Derivation from the hypervirial theorem

A remarkable consequence of Eq. (1) is that any vector field ${\mathbf B}({\mathbf \Gamma})$ that depends only on the $3N$ 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, $f({\mathbf \Gamma},t)$, is

$$\frac{df}{dt} = \frac{\partial f}{\partial t} + \{f,\ensuremath{{\mathcal H}({\mathbf \Gamma})}\xspace \},$$ (11)

where

$$\{f,\ensuremath{{\mathcal H}({\mathbf \Gamma})}\xspace \} = \sum_... ...thbf \Gamma})}\xspace }{\partial q_i} \, \frac{\partial f}{\partial p_i}\right]$$ (12)

is the Poisson bracket of $f$ and $\ensuremath{{\mathcal H}({\mathbf \Gamma})}\xspace$. Provided that (1) $f$ does not depend explicitly on time and (2) $f({\mathbf \Gamma})$ remains bounded in the course of time, the time average of Eq. (11) yields the classical hypervirial theorem,

$$\left< \{f,\ensuremath{{\mathcal H}({\mathbf \Gamma})}\xspace \} \right> = 0$$ (13)

or, equivalently,

$$\sum_{i=1}^{3N} \left< \frac{\partial \mathcal{H}}{\partial p_i} ... ...bf \Gamma})}\xspace }{\partial q_i} \, \frac{\partial f}{\partial p_i} \right>.$$ (14)

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

$$f({\mathbf \Gamma}) = p_\ell \, Q(\{q_i\}),$$ (15)

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

$$\sum_{i=1}^{3N} \left< \frac{\partial \ensuremath{{\mathcal H}({\... ... Q \, \frac{\partial \ensuremath{V(\{q_i\})}\xspace }{\partial q_\ell} \right>.$$ (16)

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

$$\sum_{i=1}^{3N} \left< \frac{\partial \ensuremath{{\mathcal H}({\mathbf \Gamma})}\xspace }{\partial p_i} \, \frac{\partial f}{\partial q_i} \right> = \sum_{i=1}^{3N} \left< \frac{\partial \ensuremath{{\mathcal H}(... ...{\partial p_i} \, p_\ell \right> \left< \frac{\partial Q}{\partial q_i} \right>$$ (17)

$$= \left< p_i \, \frac{\partial \ensuremath{{\mathcal H}({\mathbf ... ...\xspace }{\partial p_i} \right> \left< \frac{\partial Q}{\partial q_i} \right>,$$ (18)

where terms with $i \ne \ell$ vanish. Now, $\left< p_i\frac{\partial {\mathcal H}}{\partial p_i} \right> = k_B T$ is the standard kinematic definition of the temperature. Substituting this, Eq. (16) and Eq. (18) into Eq. (14) yields

$$k_B T = \left< Q \, \frac{\partial \ensuremath{V(\{q_i\})}\xspace }{\partial q_\ell} \right> \left< \frac{\partial Q}{\partial q_i} \right>^{-1},$$ (19)

which is a special case of Eq. (1) when ${\mathbf B}({\mathbf \Gamma})= \mathbf{B}(\{q_i\})$.

Hirschfelder (6) chose $Q = q_i^s$ to obtain a hierarchy of hypervirial temperatures characterized by the index $s$. Choosing instead $Q = F_i^s$ 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 $Q = q_i$ in Eq. (19) to obtain $k_B T = \left< q_i \frac{\partial {\mathcal H}}{\partial q_i} \right>$, which is equivalent to the Clausius virial theorem, $k_B T = \left< r_i F_i \right>$, for systems with pairwise additive interactions. Next, we choose $f({\mathbf \Gamma}) = q_i \, P(\{p_i\})$, where $P$ 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

$$k_B T = \left< P \, \frac{\partial \ensuremath{K(\{p_i\})}\xspace }{\partial p_\ell} \right> \left< \frac{\partial P}{\partial p_i} \right>^{-1}.$$ (20)

Combining this with Eq. (19) yields

$$k_B T = \frac{\left< QP \left( \frac{\partial \ensuremath{V(\{q_i\})}\x... ...\frac{\partial Q}{\partial q_i} + Q \, \frac{\partial P}{\partial p_i} \right>}$$ (21)

$$= \frac{\left< QP \, \nabla_i \ensuremath{{\mathcal H}({\mathbf \Gamma})}\xspace \right>}{\nabla_i (QP)},$$ (22)

where, once again, we have exploited the statistical independence of the coordinates and momenta. Because this holds for any choice of $Q$ and $P$, it holds for any sum of products of the form $\sum_m Q_m P_m$. Thus, Eq. (22) is equivalent to Eq. (1) for any choice of ${\mathbf B}({\mathbf \Gamma})$ whose components can be expressed as Taylor series in the coordinates and momenta.