Ultra-relativistic gas average energy as E(T)

[prehisto]

Homework Statement

Ultra-relativistic particle with energy epsilon and momentum p is described by epsilon=cp. Find average energy E(T) of gas which consists of particles described above.

Homework Equations

So i think i should use this kind of equations(please use the link below)
http://postimg.org/image/wwhct3lev/full/
So i know the general form of Partition integral and i think i could calculate the average energy using Helmholtz formula.
The problem now is how to represent the Partition integral according to my problem.

could someone please help me?
P.s. I apologize for using image instead of Latex( which does not work for me)

 

[_coyote_]

Any help will be greatly appreciated. Thank you very much!A:The partition function is the integral over all microstates of the Boltzmann factor. In this case, the Boltzmann factor is just 1 since the particles are ultra-relativistic. Thus, the partition function is just the number of microstates. The single particle energy is $\epsilon = cp$, so each particle has $c$ energy levels. The total number of microstates is $N=c^N$.The average energy is then$$E = \frac{\partial \ln Z}{\partial (-1/T)} = \frac{N}{Z}\frac{\partial Z}{\partial (-1/T)} = \frac{N k_B T}{Z} \frac{\partial Z}{\partial T} = \frac{N c k_B T}{Z}$$where we have used a substitution of $-1/T$ for $\beta$ in the first equality and the fact that $\partial Z/\partial T = -\partial Z/\partial \beta$ in the third equality.Plugging in our expression for $Z$ and rearranging gives$$E = c N k_B T$$