# Tsallis Entropy and the Transition to Scaling in Fragmentation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1

_{i}is the probability of finding the system in the microscopic state i, k is Boltzmann’s constant, and W is the total number of microstates.

_{i}the parts or fragments in which the object has been divided, its entropy S obeys S(∪ A

_{i}) < Σ

_{i}S(A

_{i}), defining a “superextensivity” [7] in this system . This suggests that it may be necessary to use non-extensive statistics, instead of the BG statistics. This kind of theory has already been proposed by Tsallis [8], who postulated a generalized form of entropy, given by

_{q}(p) is defined in [7] as

_{q}→ S when q → 1, recovering BG statistics.

_{m}, we can define a dimensionless volume $v=\frac{V}{{V}_{m}}$. Then, the constraints to impose are

_{1}and α

_{2}are determined from eqs.4 and 5. The extremization of L(p

_{i}; α

_{1}; α

_{2}) leads to:

^{1/3}. Then the probability density is:

## References

- Sotolongo-Costa, O.; Moreno, Y.; Lloveras, J.; Antoranz, J. C. Phys Rev. Lett.
**1996**, 76, 42. [CrossRef] [PubMed] [Green Version] - Ishii, T.; Matsushita, M. J. Phys. Soc Jpn.
**1992**, 61, 3474. - Sotolongo-Costa, O. Fractal viewpoint of transition to scaling in atomization. In Procs. of the Joint Meeting of the Portuguese, British, Spanish and Swedish Section of the Combustion Institute; Funchal, Madeira Ed M. Heitor; 1996; pp. 18.9.1–18.9.4. [Google Scholar]
- Sotolongo-Costa, O.; Grau-Crespo, R.; Trallero-Herrero, C. Revista Mexicana de Fisica.
**1998**, 44, 441. [Google Scholar] - Li, X.; Tankin Combust, R. S. Sci. and Tech.
**1987**, 56, 65. - Englman, R.; Rivier, N.; Jaeger, Z. Phil. Mag.
**1987**, 56, 751. - Tsallis, C. Brazilian. J. Phys.
**1999**, 29, 1. - Tsallis, C. J. Stat. Phys.
**1988**, 52, 476. - Williams, F. Combustion Theory. Addison Wesley, 1965. [Google Scholar]

© 22000 by the author. Reproduction for noncommercial purposes permitted.

## Share and Cite

**MDPI and ACS Style**

Sotolongo-Costa, O.; Rodriguez, A.H.; Rodgers, G.J.
Tsallis Entropy and the Transition to Scaling in Fragmentation. *Entropy* **2000**, *2*, 172-177.
https://doi.org/10.3390/e2040172

**AMA Style**

Sotolongo-Costa O, Rodriguez AH, Rodgers GJ.
Tsallis Entropy and the Transition to Scaling in Fragmentation. *Entropy*. 2000; 2(4):172-177.
https://doi.org/10.3390/e2040172

**Chicago/Turabian Style**

Sotolongo-Costa, Oscar, Arezky H. Rodriguez, and G. J. Rodgers.
2000. "Tsallis Entropy and the Transition to Scaling in Fragmentation" *Entropy* 2, no. 4: 172-177.
https://doi.org/10.3390/e2040172