Blundell Thermal Physics · Problem 25.3

Relativistic gases

Physics — Statistical Mechanics · general-dispersion-relation-partition-function

Problem

Consider a general dispersion relation of the form

E = \alpha p^s,\tag{25.33}

where

p

is the momentum and

\alpha

and

s

are constants. Using the result of the previous question, show that the density of states as a function of energy is

g(E)\,\mathrm{d}E = \frac{gVD\pi^{D/2}}{h^D \alpha^{D/s} s\,\Gamma\!\left(\frac{D}{2}+1\right)} E^{\frac{D}{s}-1}\,\mathrm{d}E.\tag{25.34}

Hence show that the single-particle partition function takes the form

Z_1 = \frac{V}{\lambda^D},\tag{25.35}

where

\lambda

is given by

\lambda = \frac{h}{\pi^{1/2}}\left(\frac{\alpha}{k_\mathrm{B}T}\right)^{1/s}\left[\frac{\Gamma\!\left(\frac{D}{2}+1\right)}{\Gamma\!\left(\frac{D}{s}+1\right)}\right]^{1/D}.\tag{25.36}

Show that this result for three dimensions (

D=3

) agrees with (i) the non-relativistic case when

s=2

and (ii) the ultrarelativistic case when

s=1