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Finite Bernoulli fluctuations and matching exponential rates

1 · Prerequisites

2 · Summary

This page first defines variance for finite real random variables, then proves exact conditional statements for three finite model families: Bernoulli product counts with polynomial and exponential concentration bounds; a size-dependent global three-sector model whose positive-temperature limit has a mean-energy jump and an explicit canonical/shell inequivalence witness; and a symmetric walk whose fixed-time marginals converge under diffusive scaling, alongside periodic and finite-state counterexamples to mixing and coarse-entropy monotonicity.

The sampling laws and state preparations are stipulated inputs. No measured counts, apparatus behavior, laboratory independence, equilibration, deterministic collision limit, or empirical inference is asserted. Finite model probabilities are not observations, and low probability is not logical impossibility.

For the sector model, set the dimensionless horizontal coordinate $\hat{β} = β ϵ$ . The exact limiting normalized log partition is $ψ (\hat{β}) = \max {0, \log 4 - 2 \hat{β}}$ on the positive-temperature branch; the plotted corner is at $\hat{β} = \log 2$ . The drawing is a rendering of this formula only; the proof and one-sided slopes are established in Explicit nonadditive transition and ensemble inequivalence.

\pgfmathsetmacro{\betac}{ln(2)}
\pgfmathsetmacro{\psimax}{ln(4)}
\begin{tikzpicture}[x=1.5cm,y=1cm]
  \draw[->] (-0.08,0) -- (2.1,0) node[right] {$\widehat\beta$};
  \draw[->] (0,-0.08) -- (0,\psimax+0.25) node[above] {$\psi$};
  \draw[thick] (0,\psimax) -- (\betac,0) -- (2.0,0);
  \draw[dashed] (\betac,0) -- (\betac,0.65);
  \node[left] at (0,\psimax) {$\log4$};
  \node[below] at (\betac,0) {$\log2$};
\end{tikzpicture}

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-6-luna)Open item page →

Variance of a finite real random variable

Definition

Let $(Ω, w)$ be a finite probability space: $Ω$ is a finite set, each weight $w (ω)$ is nonnegative, and $\sum_{ω \in Ω} w (ω) = 1$ . Let $X : Ω \to R$ be a real random variable and let $E [X] = \sum_{ω \in Ω} X (ω) w (ω)$ . Its variance is

$Var (X) : = E [(X - E [X])^{2}] = \sum_{ω \in Ω} (X (ω) - E [X])^{2} w (ω) .$

All sums are finite, so the definition requires no additional integrability or moment hypothesis.

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Finite Bernoulli fluctuations and matching exponential rates

Physical scope

Exact bounds for the explicitly adopted finite IID Bernoulli product law; no laboratory sampling, observations, calibration, or empirical inference is asserted.

Statement

Let $n \geq 1$ be an integer, let $p \in (0, 1)$ , and give $Ω_{n} = {0, 1}^{n}$ the explicitly stipulated product probability whose singleton weights are $P_{p} ({(x_{1}, \dots, x_{n})}) = \prod_{i = 1}^{n} p^{x_{i}} (1 - p)^{1 - x_{i}};$ for an event $A \subseteq Ω_{n}$ , $P_{p} (A)$ is the sum of these weights over $x \in A$ . Write $X_{i} (x) = x_{i}$ , $K = \sum_{i = 1}^{n} X_{i}$ , and $\overline{X} = K / n$ . These variables and all probabilities are dimensionless. The product law is an adopted sampling model; independence is an assumption of that law, not an inference about any apparatus or equilibrium system.

For $a \in (0, 1)$ define the Bernoulli relative-entropy rate $D (a ∥ p) = a \log \frac{a}{p} + (1 - a) \log \frac{1 - a}{1 - p},$ and set $D (0 ∥ p) = - \log (1 - p)$ and $D (1 ∥ p) = - \log p$ . Then $E K = n p, Var (K) = n p (1 - p),$ and for every dimensionless $ε > 0$ , $P_{p} (∣ \overline{X} - p ∣ \geq ε) \leq \frac{p (1 - p)}{n ε^{2}}, P_{p} (∣ \overline{X} - p ∣ \geq ε) \leq 2 e^{- 2 n ε^{2}} .$ For $p < a < 1$ and $0 < a < p$ , respectively, $P_{p} (K \geq n a) \leq e^{- n D (a ∥ p)}, P_{p} (K \leq n a) \leq e^{- n D (a ∥ p)} .$ For each integer $0 \leq k \leq n$ , putting $a = k / n$ gives the point-mass bounds $\frac{1}{n + 1} e^{- n D (a ∥ p)} \leq P_{p} (K = k) \leq e^{- n D (a ∥ p)} .$ Consequently, for every sequence of integers $k_{n} \in {0, \dots, n}$ with $k_{n} / n \to a \in (0, 1)$ , $\lim_{n \to \infty} \frac{1}{n} \log P_{p} (K = k_{n}) = - D (a ∥ p) .$ All conclusions are conditional calculations for this finite product model and make no claim that physical outcomes are independent or that a small model probability is a logical contradiction.

Facts & Assumptions

Given: The stated finite product law, $n \geq 1$ , $p \in (0, 1)$ , and the definitions of $K$ , $\overline{X}$ , and $D (a ∥ p)$ in the Statement.

[F1]

Declared example preparations and thermodynamic interpretation: consuming the named finite model may state its product preparation as a separate explicit probability law; such a preparation is an adopted assumption, not evidence that a laboratory realizes it.

[F2]

Finite probability spaces, outcome weights, events, and event probabilities: a finite probability space is a finite outcome set with nonnegative weights summing to one; event probabilities are sums of those weights.

[F3]

Real random variables on finite probability spaces and their finite distributions: a real random variable on a finite probability space is a function from the outcome set to $R$ .

[F4]

Expectation of a real random variable on a finite probability space: $E Y = \sum_{x \in Ω_{n}} P_{p} (x) Y (x)$ .

[F5]

The moment generating function $M_{X} (t) = E [e^{t X}]$ on a finite probability space: for a finite real random variable $Y$ , $M_{Y} (s) = E [\exp (s Y)]$ for every real $s$ .

[F6]

The real exponential function and the number $e$ by a power series: $\exp$ is defined by its everywhere-convergent real power series.

[F7]

The natural logarithm as the inverse of the exponential function: $\log x$ is the unique real $y$ with $\exp (y) = x$ , for $x > 0$ .

[F8]

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

[F9]

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

[F10]

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 $f (b) - f (a) = f^{'} (c) (b - a)$ : if $g$ is continuous on a closed interval and differentiable in its interior, then $g (b) - g (a) = g^{'} (c) (b - a)$ for some interior $c$ .

[F11]

The logarithm grows more slowly than every positive real power: for every $α > 0$ , $\log x / x^{α} \to 0$ as $x \to + \infty$ .

[F12]

Variance of a finite real random variable: on a finite probability space, $Var (Y) = E [(Y - E [Y])^{2}]$ .

Proof

Given: The finite IID Bernoulli model and all hypotheses in the Statement.

1.1givenF1F2F3F4F12algebra

The total weight is $\prod_{i = 1}^{n} (p + (1 - p)) = 1$ , so the stipulated weights form a finite probability law. Summing over all coordinates except $i$ gives $P (X_{i} = 1) = p$ and $P (X_{i} = 0) = 1 - p$ ; the product factorization also gives independence. For $0 \leq k \leq n$ put $c_{n, k} : = # {S \subseteq {1, \dots, n} : ∣ S ∣ = k}$ . The set of coordinates with value $1$ gives a bijection between outcomes with $K = k$ and such subsets, so $P (K = k) = c_{n, k} p^{k} (1 - p)^{n - k}$ . By [F4] and [F12], $E X_{i} = p$ and $Var (X_{i}) = E [(X_{i} - p)^{2}] = p (1 - p)$ . For $i \neq j$ , the product weights factor the finite sum, giving $E [(X_{i} - p) (X_{j} - p)] = E (X_{i} - p) E (X_{j} - p) = 0$ . Expanding $K$ and $(K - n p)^{2}$ therefore yields $E K = n p$ and $Var (K) = n p (1 - p)$ .

1.2givenF5F6algebra

The finite-sum definition [F5] and product weights give $M_{K} (s) = \sum_{x \in Ω_{n}} P_{p} (x) e^{s \sum_{i} x_{i}} = \prod_{i = 1}^{n} ((1 - p) + p e^{s}) = (1 - p + p e^{s})^{n} .$ This is a finite factorization, so no limiting interchange is involved.

1.3givenF7F8F9F10F11algebra

On $0 < a < 1$ , differentiating the displayed expression for $D$ with [F9] gives $D (p ∥ p) = 0$ , $\partial_{a} D (a ∥ p) ∣_{a = p} = 0$ , and $\partial_{a}^{2} D (a ∥ p) = 1 / [a (1 - a)] \geq 4$ , since $a (1 - a) \leq 1 / 4$ . Let $G (a) = D (a ∥ p) - 2 (a - p)^{2}$ . Then $G (p) = G^{'} (p) = 0$ and $G^{''} \geq 0$ . For $a > p$ , the mean value theorem applied first to $G^{'}$ and then to $G$ gives $G^{'} (a) \geq 0$ and $G (a) \geq 0$ ; for $a < p$ , the same two applications give $G^{'} (a) \leq 0$ and again $G (a) \geq 0$ . Thus $D (a ∥ p) \geq 2 (a - p)^{2}$ for interior $a$ . At $a ↓ 0$ , the only noncontinuous-looking term is $a \log a = - \log (1 / a) / (1 / a)$ , which tends to zero by [F11]; at $a ↑ 1$ the analogous term is $(1 - a) \log (1 - a)$ . The remaining factors are continuous, so the endpoint extensions preserve the inequality.

2.1givenstep 1.1algebra

Fix $1 \leq k \leq n - 1$ and let $a = k / n$ . For $0 \leq j \leq n$ put $q_{j} = c_{n, j} a^{j} (1 - a)^{n - j}$ . These are the count probabilities under the finite product weights with parameter $a$ ; their sum is $1$ by factoring the total weight as $(a + (1 - a))^{n}$ . For each $0 \leq j < n$ , count pairs $(S, T)$ with $S \subset T \subseteq {1, \dots, n}$ , $∣ S ∣ = j$ , and $∣ T ∣ = j + 1$ . Choosing $S$ first gives $c_{n, j} (n - j)$ pairs, while choosing $T$ first gives $c_{n, j + 1} (j + 1)$ ; hence $c_{n, j + 1} / c_{n, j} = (n - j) / (j + 1)$ . Therefore $\frac{q_{j + 1}}{q_{j}} = \frac{n - j}{j + 1} \frac{a}{1 - a} .$ The factor $(n - j) / (j + 1)$ decreases strictly with $j$ , since $(n - j - 1) (j + 1) < (n - j) (j + 2)$ for $j < n - 1$ . At $j = k - 1$ the ratio is $(n - k + 1) / (n - k) > 1$ , and at $j = k$ it is $k / (k + 1) < 1$ . Thus the sequence increases through $q_{k}$ and decreases after it, so $q_{k}$ is the largest of its $n + 1$ masses. Their sum is $1$ , whence $1 / (n + 1) \leq q_{k} \leq 1$ .

2.2givenF4step 1.1algebra

Pointwise on $Ω_{n}$ , $1_{{∣ \overline{X} - p ∣ \geq ε}} \leq (\overline{X} - p)^{2} / ε^{2}$ . Summing this inequality with the finite weights and using step 1.1 gives the first probability bound.

2.3givenF4F5F6F7F8step 1.2algebra

Suppose first $p < a < 1$ and put $s = \log ⁣ (a (1 - p) / (p (1 - a))) > 0$ . On ${K \geq n a}$ , $e^{s K} \geq e^{s n a}$ , so the finite expectation of the nonnegative random variable $e^{s K}$ is at least $e^{s n a} P_{p} (K \geq n a)$ . By step 1.2, $P_{p} (K \geq n a) \leq e^{- s n a} (1 - p + p e^{s})^{n} = e^{- n D (a ∥ p)},$ where $e^{s} = a (1 - p) / (p (1 - a))$ , $1 - p + p e^{s} = (1 - p) / (1 - a)$ , and [F7]–[F8] simplify the logarithm. If $0 < a < p$ , the same value of $s$ is negative; on ${K \leq n a}$ one again has $e^{s K} \geq e^{s n a}$ , and the same substitution yields $P_{p} (K \leq n a) \leq e^{- n D (a ∥ p)}$ . This proves both exponential tail bounds without using a general concentration theorem as a premise.

3.1givenF7F8step 1.1step 2.1algebra

Direct division of the two point probabilities, using the same subset count $c_{n, k}$ from step 1.1, gives $\frac{P_{p} (K = k)}{q_{k}} = {(\frac{p}{a})}^{k} {(\frac{1 - p}{1 - a})}^{n - k} .$ Since $k = n a$ and $n - k = n (1 - a)$ , [F7]–[F8] give the logarithm of this ratio as $n a \log (p / a) + n (1 - a) \log ((1 - p) / (1 - a)) = - n D (a ∥ p)$ , so the ratio is $e^{- n D (a ∥ p)}$ . Combining with step 2.1 proves the claimed bounds for $1 \leq k \leq n - 1$ . For $k = 0$ and $k = n$ , direct evaluation gives $P_{p} (K = 0) = (1 - p)^{n} = e^{- n D (0 ∥ p)}$ and $P_{p} (K = n) = p^{n} = e^{- n D (1 ∥ p)}$ , which satisfy the same inequalities.

3.2givenstep 2.3step 1.3algebra

If $0 < ε < 1$ , split the deviation event into the upper threshold $a_{+} = p + ε$ and lower threshold $a_{-} = p - ε$ . For an interior threshold apply step 2.3 and then step 1.3; at $a_{+} = 1$ or $a_{-} = 0$ use the exact endpoint probability $p^{n}$ or $(1 - p)^{n}$ and the endpoint case of step 1.3; a threshold outside $[0, 1]$ gives an empty event. Each nonempty tail is at most $e^{- 2 n ε^{2}}$ , so their sum is at most $2 e^{- 2 n ε^{2}}$ . If $ε \geq 1$ , the deviation event is empty because $0 \leq \overline{X} \leq 1$ and $0 < p < 1$ .

4.1givenF7F11step 3.1∎

If $k_{n} / n \to a \in (0, 1)$ , then $D (k_{n} / n ∥ p) \to D (a ∥ p)$ by continuity on $(0, 1)$ . Taking logarithms of step 3.1 and dividing by $n$ bounds $n^{- 1} \log P_{p} (K = k_{n})$ between $- D (k_{n} / n ∥ p) - \log (n + 1) / n$ and $- D (k_{n} / n ∥ p)$ . By [F11], $\log (n + 1) / n \to 0$ , so the squeeze theorem gives the stated rate. These probabilities remain conditional on the explicitly stipulated product law; no finite sample or measurement is asserted.

Physics theoremStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

Explicit nonadditive transition and ensemble inequivalence

Physical scope

A specified finite global three-sector model with N-dependent multiplicities; its positive-temperature canonical limit has a jump in mean energy per member and a middle-sector ensemble-inequivalence witness. No volume density, short-range model, or empirical system is asserted.

Statement

Fix $N \geq 1$ and $ϵ > 0$ joules. Let $Ω_{N} = A_{N} ⊔ B_{N} ⊔ C_{N}$ be disjoint finite sectors with $∣ A_{N} ∣ = 1$ , $∣ B_{N} ∣ = 1$ , and $∣ C_{N} ∣ = 4^{N}$ , and define the energy $H_{N}$ to be $0$ , $N ϵ$ , and $2 N ϵ$ on those sectors, respectively. The model is global and size-dependent. Here product-additive means that one fixed finite one-site alphabet $Γ$ supplies $Ω_{N} = Γ^{N}$ for every $N$ , and one fixed one-site energy function $h : Γ \to R$ (with values measured in joules) satisfies $H_{N} (γ_{1}, \dots, γ_{N}) = \sum_{i = 1}^{N} h (γ_{i})$ . The present family is non-product-additive; no local, short-range, or fluid interpretation is assumed.

Use counting measure on each $Ω_{N}$ as the reference measure, so every individual state has weight one. For a real inverse-energy parameter $β$ in joules $^{- 1}$ , define the finite exponential-tilt probability by $Z_{N} (β) = \sum_{ω \in Ω_{N}} e^{- β H_{N} (ω)} = 1 + e^{- β N ϵ} + 4^{N} e^{- 2 β N ϵ}, P_{β, N} (ω) = \frac{e^{- β H_{N} (ω)}}{Z_{N} (β)} .$ For $β > 0$ this is the adopted canonical preparation. For $β \leq 0$ it is only the same finite counting formula used algebraically; no negative-temperature reservoir is asserted. Also, for a shell half-width $Δ$ in joules with $0 \leq Δ < N ϵ$ , define $S_{N, Δ} = {ω : ∣ H_{N} (ω) - N ϵ ∣ \leq Δ}$ and give it the uniform counting probability $Q_{N, Δ} (ω) = ∣ S_{N, Δ} ∣^{- 1}$ for $ω \in S_{N, Δ}$ .

For every real $β$ , the finite exponential-tilt probability of $B_{N}$ obeys $P_{β, N} (B_{N}) \leq \frac{1}{2^{N + 1} + 1},$ whereas the shell probability of $B_{N}$ is $1$ . Define the dimensionless normalized log partition function $ψ_{N} (β) = N^{- 1} \log Z_{N} (β)$ . Then $ψ_{N} (β) ⟶ ψ (β) : = \max {0, \log 4 - 2 β ϵ}, 0 \leq ψ_{N} (β) - ψ (β) \leq \frac{\log 3}{N} .$ For $β > 0$ , define the canonical mean energy per member by $u_{N} (β) : = N^{- 1} E_{P_{β, N}} [H_{N}]$ and, when the limit exists, set $u (β) : = \lim_{N \to \infty} u_{N} (β)$ . The corner occurs at the positive inverse temperature $β_{c} = (\log 2) / ϵ$ . The limit $u (β)$ exists for every $β > 0$ and equals $2 ϵ$ for $0 < β < β_{c}$ , $ϵ$ at $β = β_{c}$ , and $0$ for $β > β_{c}$ . In this item, a first-order transition point means a positive- $β$ point where $u$ has finite but unequal one-sided limits; thus the jump at $β_{c}$ is first order under this explicit convention. Every finite- $N$ function $ψ_{N}$ is real analytic. This is one explicit global finite-state countermodel, not a universal criterion for phase transitions or ensemble equivalence in short-range matter.

Facts & Assumptions

Given: The stated finite sectors, their counting measure, the energy function, the dimensionful parameter $ϵ > 0$ , and the finite exponential-tilt and shell laws in the Statement.

[A1]

For the canonical branch, use the adopted preparation convention only when $β > 0$ ; the $β \leq 0$ expression is a finite algebraic extension, not a physical negative-temperature claim.

[A2]

The shell comparison is the separately specified uniform probability on $S_{N, Δ}$ ; it is not derived from dynamics or from a general equivalence principle.

[F1]

Declared example preparations and thermodynamic interpretation: a canonical probability is the normalized $e^{- β H}$ weight relative to a specified reference measure when $β > 0$ and the partition function is finite and positive.

[F2]

The natural logarithm as the inverse of the exponential function: for $x > 0$ , $\log x$ is the unique real $y$ with $\exp (y) = x$ .

[F3]

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

[F4]

The real exponential function and the number $e$ by a power series: $\exp$ is the everywhere-convergent real power series.

[F5]

A real-analytic function on an open subset of $R$ is locally represented by a convergent real power series: a function is real analytic on an open real domain when it has a convergent power-series representation on a neighbourhood of every point.

[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 in the interior of its radius of convergence.

[F7]

Real-analytic functions are closed under sums, products and compositions, and under quotients where the denominator is nonzero: sums, products, nonvanishing quotients, and compatible compositions of real-analytic functions are real analytic.

[F8]

Real analytic inverse and implicit functions: a real-analytic map with invertible derivative has a real-analytic local inverse.

[F9]

The exponential function is smooth and $(\exp)^{'} = \exp$ : $\exp^{'} (x) = \exp (x) > 0$ for every real $x$ .

[F10]

Finite probability spaces, outcome weights, events, and event probabilities: a finite probability space is a finite outcome set with nonnegative weights summing to $1$ , and event probabilities are sums of the weights.

[F11]

Real random variables on finite probability spaces and their finite distributions: a real random variable on a finite probability space is a function from the outcome set to $R$ .

[F12]

Expectation of a real random variable on a finite probability space: $E [Y] = \sum_{ω \in Ω} Y (ω) w (ω)$ .

Proof

Given: The exact model data and law in the Statement; $β$ is positive for the canonical physical interpretation and may be any real number in the separate finite algebraic claims.

1.1givenA1F1F10algebra

If this family were product-additive under the definition in the Statement, its fixed alphabet would have $∣ Γ ∣ = ∣ Ω_{1} ∣ = 6$ , forcing $∣ Ω_{2} ∣ = ∣ Γ ∣^{2} = 36$ . But $∣ Ω_{2} ∣ = 1 + 1 + 4^{2} = 18$ . Thus no such product decomposition exists. The three sector weights in the finite exponential tilt are $1$ , $x$ , and $4^{N} x^{2}$ , where $x = e^{- β N ϵ} > 0$ ; they sum to the displayed positive finite $Z_{N} (β)$ , so division by $Z_{N}$ gives a finite probability law as in [F10]. For $β > 0$ , [A1] and [F1] identify this normalized counting weight with the adopted canonical branch; for $β \leq 0$ it remains only the stated algebraic expression.

1.2givenA2algebra

The energies of $A_{N}$ and $C_{N}$ differ from $N ϵ$ by exactly $N ϵ$ . The condition $Δ < N ϵ$ excludes both sectors from $S_{N, Δ}$ , while $B_{N} \subseteq S_{N, Δ}$ . Thus the shell is exactly the one-state set $B_{N}$ and its uniform counting probability assigns $B_{N}$ probability one.

1.3givenF2F3algebra

Put $M_{N} = \max {1, 4^{N} x^{2}}$ . If $x \leq 1$ , then $x \leq M_{N}$ ; if $x > 1$ , then $4^{N} x^{2} \geq x$ since $N \geq 1$ . Hence $M_{N} \leq Z_{N} \leq 3 M_{N}$ . By [F2]–[F3], $\frac{1}{N} \log M_{N} = \max {0, \log 4 - 2 β ϵ} = ψ (β),$ so taking logarithms in $M_{N} \leq Z_{N} \leq 3 M_{N}$ gives $0 \leq ψ_{N} - ψ \leq (\log 3) / N$ , and consequently $ψ_{N} \to ψ$ .

1.4givenF4F6F7algebra

By [F4], $\exp$ is a power series converging for every real argument, so [F6] makes $\exp$ real analytic on $R$ . The affine polynomial $β \mapsto - β N ϵ$ is real analytic, hence [F7] makes $β \mapsto e^{- β N ϵ}$ real analytic as well.

2.1givenF10step 1.1algebra

Since $(1 - 2^{N} x)^{2} \geq 0$ , $1 + 4^{N} x^{2} \geq 2^{N + 1} x$ . Therefore $Z_{N} = 1 + x + 4^{N} x^{2} \geq (2^{N + 1} + 1) x$ , and division by $Z_{N} > 0$ gives $P_{β, N} (B_{N}) \leq (2^{N + 1} + 1)^{- 1}$ .

2.2givenF2F3step 1.3algebra

The equality $\log 4 = 2 \log 2$ from [F3] gives the unique corner $β_{c} = (\log 2) / ϵ > 0$ . On $0 < β < β_{c}$ , the larger branch is $\log 4 - 2 β ϵ$ and has slope $- 2 ϵ$ ; on $β > β_{c}$ the larger branch is $0$ and has slope $0$ . The one-sided slopes differ.

2.3givenF2F7step 1.4algebra

The exponential is positive, so $Z_{N} = 1 + e^{- β N ϵ} + 4^{N} e^{- 2 β N ϵ} > 0$ for every real $β$ . Its expression as a sum and product of the analytic function in step 1.4 is real analytic by [F7].

3.1givenF11F12step 2.2algebra

By [F12], with [F11] identifying $H_{N}$ as a finite real random variable, the mean energy per member is $u_{N} (β) = ϵ (x + 2 \cdot 4^{N} x^{2}) / Z_{N}$ for $β > 0$ . If $0 < β < β_{c}$ , divide numerator and denominator by the high-sector weight $4^{N} x^{2}$ : the low-to-high ratio is $e^{- N (\log 4 - 2 β ϵ)} \to 0$ and the middle-to-high ratio is $e^{N (β ϵ - \log 4)} \leq 2^{- N} \to 0$ , so $u_{N} (β) \to 2 ϵ$ . If $β > β_{c}$ , both non-low weights $x = e^{- β N ϵ}$ and $4^{N} x^{2} = e^{N (\log 4 - 2 β ϵ)}$ tend to zero, so $u_{N} (β) \to 0$ . At $β = β_{c}$ , the three sector weights are $1$ , $2^{- N}$ , and $1$ , so $u_{N} (β_{c}) = ϵ (2^{- N} + 2) / (2 + 2^{- N}) \to ϵ$ . These limits give the stated piecewise $u$ and its unequal one-sided limits.

3.2givenF2F8F9step 1.4step 2.3algebra

Fix $β_{0} \in R$ and put $z_{0} = Z_{N} (β_{0}) > 0$ . By step 1.4, $\exp$ is real analytic; [F9] gives $\exp^{'} (\log z_{0}) = z_{0} \neq 0$ , so [F8] gives a real-analytic local inverse near $z_{0}$ . By the inverse definition [F2], that local inverse is $\log$ on this neighborhood, hence $\log$ is real analytic near $z_{0}$ .

4.1givenF5F7step 2.2step 2.3step 3.2algebra∎

By [F5] the target property is real analyticity, and [F7] makes the composition $N^{- 1} \log Z_{N}$ real analytic using steps 2.3 and 3.2. Hence every finite- $N$ function $ψ_{N}$ is real analytic. The derivative jump in step 2.2 belongs to the limiting function, not to a finite- $N$ singularity.

Remarks

The uniform shell and the positive-temperature canonical law are two explicitly specified preparations. The strict middle-sector probability bound is uniform in all real $β$ , including $N$ -dependent choices. It establishes one ensemble-inequivalence observable in this global finite-sector sequence; it does not establish behavior for additive or short-range systems.

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Diffusion-scale walk and recurrence/coarse-entropy counterexamples

Physical scope

A finite IID symmetric walk under diffusive scaling, an explicitly defined periodic rotor, and a four-state measure-preserving permutation. Results are conditional model calculations; no collision-to-diffusion limit, path-space convergence, experimental observation, or universal entropy law is asserted.

Statement

Assume the Axiom of Choice. The following are separate calculations for three explicitly defined models.

Diffusive-scale walk. Let $D > 0$ have units m $^{2}$ s $^{- 1}$ , fix $t \geq 0$ seconds, and let $(δ t_{j})_{j \geq 1}$ be any positive sequence tending to zero, with $ℓ_{j} = \sqrt{2 D δ t_{j}}$ metres and $n_{j} = ⌊ t / δ t_{j} ⌋$ . For each $j$ , on the finite product space $Ω_{j} = {- 1, 1}^{n_{j}}$ with uniform probability, let $Ξ_{i}$ be the coordinate increments and $S_{j} = ℓ_{j} \sum_{i = 1}^{n_{j}} Ξ_{i} .$ For the dimensionless lattice index $r \in Z$ , define $u_{n} (r) : = 2^{- n} # {(x_{1}, \dots, x_{n}) \in {- 1, 1}^{n} : \sum_{i = 1}^{n} x_{i} = r},$ with the empty sum equal to zero. Then $E S_{j} = 0$ , $Var (S_{j}) = n_{j} ℓ_{j}^{2}$ , and the lattice probabilities satisfy $u_{n + 1} (r) = \frac{1}{2} u_{n} (r - 1) + \frac{1}{2} u_{n} (r + 1)$ with $u_{0} (r) = 1_{{r = 0}}$ . For every wave number $k \in R$ in m $^{- 1}$ , $φ_{S_{j}} (k) = \cos (ℓ_{j} k)^{n_{j}} .$ The laws of $S_{j}$ converge weakly on $R$ to $N (0, 2 D t)$ , the normal probability law with mean zero and variance $2 D t$ (in m $^{2}$ ). The conclusion is fixed-time marginal convergence only.

Periodic rotor. Let $T : = R / (2 π Z)$ with its quotient topology, and let the phase space $X = T \times R$ have its metrizable product topology. Let $q \in T$ be a dimensionless angle, $p \in R$ have units kg m $^{2}$ s $^{- 1}$ , and $I > 0$ have units kg m $^{2}$ . Define the model energy $H (q, p) = p^{2} / (2 I)$ and, for time $s \in R$ measured in seconds, the flow $Φ_{s} (q, p) = (q + p s / I m o d 2 π, p) .$ For each $p_{0} \neq 0$ , the orbit is periodic with period $T = 2 π I / ∣ p_{0} ∣$ seconds. Let $ν$ be normalized arc length on $T$ , define the continuous inclusion $ι_{p_{0}} (q) = (q, p_{0})$ , and for each Borel $E \subseteq X$ set $μ_{0} (E) = ν (ι_{p_{0}}^{- 1} (E))$ . Thus $μ_{0}$ is the uniform angle probability on the fixed- $p_{0}$ orbit. For this invariant probability, mixing means $μ_{0} (B \cap Φ_{s}^{- 1} C) \to μ_{0} (B) μ_{0} (C)$ for all Borel $B, C \subseteq X$ as $s \to \infty$ . The rotor is not mixing: for the half-circle arc $A = {([q], p_{0}) : q \in [0, π)}$ , where $[q] = q + 2 π Z \in T$ , the mixing correlation fails along times $s = m T$ with positive integers $m$ . For any $q_{0} = [θ_{0}] \in T$ , the point-supported initial law $δ_{(q_{0}, p_{0})}$ (unit mass at $(q_{0}, p_{0})$ ) has no weak limit as $s \to \infty$ ; the bounded continuous witness $g ([θ], p) = \cos (θ - θ_{0})$ is well-defined because cosine is $2 π$ -periodic.

Finite coarse-entropy model. Let $Ω = {0, 1, 2, 3}$ with uniform reference probability and permutation $T (0) = 1, T (1) = 2, T (2) = 3, T (3) = 0$ . Start from $π_{0} = (1 / 2, 1 / 2, 0, 0)$ , and evolve by pushforward under $T$ . For the coarse partition ${{0, 1}, {2, 3}}$ , let $(r_{1}, r_{2})$ be any probability vector with $r_{i} \geq 0$ and $r_{1} + r_{2} = 1$ , and define its coarse Shannon entropy by $H_{c} (r_{1}, r_{2}) = - \sum_{i : r_{i} > 0} r_{i} \log r_{i}$ in nats. Define fine entropy analogously for a probability vector on the four states, using $0 \log 0 = 0$ . The coarse entropies at times $0, 1, 2$ are $0, \log 2, 0$ , while the fine entropy is $\log 2$ at all three times. Thus deterministic invertibility and preservation of the uniform reference measure do not imply monotone coarse entropy.

The smooth-flow result Fine entropy under a specified incompressible periodic flow has different hypotheses: it concerns positive $C^{1}$ densities on a flat torus transported by supplied jointly $C^{2}$ volume-preserving diffeomorphisms and preserves their dimensionless fine differential entropy. It supplies no monotonicity conclusion for coarse entropy in a finite permutation. These models report no observations and prove no diffusion limit from deterministic collisions.

Facts & Assumptions

Given: The Axiom of Choice; the finite product walk laws; the stated rotor flow and dimensions; the four-state permutation and initial probability vector; and the definitions of the entropies in the Statement.

[F1]

Declared example preparations and thermodynamic interpretation: product preparations are separately specified probability laws in consuming model statements, not consequences of flow and not assertions that a laboratory realizes them.

[F2]

Finite probability spaces, outcome weights, events, and event probabilities: a finite probability space is a finite set with nonnegative outcome weights summing to one.

[F3]

Real random variables on finite probability spaces and their finite distributions: a finite real random variable is a function from the finite outcome set to $R$ .

[F4]

Expectation of a real random variable on a finite probability space: expectation is the finite weighted sum.

[F5]

Characteristic function of a real random variable: $φ_{X} (k) = E [e^{i k X}]$ ; for a Borel law $μ$ , $φ_{μ} (k) = \int e^{i k x} μ (d x)$ .

[F6]

Standard normal and normal laws: $N (m, σ^{2})$ is the law of $m + σ Z$ for a standard normal $Z$ , including $N (m, 0) = δ_{m}$ .

[F7]

Characteristic function of a normal law: assuming AC, $N (m, σ^{2})$ has characteristic function $e^{i m k - σ^{2} k^{2} / 2}$ for every real $k$ , including $σ = 0$ .

[F8]

Characteristic function criterion for weak convergence: assuming AC, Borel probability laws on $R$ converge weakly to a specified Borel probability law if their characteristic functions converge pointwise at every real frequency.

[F9]

The Axiom of Choice: AC states that every family of nonempty sets has a choice function; the weak-convergence and normal-law suppliers explicitly require this full axiom.

[F10]

The derivatives of sine and cosine are cosine and minus sine: $\sin^{'} = \cos$ , $\cos^{'} = - \sin$ , $\sin 0 = 0$ , and $\cos 0 = 1$ .

[F11]

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

[F12]

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

[F13]

Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ : products and quotients of differentiable functions obey the usual derivative rules where the denominator is nonzero.

[F14]

The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ : derivatives of differentiable compositions are compositions of derivatives.

[F15]

Peano's form: the normalized Taylor remainder tends to zero: if $f$ is twice differentiable on an open neighbourhood of $0$ , then $f (u) = f (0) + f^{'} (0) u + \frac{1}{2} f^{''} (0) u^{2} + o (u^{2})$ .

[F16]

The one-dimensional torus and its normalized Haar integral: under countable choice, on $R / Z$ the probability measure is the pushforward of Lebesgue measure on $[0, 1)$ and is translation-invariant; rescaling the angle by $2 π$ gives normalized arc length on $R / (2 π Z)$ .

[F17]

Fine entropy under a specified incompressible periodic flow: its current same-run draft statement says that for a specified jointly $C^{2}$ diffeomorphic flow on a flat torus with identity initial map, $C^{1}$ divergence-free velocity, and positive bounded $C^{1}$ probability density, the dimensionless fine differential entropy of the transported density is finite and conserved; the item's proof is complete conditional on its stated mathematical suppliers, but independent review remains pending.

[F18]

Weak convergence of borel probability measures: weak convergence of Borel probability measures on a metric space means convergence of integrals against every bounded continuous real-valued function.

[F19]

Variance of a finite real random variable: on a finite probability space, $Var (Y) = E [(Y - E [Y])^{2}]$ .

[A1]

For each walk index $j$ , the coordinates on ${- 1, 1}^{n_{j}}$ are assigned the explicit uniform product weights $2^{- n_{j}}$ ; the different $j$ may use different finite spaces.

[A2]

The rotor law is the explicit flow in the Statement; no flow-existence theorem, ergodic hypothesis, or coupling to another system is assumed.

[A3]

The four-state probability evolves by the stated deterministic pushforward, with no stochastic reset.

Proof

Given: The definitions and model assumptions in the Statement and Facts & Assumptions; assume full AC for the weak-convergence and normal-law suppliers.

1.1givenA1F1F2F3F4F19algebra

Under the product-preparation convention [F1], for a fixed $j$ the uniform weight on ${- 1, 1}^{n_{j}}$ is a probability law, including the singleton case $n_{j} = 0$ . Each increment has mean zero and square one by [F2]–[F4]. Independence follows from the product weights, so the cross terms in $E (\sum_{i} Ξ_{i})^{2}$ vanish; hence $E S_{j} = 0$ and, by [F19], $Var (S_{j}) = n_{j} ℓ_{j}^{2}$ . Conditioning on the last increment gives $u_{n + 1} (r) = \frac{1}{2} u_{n} (r - 1) + \frac{1}{2} u_{n} (r + 1)$ , with the stated initial mass.

1.2givenA2F9F16algebra

For the rotor, write $R_{s} (q) = q + p_{0} s / I$ modulo $2 π$ . At $T = 2 π I / ∣ p_{0} ∣$ , the angle increment is $2 π$ when $p_{0} > 0$ and $- 2 π$ when $p_{0} < 0$ , so $R_{T}$ is the identity. The stated full AC includes the countable-choice assumption of [F16]. For every Borel $E \subseteq X$ , $ι_{p_{0}}^{- 1} (Φ_{s}^{- 1} E) = R_{s}^{- 1} (ι_{p_{0}}^{- 1} E)$ ; translation invariance in [F16] therefore gives $μ_{0} (Φ_{s}^{- 1} E) = μ_{0} (E)$ , and in particular $μ_{0}$ is invariant under the rotor flow.

1.3givenA3F11algebra

Under the four-state pushforward, the first three probability vectors are $π_{0} = (1 / 2, 1 / 2, 0, 0), π_{1} = (0, 1 / 2, 1 / 2, 0), π_{2} = (0, 0, 1 / 2, 1 / 2) .$ The map $T$ is a permutation, so it maps the uniform reference weights $1 / 4$ to themselves; it also leaves the multiset of the evolving fine probabilities unchanged, and each fine entropy is $- \frac{1}{2} \log \frac{1}{2} - \frac{1}{2} \log \frac{1}{2} = \log 2$ . The coarse cell probabilities for ${{0, 1}, {2, 3}}$ are $(1, 0)$ , $(1 / 2, 1 / 2)$ , and $(0, 1)$ , giving coarse entropies $0, \log 2, 0$ under the zero-term convention. This proves the stated decrease.

2.1givenA1F5F6F7F10F11F12F13F14F15step 1.1algebra

By independence and [F5], the characteristic function is $φ_{S_{j}} (k) = \prod_{i = 1}^{n_{j}} E (e^{i k ℓ_{j} Ξ_{i}}) = \cos (ℓ_{j} k)^{n_{j}} .$ Fix $k$ . Continuity of cosine and $\cos 0 = 1$ make $\cos u > 0$ for all sufficiently small $∣ u ∣$ . For such $u$ , put $f (u) = \log \cos u$ . By the chain rule, $f^{'} (u) = - \sin u / \cos u$ and $f^{'} (0) = 0$ . From [F10], $\sin u / u \to \sin^{'} (0) = 1$ and $\cos u \to 1$ , so the difference quotient $f^{'} (u) / u = - \sin u / (u \cos u)$ tends to $- 1$ ; hence $f^{''} (0) = - 1$ . The same derivative rules make $f$ twice differentiable on a neighbourhood of zero, so [F15] gives $\log \cos u = - u^{2} / 2 + o (u^{2})$ . For $t > 0$ , $n_{j} δ t_{j} \to t$ because $0 \leq t - n_{j} δ t_{j} < δ t_{j}$ , and therefore $n_{j} \log \cos (ℓ_{j} k) = - \frac{n_{j} ℓ_{j}^{2} k^{2}}{2} + n_{j} o (ℓ_{j}^{2}) ⟶ - D t k^{2} .$ Indeed $n_{j} ℓ_{j}^{2} = 2 D n_{j} δ t_{j} \to 2 D t$ , so $n_{j} ℓ_{j}^{2}$ is bounded, while $o (ℓ_{j}^{2}) / ℓ_{j}^{2} \to 0$ ; their product proves the remainder tends to zero. For $t = 0$ , $n_{j} = 0$ and every law is $δ_{0}$ . Thus in all cases $φ_{S_{j}} (k) \to e^{- D t k^{2}} = φ_{N (0, 2 D t)} (k)$ for every real $k$ , using [F6]–[F7] for the target law.

2.2givenF16F18step 1.2algebra

The measure definition gives $μ_{0} (A) = ν ([0, π)) = 1 / 2$ . At each $s = m T$ , $Φ_{s}$ is the identity on the support of $μ_{0}$ , so $μ_{0} (A \cap Φ_{s}^{- 1} A) = 1 / 2 \neq 1 / 4 = μ_{0} (A)^{2}$ , contradicting the mixing condition stated in the Statement. For $q_{0} = [θ_{0}]$ , the function $g ([θ], p) = \cos (θ - θ_{0})$ is well-defined and continuous on the quotient because cosine is continuous and $2 π$ -periodic, and it is bounded by $1$ . Its values are $1$ at times $m T$ and $- 1$ at times $m T + T / 2$ , so its expectation has no limit. By [F18], weak convergence would force convergence of this expectation, so the evolving point laws have no weak limit.

2.3givenF17step 1.3

The mathematical result [F17] assumes a smooth diffeomorphic flow on a flat torus, divergence-free velocity, and a positive bounded $C^{1}$ density. The atomic four-state law in step 1.3 is not such a density or flow; its coarse-entropy decrease neither invokes nor contradicts that fine differential-entropy result. It establishes no monotonicity law for other coarse-graining procedures.

3.1givenF7F8F9step 1.1step 2.1

The right-hand side of step 2.1 is the characteristic function of the specified Borel probability $N (0, 2 D t)$ . Apply the reverse implication of [F8] to the Borel laws of $S_{j}$ to obtain weak convergence. The only non-finite-choice strength used here is the full AC explicitly carried by [F7]–[F9]; the finite product construction is in step 1.1 and its pointwise characteristic calculation is in step 2.1.

4.1givenA1A2A3step 1.1step 2.1step 3.1step 1.2step 1.3∎

The walk limit in steps 1.1–2.1 concerns only fixed-time marginal laws under the stated independent-increment scaling; no path-space limit, deterministic collision limit, experiment, observed frequency, or empirical inference is part of the hypotheses or conclusion.

Remarks

The full Axiom of Choice includes the countable-choice assumption of the normalized torus-measure definition [F16] and is also required by the imported normal-law and weak-convergence interfaces [F7]–[F9]. The recurrence, finite permutation, and finite product calculations themselves are choice-free. A finite histogram is not an underlying probability law, and no histogram is supplied as one here.

5 · Examples, experiments and counterexamples

None yet.

Finite Bernoulli fluctuations and matching exponential rates

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

Variance of a finite real random variable

Definition

Finite Bernoulli fluctuations and matching exponential rates

Statement

Facts & Assumptions

Proof

Explicit nonadditive transition and ensemble inequivalence

Statement

Facts & Assumptions

Proof

Remarks

Diffusion-scale walk and recurrence/coarse-entropy counterexamples

Statement

Facts & Assumptions

Proof

Remarks

5 · Examples, experiments and counterexamples

Sources