§ IV. Derivation from the hypervirial theorem
A remarkable consequence of Eq. (1) is that any vector field that depends only on the configurational degrees of freedom gives rise to a functionally distinct but thermodynamically equivalent definition of the temperature that depends only on configurational coordinates. Here we show that this insight emerges transparently from the hypervirial theorem (6); (7), and that Eq. (1) itself can be derived from this starting point.
The Hamiltonian equation of motion for an arbitrary dynamical variable, , is
is the Poisson bracket of and . Provided that (1) does not depend explicitly on time and (2) remains bounded in the course of time, the time average of Eq. (11) yields the classical hypervirial theorem,
If we restrict to be a homogeneous first-degree function of the momenta,
while still allowing to be an arbitrary function of coordinates, the right hand side of Eq. (14) becomes
The coordinates and momenta are statistically uncorrelated in equilibrium, so that
which is a special case of Eq. (1) when .
A similar line of reasoning provides a straightforward derivation of the most general result, Eq. (1). We begin by choosing in Eq. (19) to obtain , which is equivalent to the Clausius virial theorem, , for systems with pairwise additive interactions. Next, we choose , where is an arbitrary function of the momenta, and substitute into Eq. (14). The following steps are analogous to those used in deriving Eq. (19) and yield
Combining this with Eq. (19) yields
where, once again, we have exploited the statistical independence of the coordinates and momenta. Because this holds for any choice of and , it holds for any sum of products of the form . Thus, Eq. (22) is equivalent to Eq. (1) for any choice of whose components can be expressed as Taylor series in the coordinates and momenta.