# Equipartition theorem for the (relativistic) case E = c|q|

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Consider a classical "degree of freedom" that is linear rather than quadratic: E = c|q| for some constant c. (An example would be the kinetic energy of a highly relativistic particle in one dimension, written in terms of its momentum.) Repeat the derivation of the equipartition theorem for this system, and show that the average energy is kT.

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#### Solution Preview

The partition function is:

Z = Sum over all states r of exp[-beta E_{r}]

In the classical limit you can replace the summation by an integration. To do that you need to know how many states there are inside a volume element in a region of momentum and ordinary space. The number of states for a particle inside an n-dimensional ...

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