Thus, the RMPE represents the net contribution to sex due to spatial correlations involving three, four, or more particles.  A heuristic "ordering" criterion identifies the vanishing of the RMPE as an underlying signature of an impending structural or thermodynamic transition of the system from a less ordered to a more spatially organized condition (freezing is a typical example).  Regardless of this, the knowledge of the RMPE is important to assess the impact of non-pair multiparticle correlations on the entropy of the fluid.  Recently, an accurate and simple proposal for the thermodynamic and structural properties of a hard-sphere fluid in fractional dimension 1<d<3 has been proposed (Santos, A.; López de Haro, M. Phys. Rev. E 2016, 93, 062126).  The aim of this work is to use this approach to evaluate the RMPE as a function of both d and the packing fraction ϕ.  It is observed that, for any given dimensionality d, the RMPE takes negative values for small densities, reaches a negative minimum Δsmin at a packing fraction ϕmin, and then rapidly increases, becoming positive beyond a certain packing fraction ϕ0. 
THERM