ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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Two indistinguishable hard-core particles in an interval

Physical scope

A finite one-dimensional positional toy model with two classical identical particles and a hard-core exclusion; it is a model calculation, not a reported measurement.

Example

Take a dimensionless interval $Λ = [0, L]$ with $0 < a < L$ , two identical particles, and the hard-core pair potential $φ (r) = + \infty$ for $∣ r ∣ < a$ and $0$ otherwise. At any positive temperature the configurational canonical partition function is independent of $T$ and equals $Z_{Λ, 2} = \frac{1}{2} (L - a)^{2} .$ The factor $1 / 2$ is the adopted indistinguishable positional measure. This finite-volume calculation has no momentum variables or experimental data.

Facts & Assumptions

[F1]

Stable regularly tempered classical positional preparation: for two positions the hard-core pair energy is infinite precisely on forbidden separations $∣ x_{1} - x_{2} ∣ < a$ .

[A1]

Indistinguishable-particle positional measure: integrate symmetric positional weights with $d μ_{Λ, 2} = \frac{1}{2} d x_{1} d x_{2}$ .

[A2]

Finite-volume equilibrium ensemble sampling: at exact $N = 2$ , finite-energy pairs carry weight $e^{- β H_{2}}$ and forbidden infinite-energy pairs carry weight zero, integrated against $μ_{Λ, 2}$ .

Verification

Given: The interval, hard-core radius, exact particle number, and the adopted positional measure.

1.1givenF1algebra

The allowed labelled pairs form two disjoint triangles, $x_{2} \geq x_{1} + a$ and $x_{1} \geq x_{2} + a$ . The first has area $\int_{0}^{L - a} (L - a - x_{1}) d x_{1} = (L - a)^{2} / 2$ , and the second has the same area by exchanging coordinates.

2.1givenA1A2step 1.1∎

Their total allowed area is $(L - a)^{2}$ . On it $H_{2} = 0$ and the weight is $1$ ; on the excluded band the adopted finite-energy convention assigns weight zero, without evaluating an exponential at $+ \infty$ . Multiplying the allowed area by the measure factor $1 / 2$ from [A1] gives $Z_{Λ, 2} = (L - a)^{2} / 2$ , as asserted.

Two indistinguishable hard-core particles in an interval

Example

Facts & Assumptions

Verification

Depends on

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