Pressure and work virial theorems

As noted earlier, the quantity $$-\partial H/\partial V$$ is a measure of the instantaneous value of the internal pressure $$P_{\rm int}$$. Let us look at the ensemble average of this quantity

 $$\langle P_{\rm int} \rangle$$ $$-{1 \over \Delta}C_N\int_0^{\infty}dVe^{-\beta PV} \int d{\rm x}{\partial H \over \partial V}e^{-\beta H({\rm x})}$$ $${1 \over \Delta} C_N\int_0^{\infty}dVe^{-\beta PV} \int d{\rm x}kT {\partial \over \partial V}e^{-\beta H({\rm x})}$$ $${1 \over \Delta} \int_0^{\infty}dVe^{-\beta PV} kT {\partial \over \partial V} Q(N,V,T)$$

Doing the volume integration by parts gives

 $$\langle P_{\rm int} \rangle$$ $${1 \over \Delta} \left[e^{-\beta PV} kT Q(N,V,T) \right]\vert _0^{\infty} - {1 \over \Delta } \int _0^{\infty}dVkT \left({\partial \over \partial V} e^{-\beta PV} \right) Q(N,V,T)$$ $$P{1 \over \Delta} \int_0^{\infty}dVe^{-\beta PV} Q(N,V,T) = P$$

Thus,
$\langle P_{\rm int}\rangle = P$

This result is known as the pressure virial theorem. It illustrates that the average of the quantity $$-\partial H/\partial V$$ gives the fixed pressure $$P$$ that defines the ensemble.

Another important result comes from considering the ensemble average $$-\partial H/\partial V$$

$\langle P_{\rm int} V\rangle = {1 \over \Delta} \int_0^{\infty}dVe^{-\beta PV} kTV {\partial \over \partial V}Q(N,V,T)$

Once again, integrating by parts with respect to the volume yields

 $$\langle P_{\rm int}V\rangle$$ $${1 \over \Delta} \left[e^{-\beta PV} kTV Q(N,V,T) \right]\vert _0^{\infty} - {1 \over \Delta} \int _0^{\infty}dVkT \left({\partial \over \partial V}Ve^{-\beta PV} \right)Q(N,V,T)$$ $${1 \over \Delta} \left[-kT \int_0^{\infty}dVe^{-\beta PV} Q(V) + P \int_0^{\infty}dVe^{-\beta PV} VQ(V)\right]$$ $$-kT + P\langle V \rangle$$

or
$\langle P_{\rm int} V\rangle + kT = P\langle V\rangle$

This result is known as the work virial theorem. It expresses the fact that equipartitioning of energy also applies to the volume degrees of freedom, since the volume is now a fluctuating quantity.