Blundell Thermal Physics · Problem 29.3

Bose–Einstein and Fermi–Dirac distributions

Physics — Statistical Mechanics · fermi-dirac-distribution

Problem

By rewriting the Fermi--Dirac function

f(E) = \frac{1}{\mathrm{e}^{\beta(E-\mu)} + 1} \tag{29.37}

f(E) = \frac{1}{2}\left(1 - \tanh\frac{1}{2}\beta(E - \mu)\right), \tag{29.38}

show that

f(E)

is symmetric about

E = \mu

and sketch it. Find simplified expressions for

f(E)

when (i)

E \ll \mu

, (ii)

E \gg \mu

, and (iii)

E

is very close to

\mu