Entropic measure of spatial disorder for systems of finite-sized objects

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Abstract

We consider the relative configurational entropy per cell SΔ as a measure of the degree of spatial disorder for systems of finite-sized objects. It is highly sensitive to deviations from the most spatially ordered reference configuration of the objects. When applied to a given binary image it provides the quantitatively correct results in comparison to its point object version. On examples of simple cluster configurations, two-dimensional Sierpiński carpets and population of interacting particles, the behaviour of SΔ is compared with the normalized information entropy H′ introduced by Van Siclen [Phys. Rev. E 56 (1997) 5211]. For the latter example, the additional middle-scale features revealed by our measure may indicate for the traces of self-similar structure of the weakly ramified clusters. In the thermodynamic limit, the formula for SΔ is also given.

Introduction

The problem of finding static morphological measures suitable to the quantitative characterization of complex microstructures was considered from alternative viewpoints using more or less subtle mathematics. The family of so-called Minkowski functionals [1] defined within the integral geometry approach or much more specialized measure of disorder of labyrinthine patterns [2], [3] as well as general n-point distribution function formalism [4] are examples of methods rather difficult in practical applications. Thus, much research effort has also been devoted for developing simple tools for searching correlation between the macroscopic properties and microstructure attributes of the medium. Recently, a comprehensive review devoted to that point for porous structures has been given by Hilfer [5]. Also, the similar problem of determining the effective properties of random heterogeneous media from the morphology was shortly reviewed by Torquato [6].

Here, we will focus on such measure that can be easily applied to a digitized image of the microstructure or computer generated pixel distributions. Usually, after subdivision of binary image into equal square cells the analyzed objects are approximated by a distribution of point markers [7], [8]. By point objects we understand objects small enough in comparison to the cell size k×k expressed in pixels. This model situation has been assumed in [9], [10], [11], [12]. However, it has been shown [13] that this idealized for black–white micrographs method needs a modification in the case of finite-sized objects like pixels. Such a modification leads to quantitatively correct results. Nevertheless, from a qualitative point of view using of the point measure for binary images is still acceptable. It should be also stressed that there are applications suitable for point measures only.

Recently, a novel approach based on the adaptation of Shannon information entropy has been developed as the “local porosity entropy” [14] and the “configuration entropy” [15], [16] concepts. These two entropic measures, worked out to characterize random microstructures represented in micrographs or digitized images, were found to be rigorously connected [17]. Then Van Siclen's interesting study [18], briefly outlined in Appendix A, has proposed the quantitative characterization of microstructure inhomogeneity by the “normalized information entropy” H′. Van Siclen's work has motivated us to refresh our idea of using a linear transformation of configurational entropy, mentioned in Ref. [11], as a physical measure of the degree of spatial inhomogeneity. However, only the variant for point objects has been recently reported [19].

The purpose of this paper is to provide the entropic measure SΔ applicable for systems of equally sized objects. Binary images, where black pixels play the role of indistinguishable “particles”, are the systems of interest. Our proposition, specified in Section 2, modifies the point object measure [12], [19] to the case of finite-sized objects. It differs from the other entropic approaches mentioned above. The measure is obtained for every length scale by subtracting the configurational entropy for a given arrangement of black pixels from the entropy corresponding to the most spatially ordered reference configuration and dividing the difference by the number of cells. For a given number of objects, such an approach provides a sensitive and quantitatively correct comparative characterization of digitized two-phase microstructures at every length scale. In the thermodynamic limit, the simple expression for this measure is also obtained.

To illustrate the basic features of SΔ and for their comparison with H′, the simulated distributions beginning from the simple cluster arrangements through two-dimensional Sierpiński carpets up to the population of interacting particles (chosen from Van Siclen's paper [18]), are presented in Section 3. While for the random compact aggregates of particles and weakly ramified clusters of the interacting particles, the length scale at which the first well shaped peak of SΔ appears corresponds to the first maximum in H′ indicating for clustering of the objects, the sequential SΔ peaks of various heights for Sierpiński carpets and partially for the population of interacting particles, contain more intricate information in comparison with smoothly shaped H′. In the final section we make concluding remarks and indicate some open problems.

Section snippets

Relative configurational entropy per cell

Let a binary image of size L×L in pixels be treated as a set of n indistinguishable finite-sized objects, that is black pixels of size 1×1 representing “particles” of a system and randomly distributed in χ numbered lattice cells of size k×k. The image area equals to the total number χ0χk2 of the unit cells of size 1×1. For the nontrivial binary images the particle number n satisfies the inequality 0<n<χ0 and hence, the particle fraction ϕn/χ0 holds the relation 0<ϕ<1. For each length scale k≡(

Numerical examples

Here, we will consider a few simple patterns to show the entropic measure sensitivity to clustering processes, also to gain insight into the spatial disorder of some fractals and system of interacting particles. As the first example, let us show in Figs. 1a and b two specific patterns of n=180 one-pixel particles grouped into 15 and 5 clusters in linear size L=25 square grid. Each of the bigger clusters is composed of three smaller ones. In Fig. 1c the values of SΔ(k) corresponding to the

Concluding remarks

The present work shows that the relative configurational entropy per cell SΔ(k) given by (4) can be considered as alternative, qualitatively correct and highly sensitive measure of spatial disorder at every length scale for systems of finite-sized objects. For the Sierpiński carpets and the population of interacting particle, our measure compared with the normalized information entropy H′(k) reveals additional distinctive features of the complex microstructures. On the basis of results for the

Acknowledgements

I thank Clinton DeW. Van Siclen for helpful correspondence.

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