Blundell Thermal Physics · Problem 25.2

Relativistic gases

Physics — Statistical Mechanics · density-of-states-d-dimensions

Problem

D

dimensions, show that the density of states of particles with spin degeneracy

g

in a volume

V

g(k)\,\mathrm{d}k = \frac{gV D\pi^{D/2} k^{D-1}\,\mathrm{d}k}{\Gamma\!\left(\frac{D}{2}+1\right)(2\pi)^D}.\tag{25.31}

You may need to use the fact that the volume of a sphere of radius

r

D

dimensions is (see Appendix C.8)

\frac{\pi^{D/2}r^D}{\Gamma\!\left(\frac{D}{2}+1\right)}.\tag{25.32}