InterJournal Complex Systems, 597 Status: Submitted |
Manuscript Number: [597] Submission Date: 20628 |
Diversity, Cluster Entropy and Complexity on Randomly Occupied Lattices |
Subject(s): CX.05, CX.07, CX.23
Category: Article
Abstract:
Diversity is an important characteristic that can be observed in many natural phenomena. The concept of diversity has been used to describe the complexity of different systems. In the randomly occupied lattices model, the term “complexity” has been associated with diversity in length scales, which clusters generated in this model, can assume. Moreover, diversity can also be referred to other properties of the system, such as the size, shape or configuration assumed by the clusters. Thus, ``cluster diversity'' is defined as the differentiation of clusters in respect to their size or Lattice Animals (LA). Entropy is a fundamental concept in physics and has been closely connected to complexity. It is related with the information content and order/disorder of a system. Here, we consider ``cluster entropy'', which is defined using the probability that an occupied site belongs to a cluster of size s, or being part of a specific free or fixed LA 200. This definition can be associated to the information entropy and the configurational or local porosity entropy. However, these entropies are based on the information content of a sliding m X m square region, thus being basically a local or short-range measurement. Conversely, our definition of entropy depends on the structure of the clusters, which are not limited to a local region and is capable of spanning over the whole lattice. Using computer simulation, we analyze the cluster diversity and cluster entropy of the system, which leads to the determination of probabilities associated with the maximum of these functions. We show that these critical probabilities are associated with the percolation transition and to the complexity of the system. In addition, we derived from these measurements some interesting scaling relations.
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