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I would like to set up the following problem. Assume I have a box of volume $V$ with $N$ noninteracting particles in it. The energy of each particle can be $\mathcal{E}_i$ such that $\sum_i \mathcal{E}_i = E$. I want to calculate how many microstates correspond to this macrostate of $(E, V, N)$.

Here is my approach so far. I will use the trick where we divide the phase space into boxes of size $h^3$. All the particles can be interchanged and we they are distinguishable so we have

$\Omega(E,V,N) = \frac{1}{N! h^3}\int_V d^N\mathbf{q} \int_{\sum_{i=1}^N p^2_i = 2mE} d^N\mathbf{p}$.

The second term is where I'm not sure how to proceed. How do I express the momentum integral while respecting the constraint that the total energy is exactly $E$?

The larger context: I hope to eventually arrive at some expression for $\Omega$ that let's me calculate $S = k\ln\Omega (E,V,N)$. My goal is to show that $\frac{\partial S}{\partial E} = \frac{1}{k_B T}$ is satisfied and connect to the statistical mechanical definition of temperature.

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marked as duplicate by Community Jan 13 at 0:42

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