Classical Virial Theorem (canonical ensemble derivation)

Again, let and be specific components of the phase space vector . Consider the canonical average

given by

But

Thus,

Several cases exist for the surface term :

1.

a momentum variable. Then, since , evaluated at clearly vanishes.

2.

and as , thus representing a bound system. Then, also vanishes at .

3.

and as , representing an unbound system. Then the exponential tends to 1 both at , hence the surface term vanishes.

4.

and the system is periodic, as in a solid. Then, the system will be represented by some supercell to which periodic boundary conditions can be applied, and the coordinates will take on the same value at the boundaries. Thus, H and will take on the same value at the boundaries and the surface term will vanish.

5.

, and the particles experience elastic collisions with the walls of the container. Then there is an infinite potential at the walls so that at the boundary and at the bondary.

Thus, we have the result

The above cases cover many but not all situations, in particular, the case of a system confined within a volume V with reflecting boundaries. Then, surface contributions actually give rise to an observable pressure (to be discussed in more detail in the next lecture).