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

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. This new approach permits 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. These functions allow describing the system in terms of a eigenstates. A dissipative term is introduced into the model, as a relevant term to study 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.