# § III. Hyperconfigurational temperatures

Choosing with in Eq. (1) yields a hierarchy of so-called hypervirial temperatures (6), which reduce to Clausius' virial temperature for . By the same token, we propose that with , yields the set of “hyperconfigurational temperatures” (5),

 (9)

of which is equivalent to the standard configurational temperature. Here, is the magnitude of the -th element in the set of components of the forces on the particles. Because is non-negative, is well defined for any positive real value of . Negative values of would yield diverging temperatures because at least some of the 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, in dimensions, each term of the denominator, , 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 , and for . 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

 (10)

Setting or 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, and , respectively.