As noted earlier, the quantity is a measure of the instantaneous value of the internal pressure . Let us look at the ensemble average of this quantity
Doing the volume integration by parts gives
This result is known as the pressure virial theorem. It illustrates that the average of the quantity gives the fixed pressure P that defines the ensemble.
Another important result comes from considering the ensemble average
Once again, integrating by parts with respect to the volume yields
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.