(a) Show that the momentum distribution of particles in a relativistic Boltzmannian gas, with \(\varepsilon=c\left(p^{2}+m_{0}^{2} c^{2}ight)^{1 / 2}\), is given by

\[
f(\boldsymbol{p}) d \boldsymbol{p}=C e^{-\beta c\left(p^{2}+m_{0}^{2} c^{2}ight)^{1 / 2}} p^{2} d p,
\]

with the normalization constant

\[
C=\frac{\beta}{m_{0}^{2} c K_{2}\left(\beta m_{0} c^{2}ight)},
\]

\(K_{v}(z)\) being a modified Bessel function.

(b) Check that in the nonrelativistic limit \(\left(k T \ll m_{0} c^{2}ight.\) ) we recover the Maxwellian distribution,

\[
f(\boldsymbol{p}) d \boldsymbol{p}=\left(\frac{\beta}{2 \pi m_{0}}ight)^{3 / 2} e^{-\beta p^{2} / 2 m_{0}}\left(4 \pi p^{2} d pight)
\]

while in the extreme relativistic limit \(\left(k T \gg m_{0} c^{2}ight)\) we obtain

\[
f(\boldsymbol{p}) d \boldsymbol{p}=\frac{(\beta c)^{3}}{8 \pi} e^{-\beta p c}\left(4 \pi p^{2} d pight)
\]

(c) Verify that, quite generally,

\[
\langle p uangle=3 k T
\]