Dynamics in fractal spaces

TL;DR

This paper explores the relationship between the Fokker-Planck Equation and its fractal version, deriving connections to Tsallis statistics, and demonstrates that the fractal equation effectively models dynamics in fractal systems like quark-gluon plasma.

Contribution

It introduces a link between fractal derivatives and the Plastino-Plastino Equation, connecting entropic index with fractal dimension, and compares their effectiveness in modeling complex system dynamics.

Findings

FFPE captures fractal system dynamics more accurately than traditional FPE.

The derived PPE relates entropic index to fractal geometric properties.

Numerical solutions show similar results for both equations in complex systems.

Abstract

This study investigates the interconnections between the traditional Fokker-Planck Equation (FPE) and its fractal counterpart (FFPE), utilizing fractal derivatives. By examining the continuous approximation of fractal derivatives in the FPE, it derives the Plastino-Plastino Equation (PPE), which is commonly associated with Tsallis Statistics. This work deduces the connections between the entropic index and the geometric quantities related to the fractal dimension. Furthermore, it analyzes the implications of these relationships on the dynamics of systems in fractal spaces. In order to assess the effectiveness of both equations, numerical solutions are compared within the context of complex systems dynamics, specifically examining the behaviours of quark-gluon plasma (QGP). The FFPE provides an appropriate description of the dynamics of fractal systems by accounting for the fractal nature of the momentum space, exhibiting distinct behaviours compared to the traditional FPE due to the system's fractal nature. The findings indicate that the fractal equation and its continuous approximation yield similar results in studying dynamics, thereby allowing for interchangeability based on the specific problem at hand.