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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.

Explicit nonadditive transition and ensemble inequivalence

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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