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This is problem 3.18, from the book Statistical Mechanics by Pathria. Show that for a system in the canonical ensemble $$\left\langle \left(\Delta E\right)^{3}\right\rangle =k^{2}\left\{ T^{4}\left(\frac{\partial C_{V}}{\partial T}\right)_{V}+2T^{3}C_{V}\right\} $$ Verify that for an ideal gas $$\left\langle \left(\frac{\Delta E}{U}\right)^{2}\right\rangle =\frac{2}{3N},\qquad\left\langle \left(\frac{\Delta E}{U}\right)^{3}\right\rangle =\frac{8}{9N^{2}}$$ I have no Idea how to solve it. I found that $$\left\langle \left(\Delta E\right)^{3}\right\rangle \equiv\left\langle \left(E-\left\langle E\right\rangle \right)^{3}\right\rangle =\left\langle E^{3}\right\rangle -3\left\langle E\right\rangle \left\langle (\Delta E)^{2}\right\rangle -\left\langle E\right\rangle ^{3}$$ And using the fact that $$\int\left[U-H(q,p)\right]e^{-\beta H(q,p)}d\omega=0$$ I obtain $$\frac{\partial^{2}U}{\partial\beta^{2}}=\left\langle \left(\Delta E\right)^{3}\right\rangle $$ But, now I'm stuck. How can I link that who results to show the first statement?

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up vote 3 down vote accepted

First, using the partition function $Z=\sum_n e^{-\beta E_n}$, you have $$ \langle E\rangle=\frac{1}{Z}\partial_{-\beta}Z, $$$$ \langle E^2\rangle=\frac{1}{Z}\partial_{-\beta}^2Z, $$$$ \langle E^3\rangle=\frac{1}{Z}\partial_{-\beta}^3Z. $$ Second, you can show $$ \langle E\rangle=\partial_{-\beta} \ln Z, $$$$ \langle (\Delta E)^2\rangle=\langle (E-\langle E\rangle)^2\rangle=\partial_{-\beta}^2 \ln Z, $$$$ \langle (\Delta E)^3\rangle=\langle (E-\langle E\rangle)^3\rangle=\partial_{-\beta}^3 \ln Z. $$ Well, it just indicates that $\langle E^n\rangle$'s are disconnected correlation functions, while $\langle (\Delta E)^n\rangle$'s connected. And at last: $$ \langle (\Delta E)^3\rangle=\partial_{-\beta}^2 U=(k_B T^2\partial_T)^2 E=k_B^2 T^2\partial_T(T^2 C_V)=k_B^2(T^4\partial_T C_V+2T^3C_V). $$ Aha, trumpets!

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