On avalanche-like perturbations of relaxed power-law distributions: Richardson’s law of warfare as a consequence of the relaxation to a Pareto-like distribution of wealth

Abstract

In monodimensional systems where a nonlinear Fokker-Planck equation [G.A. Casas et al., Phys. Rev. E 86, 061136 (2012)] describes relaxation to a distribution function which behaves like a power law in the tail, we show that perturbations of the relaxed distribution behave as avalanches in the sense of Self Organized Criticality [J. Nagler et al., Phys. Rev. E 60, 2706 (1999)]. We determine the power laws followed by both size and return time of these avalanches whenever the relaxed distribution is resilient against perturbation of arbitrary amplitude in the tail. We apply our results to a class of problems in econophysics [J.R. Sanchez et al., Europhys. J. 143, 241 (2007)] where the relaxed distribution of wealth is approximately described by Pareto’s principle [A. Di Vita, Europhys. J. 92, 255 (2019)]. If the destruction of wealth associated with an avalanche is identified with a war, then we retrieve Richardson’s model of arms race [L.F. Richardson, Nature 136, 1025 (1935)], prove that the largest avalanches follow Richardson’s scaling law of warfare [L.F. Richardson, J. Am. Stat. Assoc. 43, 244 (1948)] and show that the probability of war outbreak per unit time follows a Poisson law [L.F. Richardson, Nature 155, 610 (1945)], in agreement with the findings of [L. Cederman, Am. Polit. Sci. Rev. 97, 135 (2003), A. Clauset, Sci. Adv. 4 (2018), G. Martelloni et al. arXiv:1812.08071 (2018)].

Graphical abstract