Callen H.B - Thermodynamics And An Introduction To Thermostatistics · Problem 18.4-2

FLUCTUATIONS

Physics — Statistical Mechanics · sommerfeld-expansion-fermi-integrals

Problem

Derive equation 18.35 by the following sequence of operations:

a

) Denoting the integral in equation 18.35 by

I

, first integrate by parts and let

\Phi \equiv \int_0^\varepsilon \phi(\varepsilon')\,d\varepsilon'

. Then expanding

\Phi(\varepsilon)

in a power series in

(\varepsilon - \mu)

to third order, show that

I = -\sum_{m=0}^{\infty} \frac{1}{m!}\frac{d^m \Phi(\mu)}{d\mu^m} I_m

with

I_m = \int_0^{\infty}(\varepsilon - \mu)^m \frac{df}{d\varepsilon}\,d\varepsilon = -\beta^{-m}\int_0^{\infty}\frac{e^x}{\beta\mu\left(e^x+1\right)^2}x^m\,dx

b

) Show that only an exponentially small error is made by taking the lower limit of integration as

-\infty

, and that then all terms with

m

odd vanish.

c

) Evaluate the first two nonvanishing terms and show that these agree with equation 18.35.