Hyperconfigurational temperatures

Choosing ${\mathbf B}({\mathbf \Gamma})= \{q_1^s, \cdots, q_{3N}^s, 0, \cdots, 0\}$ with $s = 1, 2, 3 \cdots$ in Eq. (1) yields a hierarchy of so-called hypervirial temperatures (6), which reduce to Clausius' virial temperature for $s = 1$. By the same token, we propose that ${\mathbf B}({\mathbf \Gamma})= \{F_1^s, \cdots, F_{3N}^s, 0, \cdots, 0\}$ with $s > 0$, yields the set of ``hyperconfigurational temperatures'' (5),

$$k_B T_h^{(s)} = \frac{\left< \sum_{i=1}^{3N} F_i^{s+1} \right>}{ \left< -s\sum_{i=1}^{3N}F_i^{s-1} \partial_i F_i \right>},$$ (9)

of which $T_h^{(1)}$ is equivalent to the standard configurational temperature. Here, $F_i$ is the magnitude of the $i$-th element in the set of $3N$ components of the forces on the $N$ particles. Because $F_i$ is non-negative, $T_h^{(s)}$ is well defined for any positive real value of $s$. Negative values of $s$ would yield diverging temperatures because at least some of the $F_i$ will be vanishingly small for any system substantially larger than the range of interactions.

A simple example motivates introducing this new hierarchy of expressions. If, for example, a system is characterized by Coulomb pair interactions, $u(r) = 1/r$ in $d = 3$ dimensions, each term of the denominator, $\nabla_r^2 u(r) = r^{1-d} \, \partial_r(r^{d-1}\partial_r u(r))$, of Eqs. (6) and (7) vanishes. Consequently, the associated configurational temperature expressions in Eqs. (6), (7) and (8) diverge unphysically. The hyperconfigurational temperatures, by contrast, are still well defined with $\partial_x F_x = (1-3x^2/r^2)/r^3$, and $F_x^{s-1} \partial_x F_x + F_y^{s-1} \partial_y F_y + F_z^{s-1}\partial_z F_z\ne 0$ for $s \ne 1$. Consequently, the hyperconfigurational temperatures should apply to any system whose pair potential is continuous and differentiable. This suggests that they will be useful for studying systems whose interactions are not known a priori.

Additional useful results emerge for systems such as colloidal monolayers whose interactions are isotropic. In this case, the Cartesian coordinates may be analyzed independently

$$k_B T_{conF} = \frac{\left< \sum_{j=1}^N {\mathbf F}_j^2 \right>}{ \left< -\sum_... ...}^2) \right>}{ \left< - \sum_{j=1}^N (F^\prime_{j,x} + F^\prime_{j,y}) \right>}$$

$$= \frac{\left< \sum_{j=1}^N F_{j,x}^2 \right>}{ \left< - \sum_{j=1}... ... \sum_{j=1}^N F_{j,y}^2 \right>}{\left< - \sum_{j=1}^N F^\prime_{j,y} \right>}.$$ (10)

Setting ${\mathbf B}({\mathbf \Gamma})= \partial_x \ensuremath{V(\{q_i\})}\xspace$ or $\partial_y \ensuremath{V(\{q_i\})}\xspace$ in Eq. (1) leads to same results. We will refer to the two terms in Eq. (10), as well as analogous results for other temperature expressions, as the Cartesian components of the configurational temperature, $T_x$ and $T_y$, respectively.