Statistical Mechanics · Problem 3.27

The Grand Canonical Ensemble — R. K. Pathria

Physics — Statistical Mechanics · quantum-oscillator-entropy-density-of-states

Problem

Obtain an asymptotic expression for the quantity

\ln g(E)

for a system of

N

quantum-mechanical harmonic oscillators by using the inversion formula (3.4.7) and the partition function (3.8.15). Hence show that

\frac{S}{Nk} = \left(\frac{E}{Nh\omega} + \frac{1}{2}\right)\ln\left(\frac{E}{Nh\omega} + \frac{1}{2}\right) - \left(\frac{E}{Nh\omega} - \frac{1}{2}\right)\ln\left(\frac{E}{Nh\omega} - \frac{1}{2}\right).

[Hint. Employ the Darwin--Fowler method.]