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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.13241 |
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| _version_ | 1866912765022568448 |
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| author | Baray, Jerome |
| author_facet | Baray, Jerome |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_13241 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Zipfian universality of interaction laws: A statistical-mechanics framework for inverse power scaling Baray, Jerome Statistical Mechanics 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. |
| title | Zipfian universality of interaction laws: A statistical-mechanics framework for inverse power scaling |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2512.13241 |