37E (Statistical Physics)

a

The first law and Gibbs-Duhem relation give

d Γ = - S, d T + V, d p - N, d μ

. Extensivity gives Euler's relation

E = TS - p V + μ N

, hence

Γ = E - TS + p V - μ N = 0

g (ω) = A ω / (π c_{s}^{2})

Requiring

\int_{0}^{ω_{D}} g (ω) d ω = 2 N

gives

ω_{D} = 2 c_{s} π N / A

For

k_{B} T ≪ ℏ ω_{D}

, extend the integral to infinity:

lo g Z = - \int_{0}^{\infty} g (ω) lo g (1 - e^{- β ℏ ω}) d ω = \frac{A k _{B}^{2} T ^{2}}{π c _{s}^{2} ℏ ^{2}} \sum_{n \geq 1} \frac{1}{n ^{3}} .

Thus $X = ζ (3) / π$ .

Since

F = - k_{B} T lo g Z

S = - \partial F / \partial T = 3 ζ (3) A k_{B}^{3} T^{2} / (π c_{s}^{2} ℏ^{2})

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