Zipfian universality of interaction laws: A statistical-mechanics framework for inverse power scaling

Abstract

Inverse power-law interaction forms, such as the inverse-square law, recur across a wide range of physical, social, and spatial systems. While traditionally derived from specific microscopic mechanisms, the ubiquity of these laws suggests a more general organizing principle. This article proposes a statistical-mechanics framework in which such interaction laws emerge as macroscopic fixed points of aggregation processes involving strongly heterogeneous microscopic contributions. We consider systems where individual interaction sources exhibit heavy-tailed heterogeneity consistent with Zipf-Pareto statistics and where aggregation proceeds without intrinsic length scales. Under minimal assumptions of heterogeneity, multiplicativity, scale invariance, and stability under coarse-graining, we show that the resulting macroscopic interaction field must adopt a scale-free, power-law form. The associated exponent is not imposed a priori but emerges from effective dimensionality, symmetry, and aggregation structure. Within this framework, the inverse-square law is interpreted as a stable statistical fixed point corresponding to isotropic aggregation in an effective three-dimensional space, while deviations from this regime naturally arise from anisotropy, constrained geometries, or nontrivial effective dimensions. This perspective provides a unified interpretation of interaction laws observed in physics, spatial economics, and human geography, without invoking domain-specific microscopic mechanisms. The proposed framework reframes inverse power-law interactions as robust emergent features of Zipfian aggregation rather than as unique consequences of particular physical forces, thereby offering a common statistical explanation for their cross-disciplinary recurrence.