Probability of a state given the partition function

[Sat D]

Homework Statement

Given the partition function of an ideal monoatomic gas, what is the probability that its energy is some $cU$, where $U$ is the average energy of the ensemble

Relevant Equations

$Z = \frac{1}{N!} \left( \frac{2\pi m k_B T}{h^2} \right)^{3N/2} V^N$

If my partition function is for a continuous distribution of energy, can I simply say that the probability of my ensemble being in a state with energy $cU$ is $e^{-\beta cU} /Z$? I believe that isn't right as my energy distribution is continuous, and I need to be integrating over small intervals.

 

[vanhees71]

First think about the phase-space distribution function in terms of energy rather than momentum!

 

[Sat D]

So,
$$p(E) = \int \delta(E-cU) \frac{e^{-\beta E(\Gamma)}}{Z} d\Gamma$$
where $\Gamma$ is the phase-space.
However, how do I simplify something like this?

 

[vanhees71]

Well, just get $E(\Gamma)$.

 

[Sat D]

How do you mean?

 

[vanhees71]

Well, you should know, how the phase-space distribution function for an ideal gas looks. Then you can get the distribution function of energy by evaluating the integral you've written down.