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Finite Bernoulli fluctuations and matching exponential rates — Examples

1 · Prerequisites

2 · Summary

These worked calculations specialize the finite Bernoulli bounds, compare the two preparations for a four-member three-sector model, and evaluate one diffusive step scale, one rotor period, and the first three coarse and fine entropy values of the exact four-state permutation. They are theoretical model calculations, not sample data or reported observations.

In the diagram, arrows show the permutation $T$ , and the two node colors mark the coarse cells ${0, 1}$ and ${2, 3}$ . The initial distribution assigns probability $1 / 2$ to states $0$ and $1$ , as in A diffusive time step and a four-state entropy reversal. The drawing is schematic; the item gives the exact map, probability vectors, and entropy calculation, and the proof does not rely on the diagram.

\begin{tikzpicture}[>=stealth, every node/.style={circle,draw,minimum size=8mm}]
  \node[fill=blue!18] (s0) at (0,0) {$0$};
  \node[fill=blue!18] (s1) at (2,0) {$1$};
  \node[fill=orange!25] (s2) at (2,-2) {$2$};
  \node[fill=orange!25] (s3) at (0,-2) {$3$};
  \draw[->] (s0) -- (s1);
  \draw[->] (s1) -- (s2);
  \draw[->] (s2) -- (s3);
  \draw[->] (s3) -- (s0);
\end{tikzpicture}

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

A finite Bernoulli model at n=100 and p=1/4

Physical scope

Exact specialization of the finite IID Bernoulli product model at n=100 and p=1/4; it is not a measured data set.

Example

In the explicitly stipulated product model of Finite Bernoulli fluctuations and matching exponential rates, take $n = 100$ and $p = 1 / 4$ . We calculate model probabilities and bounds without supplying any observed trial results.

Facts & Assumptions

Given: The finite IID Bernoulli model with $n = 100$ , $p = 1 / 4$ , and $K$ the number of successes.

[F1]

Finite Bernoulli fluctuations and matching exponential rates: under the explicit product law, $E K = n p$ , $Var (K) = n p (1 - p)$ , the two tail inequalities hold, and the point-mass bound has exponent $D (a ∥ p)$ .

Verification

technique · direct

Given: The model and hypotheses in the Example and [F1].

1.1givenF1algebra

Substitution gives $E K = 25$ and $Var (K) = 100 (1 / 4) (3 / 4) = 75 / 4$ . For $ε = 1 / 10$ , the Chebyshev bound is $(3 / 16) / (100 / 100) = 3 / 16$ , while the exponential bound is $2 e^{- 2}$ . These are upper bounds on a specified model probability, not results from 100 recorded trials.

2.1givenF1algebra∎

For $k = 35$ , $a = k / n = 7 / 20$ and $D (7 / 20 ∥ 1 / 4) = \frac{7}{20} \log \frac{7}{5} + \frac{13}{20} \log \frac{13}{15} .$ The point-mass estimate in [F1] therefore reads $\frac{1}{101} e^{- 100 D (7 / 20 ∥ 1 / 4)} \leq P_{1 / 4} (K = 35) \leq e^{- 100 D (7 / 20 ∥ 1 / 4)} .$ This finite example displays the polynomial prefactor and common exponential rate without numerical fitting or empirical inference.

ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

The middle sector at N=4 and the transition point

Physical scope

Exact N=4 specialization of the stated global three-sector finite-state model; no short-range material system is asserted.

Example

Specialize the finite-sector model of Explicit nonadditive transition and ensemble inequivalence to $N = 4$ at its positive transition parameter.

Facts & Assumptions

Given: The three sectors, energies, and finite exponential-tilt law from Explicit nonadditive transition and ensemble inequivalence.

[F1]

Explicit nonadditive transition and ensemble inequivalence: at $β_{c} = (\log 2) / ϵ$ , the sector weights are $1$ , $2^{- N}$ , and $1$ ; every shell half-width $0 \leq Δ < N ϵ$ isolates $B_{N}$ , and its uniform counting probability is $Q_{N, Δ} (ω) = ∣ S_{N, Δ} ∣^{- 1}$ for $ω \in S_{N, Δ}$ , so the shell probability of $B_{N}$ is $1$ .

Verification

technique · direct

Given: $N = 4$ , $ϵ > 0$ , $β = β_{c}$ , and $0 \leq Δ < 4 ϵ$ .

1.1givenF1algebra

Since $β_{c} N ϵ = 4 \log 2$ , the three aggregate canonical sector weights are $1$ , $e^{- 4 \log 2} = 1 / 16$ , and $4^{4} e^{- 8 \log 2} = 1$ . Thus $Z_{4} = 33 / 16$ and $P_{β_{c}, 4} (B_{4}) = (1 / 16) / (33 / 16) = 1 / 33$ .

2.1givenF1algebra∎

The shell centered at $4 ϵ$ with half-width $Δ < 4 ϵ$ excludes the neighboring energy levels $0$ and $8 ϵ$ , so it contains exactly the single state in $B_{4}$ . Its uniform probability is therefore one on $B_{4}$ , compared with the canonical value $1 / 33$ .

ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

A diffusive time step and a four-state entropy reversal

Physical scope

Exact finite specializations of the stated walk, rotor, and permutation models; no observations or collision-derived diffusion law are asserted.

Example

This example evaluates one finite walk step size, one rotor period, and the first three states of the exact finite permutation model in Diffusion-scale walk and recurrence/coarse-entropy counterexamples.

Facts & Assumptions

Given: The three explicitly defined models and their units and conventions from Diffusion-scale walk and recurrence/coarse-entropy counterexamples.

[F1]

Diffusion-scale walk and recurrence/coarse-entropy counterexamples: under $ℓ^{2} = 2 D δ t$ , the walk has variance $n ℓ^{2}$ ; the rotor period is $2 π I / ∣ p_{0} ∣$ ; the finite permutation has coarse entropies $0, \log 2, 0$ and fine entropy $\log 2$ .

Verification

technique · direct

Given: $t = 1$ s, $D = 1$ m $^{2}$ s $^{- 1}$ , $δ t = 1 / 8$ s, $I = 1$ kg m $^{2}$ , and $p_{0} = 1$ kg m $^{2}$ s $^{- 1}$ .

1.1givenF1algebra

The scaling gives $ℓ = \sqrt{2 D δ t} = 1 / 2$ m and $n = ⌊ t / δ t ⌋ = 8$ , so $Var (S_{n}) = 8 (1 / 2 m)^{2} = 2$ m $^{2}$ and $φ_{S_{n}} (k) = \cos ((1 / 2 m) k)^{8}$ . This is a finite model marginal; its Gaussian limit is the separate $δ t ↓ 0$ conclusion of [F1].

2.1givenF1algebra∎

The rotor return period is $2 π I / ∣ p_{0} ∣ = 2 π$ s. The finite permutation calculation has coarse entropy $0$ at times zero and two and $\log 2$ at time one, while the fine entropy remains $\log 2$ at all three times.

Finite Bernoulli fluctuations and matching exponential rates — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

5 · Examples, experiments and counterexamples

A finite Bernoulli model at n=100 and p=1/4

Example

Facts & Assumptions

Verification

The middle sector at N=4 and the transition point

Example

Facts & Assumptions

Verification

A diffusive time step and a four-state entropy reversal

Example

Facts & Assumptions

Verification

Sources