The Principle of Maximal Heterogeneity (PMH)—the claim that maximum-entropy selectionwithin a parametric family coincides with maximal heterogeneity as formalized by majorization—has recently been justified axiomatically for monotone scale-free families. Empirically, however,heavy-tailed phenomena are often better modeled by the non-monotone (unimodal) log-normal dis-tribution. We therefore delineate the domain of validity of the PMH by analyzing the majorization(Lorenz) structure of the log-normal family and the behavior of extensive and generalized entropiesunder standard constraints.First, we show that the log-normal Lorenz curve has the closed form L(p; σ) = Φ(Φ−1(p) − σ),independent of the location parameter μ. Consequently, the family is strictly ordered by majoriza-tion: increasing the shape parameter σ yields a strictly more concentrated (more unequal, lessLorenz-heterogeneous) distribution.Second, under the common MaxEnt constraint of fixed arithmetic mean, Shannon differentialentropy is not Schur-concave along this Lorenz-ordered path: it is maximized at the interior pointσ = 1 rather than at the heterogeneity boundary. We extend this beyond Shannon by derivingclosed-form expressions for the mean-constrained R´enyi, Tsallis, and Kaniadakis entropies as func-tions of σ and classifying their maximizers. Across these frameworks, entropy fails to be strictlydecreasing in σ, demonstrating that the misalignment between Lorenz heterogeneity and differential-entropy MaxEnt is not specific to Shannon.We then examine alternative constraints (fixed geometric mean, direct constraints on log-variance)and clarify the distinct roles of (i) Lorenz/majorization inequality, (ii) information-theoretic diver-sity, and (iii) tail heaviness. A small simulation protocol (with a ready-to-run notebook) illustratesthe Lorenz ordering and entropy landscapes in finite samples and provides a template for empiricallog-normal data.
Kevin Fathi (Wed,) studied this question.