Configurational Temperatures and Interactions in Charge-Stabilized Colloid

§ IV. Derivation from the hypervirial theorem

A remarkable consequence of Eq. (1) is that any vector field \boldsymbol{B}(\boldsymbol{\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(\boldsymbol{\Gamma},t), is

\frac{df}{dt}=\frac{\partial f}{\partial t}+\{ f,H(\boldsymbol{\Gamma})\}, (11)

where

\{ f,H(\boldsymbol{\Gamma})\}=\sum _{{i=1}}^{{3N}}\,\left[\frac{\partial H(\boldsymbol{\Gamma})}{\partial p_{i}}\,\frac{\partial f}{\partial q_{i}}-\frac{\partial H(\boldsymbol{\Gamma})}{\partial q_{i}}\,\frac{\partial f}{\partial p_{i}}\right] (12)

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

\left<\{ f,H(\boldsymbol{\Gamma})\}\right>=0 (13)

or, equivalently,

\sum _{{i=1}}^{{3N}}\left<\frac{\partial\mathcal{H}}{\partial p_{i}}\frac{\partial f}{\partial q_{i}}\right>=\sum _{{i=1}}^{{3N}}\left<\frac{\partial H(\boldsymbol{\Gamma})}{\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(\boldsymbol{\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 H(\boldsymbol{\Gamma})}{\partial q_{i}}\,\frac{\partial f}{\partial p_{i}}\right>=\left<Q\frac{\partial H(\boldsymbol{\Gamma})}{\partial q_{\ell}}\right>=\left<Q\,\frac{\partial V(\{ q_{i}\})}{\partial q_{\ell}}\right>. (16)

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

\displaystyle\sum _{{i=1}}^{{3N}}\left<\frac{\partial H(\boldsymbol{\Gamma})}{\partial p_{i}}\,\frac{\partial f}{\partial q_{i}}\right> \displaystyle=\sum _{{i=1}}^{{3N}}\left<\frac{\partial H(\boldsymbol{\Gamma})}{\partial p_{i}}\, p_{\ell}\right>\left<\frac{\partial Q}{\partial q_{i}}\right> (17)
\displaystyle=\left<p_{i}\,\frac{\partial H(\boldsymbol{\Gamma})}{\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 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 V(\{ q_{i}\})}{\partial q_{\ell}}\right>\left<\frac{\partial Q}{\partial q_{i}}\right>^{{-1}}, (19)

which is a special case of Eq. (1) when \boldsymbol{B}(\boldsymbol{\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 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(\boldsymbol{\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 K(\{ p_{i}\})}{\partial p_{\ell}}\right>\left<\frac{\partial P}{\partial p_{i}}\right>^{{-1}}. (20)

Combining this with Eq. (19) yields

\displaystyle k_{B}T \displaystyle=\frac{\left<QP\left(\frac{\partial V(\{ q_{i}\})}{\partial q_{i}}+\frac{\partial K(\{ p_{i}\})}{\partial p_{i}}\right)\right>}{\left<P\,\frac{\partial Q}{\partial q_{i}}+Q\,\frac{\partial P}{\partial p_{i}}\right>} (21)
\displaystyle=\frac{\left<QP\,\nabla _{i}H(\boldsymbol{\Gamma})\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 \boldsymbol{B}(\boldsymbol{\Gamma}) whose components can be expressed as Taylor series in the coordinates and momenta.