Computing high order correlations in liquids is not easy. Josh Robinson – with the help of Paddy Royall, Roland Roth and myself – has shown earlier in 2019 that with employing mostly geometrical principles one can accurately estimate the free energy of different motifs in a simple hard sphere fluid, see here 10.1103/PhysRevLett.122.068004 .

In more detailed paper we now show how this approach can be connected to classical liquid state theory and seen as an extension of so-called scaled-particle theory, where one computes the work of insertion of solutes in a fluid in order to estimate their free energy (see, for example, the Widom insertion method.)

Our approach allows us to write down a potential of mean force for interactions between a subset of* n* particles and a fluid, generalising previous methods and opening a way to accurate measurement of free energy barriers in the formation of local structural inhomogeneities in fluids, as in the formation of crystalline precursors.

The full article reference is:

Joshua F. Robinson , Francesco Turci, Roland Roth, and C. Patrick Royall, *Many-body correlations from integral geometry*, Physical Review E 100, 062126 (2019)