By
**Joao Batista Crispiniano Garcia, T.I. Jyh, T.I.Ren**

Nature is rich in phenomena involving fragmentation, when a system is divided in discrete domains as a consequence of impulsive or dynamic processes.

In recent studies of fragmentation problems by means of extensive computer simulations Gomes and Vasconcelos [1] noticed that the maximum number of fragments (Nmax) occurred after the maximum diversity of fragment sizes (Dmax) observed along the process. Subsequently it was observed [2,3] that Nmax scales with Dmax as Nmax\simDmax^{2} in a wide variety of dynamics, object topologies and dimensions.

Introduced by von Neuman and Ulman in 1948 [4] cellular automata are discrete dynamic systems that evolve obeying simple local rules. Sales et al. [5] and Garcia et al. [6] in their study of the cellular automaton "Life" due to Conway reported a similar relation involving Nmax and Dmax.

More recently Coutinho et al. [7] studying the problems of partial covering time (PCT) and random covering time (RCT) in two dimensions with Monte Carlo simulations observed (among other results) the same relation between the maximum number of disconnected unvisited regions and the corresponding maximum diversity of sizes of the unvisited regions.

It is quite intuitive to associate the complexity of a system with the variety of species observed in it. In a formal way complexity is related to the entropic information content of the dynamic, that is a quantity not trivial to measure. If the diversity can give a rough idea of the complexity of a system then the relation above mentioned can take us to some cosmological conjecture. Given a certain initial amount of mass (Mo) and the dimension of a universe (d) how much complex can this universe become? If the mass goes up so does the complexity? Does it depend on the dynamics?

The analytic relation between diversity (D) and the number of fragmented configurations (partitions) of a discrete system (w) grows logarithmically with w as D = k * ln w + C, where k = 0.88 and C = 0.39 [8].

Extensive numerical simulations [9] showed that Dmax produced by fragmentation dynamics on a system of mass Mo tends to decay with the space dimension d as Dmax^2/Mo^-d, independent of the details of the dynamics. Associating this behavior with the answer given by quantum mechanics to the problem of stability of a universe of low dimension it was suggested recently an answer to why our universe is tri-dimensional [9].

Studying the problem of diversity in a bi-dimensional net that was occupied with uniform probability using the box counting algorithm [10], Gomes et al. [11] observed unexpected oscillations of Dmax with growing Mo. This observation when associated with the problem of the evolution of the variety of biological especies on Earth suggests that similar oscillations can have occurred, e.g. the diversity of species in the Cretaceus period (\sim 65 millions of years ago) is not necessarily bigger than during the Permian (\sim 225 millions of years ago)[12].

Finally we would like to call the attention for three points of contact between entropy and diversity of a system: both are macroscopic quantities depending logarithmically on the number of microstates; both are associated with complexity or deviations of the state of maximum order; and both are interesting only in a discrete universe. The diversity has some other attractives: it is easier to be numerically implemented, it can be defined for any system and presents a close connection with fractals (systems with large diversity of scales).

* Work partially supported by CAPES, CNPq and FINEP (brazilian government agencies).