Tsallis Entropy and the Transition to Scaling in Fragmentation

Oscar Sotolongo-Costa,G Rodgers,Arezky Rodriguez

doi:10.3390/e2040172

Abstract

By using the maximum entropy principle with Tsallis entropy we obtain a fragment size distribution function which undergoes a transition to scaling. This distribution function reduces to those obtained by other authors using Shannon entropy. The treatment is easily generalisable to any process of fractioning with suitable constraints.

Highlights

As a result of developments in materials science, combustion technology, geology and many other fields of research, there has been an increase of interest in the problem of fragmentation of objects. Within this general field there is a collection of papers [1, 2, 3, 4] where a transition occurs from a “classical” distribution of fragments to a power law distribution. This transition has not been adequately explained in terms of any general principles, in [1] the representation of the fragmentation process in terms of percolation on a Bethe lattice leads to a transition to a power law in the distribution of fragment sizes
Some attempts have been made to derive the fragment size distribution function from the maximum entropy principle [5, 6], subject to some constraints which mainly came from physical considerations about the fragmentation phenomena
For a given regime of breakage, generally identified as that of the lowest energy of breakage, the fragment distribution function can be deduced through the maximum entropy principle using BG statistics

Summary

Introduction

As a result of developments in materials science, combustion technology, geology and many other fields of research, there has been an increase of interest in the problem of fragmentation of objects. Within this general field there is a collection of papers [1, 2, 3, 4] where a transition occurs from a “classical” distribution of fragments (e.g. log-normal or Rossin-Ramler- like) to a power law distribution. This transition has not been adequately explained in terms of any general principles, in [1] the representation of the fragmentation process in terms of percolation on a Bethe lattice leads to a transition to a power law in the distribution of fragment sizes.

Results

Conclusion