# § 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

(11) |

where

(12) |

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,

(13) |

or, equivalently,

(14) |

If we restrict to be a homogeneous first-degree function of the momenta,

(15) |

while still allowing to be an arbitrary function of coordinates, the right hand side of Eq. (14) becomes

(16) |

The coordinates and momenta are statistically uncorrelated in equilibrium, so that

(17) | ||||

(18) |

where terms with vanish. Now, is the standard kinematic definition of the temperature. Substituting this, Eq. (16) and Eq. (18) into Eq. (14) yields

(19) |

which is a special case of Eq. (1) when .

Hirschfelder (6) chose to obtain a hierarchy of hypervirial temperatures characterized by the index . Choosing instead yields the hyperconfigurational temperatures in Eq. (9).

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

(20) |

Combining this with Eq. (19) yields

(21) | ||||

(22) |

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.