variance $m / β$ .

[F4]

Classical particle counting and physical observables: for one operationally indistinguishable species, the labelled lift carries the symmetric permutation action and a permutation-invariant one-particle marginal is interpreted as the law of a uniformly selected particle.

[A1]

This permutation-invariant spherical-shell law is a chosen finite classical distribution on one operationally indistinguishable species; it is not a claim that an isolated gas dynamically reaches it or that it is the only microcanonical convention.

Proof

1.1givenF1A1algebra

Under the shell preparation [F1] and its explicit law [A1], $∣ P ∣ = R$ almost surely by the definition of $P$ ; hence $H (P) = R^{2} / (2 m) = E$ on every realization outside the null event $Z = 0$ . A constant random variable has variance zero.

2.1F2step 1.1

For $1 \leq k < D$ , the first $k$ momentum coordinates are the projection in [F2], so their exact Lebesgue density is the displayed $f_{D, k}$ , which vanishes outside the closed radius- $R$ ball. The event that this block has norm $R$ requires the remaining $D - k$ Gaussian coordinates to vanish; their joint density and $D - k \geq 1$ imply that this event has probability zero.

2.2givenF3step 1.1algebra

Choose $β = D / (2 E)$ . By [F3], the canonical mean energy is $D / (2 β) = E$ and its variance is $D / (2 β^{2}) > 0$ . Thus it cannot have the shell law of step 1.1, whose energy variance is zero.

3.1A1F2F3F4step 2.1∎

If $N > 1$ , then $d < D$ . Particle permutations act by orthogonal coordinate permutations on $R^{D}$ , so [F2] and [A1] make every particle block marginal identical; by the selected-particle convention [F4], their uniform mixture is the law of a uniformly selected particle. It has a density supported in the closed radius- $R$ ball, with no boundary atom by step 2.1. By [F3], the canonical Gaussian of a uniformly selected particle has strictly positive density at every point of $R^{d}$ , so its integral over the open annulus $R < ∣ p ∣ < 2 R$ is positive, while the shell probability of that annulus is zero. Therefore the one-particle marginals differ as well. For $N = 1$ , $d = D$ and the full shell law is singular, so no $k = D$ density assertion has been used.

Remarks

This result compares two stated finite probability laws. It proves no equivalence or inequivalence theorem for general thermodynamic ensembles and no dynamical equilibration claim.

5 · Examples, experiments and counterexamples

None yet.

Gaussian and gamma product tools

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

Gaussian, gamma and beta product integrals

Statement

Facts & Assumptions

Proof

Remarks

Jacobian identity for a specified smooth diffeomorphic flow

Statement

Facts & Assumptions

Proof

Remarks

Fine entropy under a specified incompressible periodic flow

Statement

Facts & Assumptions

Proof

Remarks

Gaussian covariance for positive quadratic forms and a boundary counterexample

Statement

Facts & Assumptions

Proof

Remarks

Exact spherical-shell coordinate marginal and fixed-block Gaussian limit

Statement

Facts & Assumptions

Proof

Remarks

Declared example preparations and thermodynamic interpretation

Postulate

Remarks

Confined coupled quadratic canonical systems

Statement

Facts & Assumptions

Proof

Remarks

Exact canonical and grand ideal-gas preparations

Statement

Facts & Assumptions

Proof

Remarks

Finite periodic Ising-ring correlations and exact decay

Statement

Facts & Assumptions

Proof

Remarks

Exact Tonks hard-rod pressure and convergent virial series

Statement

Facts & Assumptions

Proof

Remarks

Finite spherical-shell gas and its exact marginal

Statement

Facts & Assumptions

Proof

Remarks

5 · Examples, experiments and counterexamples

Sources