Realizability issues for iso-g ⁽²⁾ processes

Iso-g ⁽²⁾ processes in the equilibrium regime concern density-dependent interactions that identically cancel the usual density variation of many-body pair correlation functions. This paper considers the formal density expansion for effective pair potentials with this iso-g ⁽²⁾ property, showing how successive terms in that expansion can be determined iteratively. Explicit results through second density order have been obtained for two types of 'target' pair correlation functions: (a) a unit step function (hard core), and (b) unity augmented by an origin-centred Gaussian profile with numerical multiplier A. For the unit step function and the -1 ≤ A < 0 Gaussian cases, readability appears to be attainable up to a finite terminal density. However, when the Gaussian A > 0, the terminal density diverges to infinity while the effective pair interaction series is non-convergent, though possibly valid as an asymptotic expansion.