Microcanonical ensemble generalized pressure

[Mayan Fung]

In the discussion of the pressure in macrocanonical ensemble, I found in textbook that:
$dW = \bar p dV$ ($dW$ is in fact d_bar W, yet I can't type the bar)
The derivation goes like:
$\bar p = \frac{1}{Z} \sum_{r} e^{-\beta E_r} (-\frac{\partial E_r}{\partial V}) = ... = \frac{1}{\beta} \frac{\partial lnZ}{\partial V}$

However, $E = TdS - pdV$ and in macrocanonical ensemble, we have $T,V,N$ as variables. That means $\frac{\partial E_r}{\partial V} = (\frac{\partial E_r}{\partial V})_{T,N}$ but not keeping entropy $S$ constant.

I am confused about why we can use $\frac{\partial E_r}{\partial V}$ in the above derivation

 

[ORF]

Hi,

That means but not keeping entropy constant.

please, crosscheck that the partial derivative is for S and N constant

https://de.wikipedia.org/wiki/Mikrokanonisches_Ensemble#Druck
(sorry, it was the only wikipedia article with the pressure definition as partial derivative)

Regards,
ORF

 

[vanhees71]

It's just using different potentials. It turns out that $\Phi=-k_B T \ln Z$ is the Landau potential with natural variables $S$, $T$, and $\mu$. The relation with the internal energy $U$ is
$$\Phi=U-TS-\mu N.$$

For details see

https://itp.uni-frankfurt.de/~hees/publ/off-eq-qft.pdf
Sect. 2.1.4

Note that in this manuscript I use natural units with $k_B=1$ and in the relativistic context instead of a conserved particle number I use some conserved charge $Q$ (like electric chrage) to introduce a corresponding chemical potential.