On the response of power law distributions to fluctuations

Both in physical and in non-physical systems, the probability of extreme events depends on the slope of the tail of a distribution function. Prediction of this slope is often jeopardized by either poor knowledge of dynamics or statistical uncertainties. In many cases, however, the system attains a relaxed state, and extreme events correspond to large fluctuations near this state. Rather than starting from full (and often unavailable) knowledge of dynamics, we assume that a relaxed state exists and derive a necessary condition for its stability against fluctuations of arbitrary amplitude localized in the tail. In many problems, for suitably chosen variables this tail resembles either an exponential distribution or a power law. We take a q-exponential as a proxy of the tail; its slope depends on the dimensionless parameter q (q = 1 corresponds to an exponential). In turn, q-exponentials describe maxima of the non-extensive entropy Sq, and probabilities of fluctuations near a Sq = max state follow a generalized Einstein’s rule [E. Vives, A. Planes, Phys. Rev. Lett. 88, 020601 (2002)]. This rule provides the desired condition of stability, which allows us to write down a set of rules for semi-anaytical computation of the value qc of q in the relaxed state even with limited knowledge of dynamics. We apply these rules to a problem in econophysics [J.R. Sànchez, R. Lopez-Ruiz, Eur. Phys. J. Special Topics 143, 241 (2007)] and retrieve the main results of numerical solutions, namely the transition of a relaxed distribution of wealth from an exponential to a Pareto-like behaviour in the tail by suitable tuning of the relevant control parameters. A similar discussion holds for the scale parameter of lognormal distributions; we retrieve the results of [Z.N. Wu, J. Li, C.Y. Bai, Entropy 19, 56 (2017)]. Graphical abstract