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A hard core remains at zero inverse temperature

Example

In one spatial dimension, let $ϕ (x) = + \infty$ for $∣ x ∣ < a$ and $0$ otherwise, with $a > 0$ . On an interval $[0, L]$ , define the beta-zero weight by retaining the forbidden-pair exclusion as in Zero-inverse-temperature-parameter limit with forbidden pairs retained. If $L > a$ , the two-particle canonical partition is $Z (L, 2; 0) = \frac{(L - a)^{2}}{2},$ while the unrestricted ideal-gas value is $L^{2} / 2$ . Thus the retained hard-core beta-zero ensemble is not the Poisson ideal gas.

Facts & Assumptions

Given: $a > 0$ , $L > a$ , and the hard-admissibility convention.

[F1]

Superstable regular pair interactions and configurational ensemble notation: the canonical positional integral has normalization $1 / N!$ .

[F2]

Zero-inverse-temperature-parameter limit with forbidden pairs retained: the grand beta-zero weight is $z^{N} 1_{a d m i s s i b l e}$ . On the fixed- $N$ sector the canonical coefficient is the weight divided by $z^{N}$ (equivalently, set $z = 1$ ), namely $1_{a d m i s s i b l e}$ ; hence it contains no activity factor. For the free interval here, admissibility means $∣ x_{1} - x_{2} ∣ \geq a$ .

[F3]

The product measure of two sigma-finite measure spaces, 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, For sigma-finite factors, the product measure exists, has the rectangle formula, is sigma-finite, and is unique, and Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: the labeled coordinate measure is product Lebesgue measure, with rectangular volume and iterated integration.

[F4]

The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$ : the canonical normalization uses $2! = 2$ .

Verification

Given: The one-dimensional hard-rod potential and the stated interval.

1.1F1F2F3F4algebra

On the ordered region $x_{1} < x_{2}$ , allowed pairs satisfy $x_{2} \geq x_{1} + a$ , so their volume is $\int_{0}^{L - a} (L - x_{1} - a) d x_{1} = (L - a)^{2} / 2$ . The coordinate-swap map preserves product Lebesgue measure, so the region $x_{2} < x_{1}$ has the same volume; the diagonal has zero product measure. Thus the full labeled allowed volume is $(L - a)^{2}$ , and division by $2!$ gives $Z (L, 2; 0) = (L - a)^{2} / 2$ .

2.1F1F2step 1.1algebra∎

Since $0 < a < L$ , $(L - a)^{2} / 2 < L^{2} / 2$ , the unrestricted ideal-gas two-particle canonical term. The strict deficit is caused by the forbidden separation band and remains at beta zero by [F2].

Depends on

Dependency tree · 2 levels

Sources

H.-O. Georgii, Large deviations and the equivalence of ensembles for Gibbsian particle systems with superstable interaction