Manuscript En02-E@2 · En02-E-v2.tex · corollary · c2769986

Corollary: Statistics and generating-function domain

The complete finite packet state · lines 448–479 · proof included or linked

En02-E@2#cor:stats, En02-E@2#eq:pgf

The vacuum probability, total-number moments, and entropy are

p_{0} = det (I + K)^{- 1}, E N_{tot} = μ, Var N_{tot} = μ + tr K^{2}, S (ρ_{K}) = a \sum g (λ_{a}),

where $g (x) = (x + 1) lo g_{2} (x + 1) - x lo g_{2} x$ , $g (0) = 0$ .

If $λ_{m a x} > 0$ , then the absolutely convergent probability-generating function is

G_{K} (s) = E s^{N_{tot}} = a \prod [1 + λ_{a} (1 - s)]^{- 1}, ∣ s ∣ < 1 + λ_{m a x}^{- 1} .

Here $s$ may be complex and $0^{0} = 1$ . If $K = 0$ , $G_{K} (s) = 1$ for every $s$ . Positivity of rational denominators alone is not a convergence criterion for negative $s$ .

Proof. For each nonvacuum normal mode the geometric series converges absolutely precisely when $∣ s z_{a} ∣ < 1$ . Multiplying finitely many such series gives (eq:pgf). Their elementary moments and entropy give the remaining formulas.

Displayed formulas in this passage

En02-E@2#eq:pgf · line 462
$G_{K} (s) = E s^{N_{tot}} = a \prod [1 + λ_{a} (1 - s)]^{- 1}, ∣ s ∣ < 1 + λ_{m a x}^{- 1} .$
unlabeled · line 451
$p_{0} = det (I + K)^{- 1}, E N_{tot} = μ, Var N_{tot} = μ + tr K^{2}, S (ρ_{K}) = a \sum g (λ_{a}),$

Corollary: Statistics and generating-function domain

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