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Controlled dilute Bose/Fermi occupation limit

Physical scope

Conditional mode-by-mode approximation for a trace-class ideal Bose or Fermi grand state with dimensionless fugacities 0<q_j≤r<1 and summable squared fugacities. The occupation and log-partition errors are bounded explicitly. This is a controlled dilute or low-fugacity limit, not a high-temperature criterion, universality statement, interacting-gas result, or empirical observation.

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), the preparation postulate Quantum statistics and model equilibrium preparation, and the diagonal grand-trace hypotheses of Explicit diagonal Fock domains and occupation trace. Let the mode set be finite or countable, with $0 < q_{j} \leq r < 1$ and $\sum_{j} q_{j}^{2} < \infty$ . Write ${\overset{ˉ}{n}}_{j}^{B} = q_{j} / (1 - q_{j})$ and ${\overset{ˉ}{n}}_{j}^{F} = q_{j} / (1 + q_{j})$ . Then $∣ {\overset{ˉ}{n}}_{j}^{B} - q_{j} ∣ \leq \frac{q_{j}^{2}}{1 - r}, ∣ {\overset{ˉ}{n}}_{j}^{F} - q_{j} ∣ \leq q_{j}^{2} .$ The total mean occupations satisfy $∣ \sum_{j} {\overset{ˉ}{n}}_{j}^{B} - \sum_{j} q_{j} ∣ \leq \frac{1}{1 - r} \sum_{j} q_{j}^{2}, ∣ \sum_{j} {\overset{ˉ}{n}}_{j}^{F} - \sum_{j} q_{j} ∣ \leq \sum_{j} q_{j}^{2} .$ The grand log partition factors, with $Ξ_{B}, Ξ_{F}$ as defined in Explicit diagonal Fock domains and occupation trace, obey $0 \leq - \log (1 - q_{j}) - q_{j} \leq \frac{q_{j}^{2}}{2 (1 - r)}, 0 \leq q_{j} - \log (1 + q_{j}) \leq \frac{q_{j}^{2}}{2} .$ Consequently, $∣ \log Ξ_{B} - \sum_{j} q_{j} ∣ \leq \frac{1}{2 (1 - r)} \sum_{j} q_{j}^{2}, ∣ \log Ξ_{F} - \sum_{j} q_{j} ∣ \leq \frac{1}{2} \sum_{j} q_{j}^{2} .$ Thus the displayed linear fugacity terms approximate both total mean occupation and log partition with a controlled aggregate error under the stated squared-summability hypothesis.

Facts & Assumptions

Given: ${A C}_{ω}$ ; adopted preparation and trace-normalized diagonal occupation law; dimensionless $q_{j} \in (0, r]$ .

[A1]

Quantum statistics and model equilibrium preparation: the species sector and normalized grand preparation are adopted assumptions, not mathematical consequences.

[F1]

Explicit diagonal Fock domains and occupation trace, Statement: under its stated summability conditions, bosonic and fermionic marginals have means $q / (1 - q)$ and $q / (1 + q)$ , and $\log Ξ$ is the sum of $- \log (1 - q)$ or $\log (1 + q)$ .

Proof

1.1givenF1algebra

For bosons, direct subtraction gives ${\overset{ˉ}{n}}^{B} - q = q^{2} / (1 - q)$ ; because $q \leq r$ , this is nonnegative and at most $q^{2} / (1 - r)$ . For fermions, $q - {\overset{ˉ}{n}}^{F} = q^{2} / (1 + q)$ , which lies between zero and $q^{2}$ .

1.2givenalgebra

The identity $- \log (1 - q) - q = \int_{0}^{q} x / (1 - x) d x$ follows by integrating the derivative of $- \log (1 - x) - x$ from zero. Since $1 / (1 - x) \leq 1 / (1 - r)$ on $[0, q]$ , the integral is between zero and $q^{2} / [2 (1 - r)]$ . Also $q - \log (1 + q) = \int_{0}^{q} x / (1 + x) d x$ , whose integrand lies between zero and $x$ , giving the Fermi bound $q^{2} / 2$ .

2.1step 1.1A1algebra

Summing the nonnegative mode errors over any finite subset preserves these bounds. Taking the increasing limit over finite subsets gives the same inequalities for the total error because $\sum_{j} q_{j}^{2} < \infty$ .

3.1step 1.2F1algebra∎

By [F1], the log partition function is the sum of the corresponding mode log factors. Sum the nonnegative factor remainders from step 1.2 over finite subsets and pass to the countable limit; squared summability makes each aggregate error finite and proves the claimed controlled first-order approximation.

Controlled dilute Bose/Fermi occupation limit

Statement

Facts & Assumptions

Proof

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