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I was told that in the grand canonical ensemble we can calculate the heat capacity at a fixed volume using

$$C_V = -\beta^2 \Big( \frac{\partial E}{\partial \beta}\Big|_{\lambda, V}- \frac{\partial E}{\partial \lambda}\Big|_{\beta, V}\frac{\partial N}{\partial \beta}\Big|_{\lambda, V}\frac{\partial \lambda}{\partial N}\Big|_{\beta, V}\Big)$$

where $\lambda = e^{\beta \mu}$ is the fugacity and $\beta = \frac{1}{Tk_B}$, where $k_B$ is Boltzmanns constant. I assume that $E$ and $N$ are being taken to be $\overline{E}$ and $\overline{N}$ here.

I'm having a difficult time showing this. I know that in the canonical ensemble $$C_V = \Big(\frac{\partial \overline{E}}{\partial T}\Big)_V = \frac{\partial \beta}{\partial T}\frac{\partial \overline{E}}{\partial \beta}\Big|_V = -\beta^2 \frac{\partial \overline{E}}{\partial \beta}\Big|_V $$ if we set $k_B = 1$. I imagine that something similar is going on here, but I can't get it to work out. This answer is probably helpful?

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To solve this problem you need to do a variable substitution. To find $C_V$, we need to know the function $E(\beta, V, N)$, but in the grand canonical ensemble we find the functions $E(\beta, V, \lambda)$ and $N(\beta, V, \lambda)$. From the latter, $\lambda$ can in principle be expressed as $\lambda(\beta, V, N)$. And then we get $$ E(\beta, V, N) = E(\beta, V, \lambda(\beta, V, N)) $$ From this equality, according to calculus, we obtain $$ C_V = -\beta^2\frac{\partial E}{\partial\beta}(\beta,V,N) = $$ $$ = -\beta^2\left( \frac{\partial E}{\partial\beta}(\beta,V,\lambda(\beta,V,N)) + \frac{\partial E}{\partial\lambda}(\beta,V,\lambda(\beta,V,N)) \frac{\partial \lambda}{\partial\beta}(\beta,V,N) \right) = $$ $$ =-\beta^2\left(\left.\frac{\partial E}{\partial\beta}\right|_{\lambda,V} + \left.\frac{\partial E}{\partial\lambda}\right|_{\beta,V} \left.\frac{\partial \lambda}{\partial\beta}\right|_{V,N} \right)\tag{1} $$ In thermodynamics, the following equality of partial derivatives of related quantities is often used $$ \left.\frac{\partial \lambda}{\partial\beta}\right|_{V,N} = -\left.\frac{\partial \lambda}{\partial N}\right|_{V,\beta} \left.\frac{\partial N}{\partial\beta}\right|_{V,\lambda} $$ Substituting this equality into (1) gives the required equality.

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  • $\begingroup$ Thank you so much! $\endgroup$
    – NewStudent
    Dec 16, 2023 at 17:49

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