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I need to calculate the partition function (of the canonical ensemble) given the hamiltonian: $$ H=\sum_{i=0}^N A\vert P_i \vert^s $$ where $s>0$. and $p_i$ are independent of one another. The partition function is given by: $$ z=\frac{1}{N!} \idotsint(\prod_i \frac{d^3 q_i d^3 p_i}{(2\pi\hbar)^3})exp(-\beta\sum_{i=0}^N A\vert P_i \vert^s) $$

from here I managed to get:

$$ z=\frac{V}{(2\pi\hbar)^3}\idotsint d^3p_i exp(-\beta A\vert P_i \vert^s) $$ but from here I'm stumped. Pretty sure the solution should involve a gamma function, but that's about it. I'd appreciate any help.

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1 Answer 1

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You can separate the integrals in $p$ as follows:

$$Z\sim\prod_i\left(\int d^3p_i\exp(-\beta A|p_i|^s)\right)$$ You can find the integral as follows

$$\int d^3p\exp(-\beta A|p|^s)=\int 4\pi p^2dp\exp(-\beta A|p|^s)=\int_0^\infty 8\pi p^2dp\exp(-\beta Ap^s)$$ Let $p^s=x\Rightarrow sp^{s-1}dp=dx$ $$\int8\pi p^2\frac{dx}{sp^{s-1}} \exp(-\beta Ax)\Rightarrow \int 8\pi x^{3/s-1}dx\exp(-\beta Ax)=8\pi\frac{\Gamma[3/s]}{(\beta A)^{3/s}}$$ $$Z\sim \left(8\pi\frac{\Gamma[3/s]}{(\beta A)^{3/s}}\right)^N$$

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