Joint probability distribution of the invariants comprising determinantal inequalities: Heuristic derivation

Joint probability distributions are derived that are expressed in terms of the determinants that form the determinantal inequalities associated with the non-negative Fourier series that represent crystal structures. The derivation involves heuristic considerations. It is therefore appropriate to test the distributions extensively by making comparisons with results obtained by other theoretical means and evaluations of the implications of the distributions. Those performed thus far on the low-order determinants (third and fourth orders) have provided satisfactory results. The determinantal probability distributions imply a general maximum determinant rule, contain a wealth of information, and provide numerous paths that may be followed for future development.