From Boscovich’s Curve to the Spectral Potential Mean-Field Model of Condensed Matter

In this study, the Boscovich curve of 1763 is reinterpreted as a mean-field potential for interacting particles in condensed matter. In a dense many-body system, each particle experiences an effective potential arising from the average distribution of all the others. This mean-field potential, which exhibits alternating maxima (energy barriers) and minima (coordination shells), thereby reducing the complexity of the N-body problem to an effective two-body radial problem, with the correlation distance r as the key variable. The relationship between the PMF and the radial distribution function g(r) is given by the Kirkwood equation UB(r) = −kT ln g(r), which provides a multi-well potential in condensed matter. Furthermore, the system is described by the Fisher density functional equation for the correlation amplitudes, −2kT ∇2ψ(r) + UB(r)ψ(r) = μψ(r) whose eigenvalues μi correspond to potential levels and whose eigenfunctions ψi are the correlation amplitudes of the coordination shell structure. Based on the multi-well potential picture, the oscillatory behavior of UB(r) is modeled analytically by a weighted sum of Lennard-Jones potentials, modulated by sigmoid functions. The parameters—well depths, widths, and coordination distances—are assigned on the basis of known structural properties of the system, derived either from experimental data or from geometric models such as FCC or HCP lattices. The radial distribution function is then reconstructed as a linear combination of the squared eigenfunctions obtained from the Fisher equation. The resulting discrete eigenvalue spectrum provides a spectral interpretation of the shell structure of condensed matter, wherein the complexity of many-body interactions is encoded in a hierarchy of correlation modes, each associated with a specific coordination shell. Unlike classical DFT—which relies on approximate excess free-energy functionals—and Ornstein–Zernike theory—which requires closure approximations—our approach provides a direct spectral interpretation of the coordination shell structure through the eigenvalue spectrum of the Fisher equation, where the PMF acts as the effective potential and the radial distribution function is reconstructed as a combination of squared eigenfunctions. The method is validated for liquid argon and FCC lattices and establishes a historical connection with Boscovich’s curve as a statistical potential.