Theoretical model to deduce a PDF with a power law tail using Extreme Physical Information

TL;DR

This paper develops a theoretical model using Extreme Physical Information to derive a probability density function with a power law tail, aiding the analysis of complex systems and rare events.

Contribution

It introduces a novel approach employing EPI to derive a PDF with a power law tail, linking tail behavior to the distribution head in complex systems.

Findings

Derived a second order differential equation for the PDF

Established a relation between power law tail and distribution head

Included a dissipative term for open system analysis

Abstract

The theory of Extreme Physical Information (EPI) is used to deduce a probability density function (PDF) of a system that exhibits a power law tail. The computed PDF is useful to study and fit several observed distributions in complex systems. With this new approach it is possible to describe extreme and rare events in the tail, and also the frequent events in the distribution head. Using EPI, an information functional is constructed, and minimized using Euler-Lagrange equations. As a solution, a second order differential equation is derived. By solving this equation a family of functions is calculated. Using these functions it is possible to describe the system in terms of eigenstates. A dissipative term is introduced into the model, as a relevant term for the study of open systems. One of the main results is a mathematical relation between the scaling parameter of the power law observed in the tail and the shape of the head.