# Expression for heat capacity at fixed volume in grand canonical ensemble

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?

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.

• Thank you so much! Dec 16, 2023 at 17:49