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Quantum occupation fluctuations and the classical approximation boundary

Physical scope

A one-mode exact comparison of Bose, Fermi, and first-order classical occupations at q=1/2, with no empirical interpretation.

Example

Take one dimensionless mode with fugacity $q = 1 / 2$ . The first-order dilute term for the mean occupation is $q = 1 / 2$ . As an explicit comparison, define the Poisson law by $P_{q} (n) = e^{- q} q^{n} / n!$ for $n \in N_{0}$ ; its mean and variance are both $q$ . The exact Bose law is geometric and the exact Fermi law is Bernoulli. This declared calculation illustrates the distinct occupation statistics and why a small-error conclusion needs a small-fugacity condition; the Poisson comparison is not asserted to follow from the linked theorem's mean and log-partition bounds.

Verification

Given: ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), $q = 1 / 2$ , and the exact occupation and variance formulas of Controlled dilute Bose/Fermi occupation limit and Explicit diagonal Fock domains and occupation trace.

[F1] Controlled dilute Bose/Fermi occupation limit, Statement: Bose and Fermi mode means are $q / (1 - q)$ and $q / (1 + q)$ , with explicit errors relative to $q$ .

[F2] Explicit diagonal Fock domains and occupation trace, Statement: the one-mode Bose and Fermi variances are $q / (1 - q)^{2}$ and $q / (1 + q)^{2}$ .

1.1givenF1algebra

Substitution of $q = 1 / 2$ gives ${\overset{ˉ}{n}}_{B} = 1$ , ${\overset{ˉ}{n}}_{F} = 1 / 3$ , while the first-order value is $1 / 2$ . Thus the Bose error is $1 / 2$ and the Fermi error magnitude is $1 / 6$ .

2.1givenF2algebra∎

For the comparison law, $E N = e^{- q} \sum_{n \geq 1} n q^{n} / n! = q$ and $E [N (N - 1)] = e^{- q} \sum_{n \geq 2} n (n - 1) q^{n} / n! = q^{2}$ , so $Var (N) = q^{2} + q - q^{2} = q = 1 / 2$ . The exact variances from [F2] are $q / (1 - q)^{2} = 2$ for bosons and $q / (1 + q)^{2} = 2 / 9$ for fermions. These are finite exact model values; since $q$ is not near zero, no small-error claim follows from this example.

Depends on

Dependency tree · 2 levels

Sources

David Tong, Statistical Physics, §§3.5–3.6, Bose and Fermi occupations