Canonical ensemble <(Delta E)^3> expression

[GrandsonOfMagnusCarl]

Homework Statement

Given: <(Delta E)^2> = k_B T^2 C_V

Show: <(Delta E)^3> = k_B^2 [T^4 (d C_V / d T)_V + 2 T^3 C_V]

Relevant Equations

stat mech, thermo

I try involving differentiating ^2 but I get an expression of different proportionality.

 

[vanhees71]

I guess, that's for an ideal gas? If so then just get the grand-canonical partition function
$$Z(\beta,\alpha,V)=V \int_{\mathbb{R}^3} \exp(-\beta p^2/(2m)+\alpha)$$
and evaluate $U=\langle E \rangle=-\partial_{\beta} Z/Z$, $\langle \Delta E^2 \rangle=\partial_{\beta}^2 Z/Z- \langle E \rangle^2$,...

 

[GrandsonOfMagnusCarl]

I guess, that's for an ideal gas? If so then just get the grand-canonical partition function
$$Z(\beta,\alpha,V)=V \int_{\mathbb{R}^3} \exp(-\beta p^2/(2m)+\alpha)$$
and evaluate $U=\langle E \rangle=-\partial_{\beta} Z/Z$, $\langle \Delta E^2 \rangle=\partial_{\beta}^2 Z/Z- \langle E \rangle^2$,...

For any canonical ensemble.
Ah, yours would be an approach. I never thought of using an explicit example system to find that resulting expression.