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Working note in Phil's Lagrange Multipliers support files. It builds H(N,λ) = f(N) + λ1 a(N) + λ2 b(N) with constraints on total number M and total energy U. It computes the first derivatives f_i = ln(N_i/g_i), a_i = 1 and b_i = ε_i, then shows H_ii = -1/N_i < 0, so H curves downward in every direction and the solution is a maximum.
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H(N,λ) ≡ f(N) + λ1a(N) + λ2 b(N)
Hi(N,λ) = fi + λ1ai + λ2 bi = ln(Ni/gi) + λ1 + λ2εi i = 1,2...m
fi = ∂f/∂Ni = (1+lngi) - Ni (1/Ni) - lnNi = lngi - lnNi = ln(Ni/gi) (6.9)
ai = ∂a/∂Ni = ∂i [ ΣjNj - M] = 1
bi = ∂b/∂Ni = ∂i [ΣjεjNj-U] = εi (6.10)
Therefore
Hii = fii + λ1aii + λ2 bii = fii
But
fi = (lngi + 1) - Ni (1/Ni) - lnNi = lngi - lnNi
fii = - 1/Ni < 0
Since Hii = - 1/Ni < 0, for every (large) N the function H(N) "cups down" in all dimensions 1,2...m. In particular, this is true for the solution N, and thus that solution is a maximum.