Show that in the relativistic case the equipartition theorem takes the form [ leftlangle m_{0} u^{2}left(1-u^{2} /

Question:

Show that in the relativistic case the equipartition theorem takes the form

\[
\left\langle m_{0} u^{2}\left(1-u^{2} / c^{2}ight)^{-1 / 2}ightangle=3 k T \text {, }
\]

where $m_{0}$ is the rest mass of the particle and $u$ its speed. Check that in the extreme relativistic case the mean thermal energy per particle is twice its value in the nonrelativistic case.

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