An information theoretic synthesis and analysis of Compton profiles

Abstract

The information theoretic technique of entropy maximization is applied to Compton profile (CP) data, employing single and double distribution moments <pⁿ≳ as constraints. Emphasis is placed on the first and second (n = 1,2) moments— the average momentum and energy, respectively. Studies are made both on semi-infinite and finite ranges of the variable q in the Compton profile J(q). The quality of resulting maximum entropy profiles is judged by how well they predict familiar CP quantities— moments, the profiles' magnitude at the origin J(0), and the width at half-maximum q0.5. Information theoretic quantities— Shannon entropies, information contents, and surprisals— are also presented. Based upon the "sum" constraint <q+q²≳ , a relation is derived which approximately predicts J(0) given q_0.5 (or vice versa) for a large class of molecular systems, i.e., J(0) = (2/N)exp[-K/4][K]^1/2[(π ^1/2/2)@(K^1/2/2)]@qL ^-1, where K = ln 2/(q²_0.5+q_0.5). The assimilation of more and more constraints (information) results in generally improved Compton profiles. The average momentum constraint contains the most information of all moment expectation values, as judged by its predictive capacity and by the information theory measures.