What is a physical measure of spatial inhomogeneity comparable to the mathematical approach?

Abstract

Abstract.A linear transformation f(S) of configurational entropy with length scale dependent coefficients as a measure of spatial inhomogeneity is considered. When a final pattern is formed with periodically repeated initial arrangement of point objects the value of the measure is conserved. This property allows for computation of the measure at every length scale. Its remarkable sensitivity to the deviation (per cell) from a possible maximally uniform object distribution for the length scale considered is comparable to behaviour of strictly mathematical measure h introduced by Garncarek et al. in [2]. Computer generated object distributions reveal a correlation between the two measures at a given length scale for all configurations as well as at all length scales for a given configuration. Some examples of complementary behaviour of the two measures are indicated.PACS. 05.90.+m Other topics in statistical physics, thermodynamics and nonlinear dynamical systems -68.55.-a Thin film structure and morphology

A generalization of the inhomogeneity measure for point distributions to the case of finite size objects

“…This weakness has been recently overcome by introducing entropic average measure (per cell) of spatial inhomogeneity S ∆ (k) ≡ (S max − S)/χ, see Refs. [5,6]. Here S max and S describe respectively the highest possible configuration entropy and the real one for a given pattern; see also Refs.…”

“…In connexion with Ref. [6], the present work is motivated by its complementary function to the paper [5], where the PO-versions of the two measures have been already compared.…”

Inhomogeneity and complexity measures for spatial patterns

“…The approach of Refs. [10,11] can be regarded as exact for investigating the degree of inhomogeneity in a phase space. When the entropic measures are used for the same binary image, that is, (1)-pixel = ÿnite-sized object, exact measure, and (2)-pixel ∼ = point marker as a punctual object, approximated measure that is, however, qualitatively correct, a simple relation between them can be established (see Eq.…”

“…In dealing with the PO-measure [10] one proceeds as follows: (i) regards black pixels as punctual objects (n i per cell), while (ii) white pixels are treated as a continuous medium.…”

Coarsened Lattice Spatial Disorder in the Thermodynamic Limit

“…Recently, the normalized information entropy, where the initial entropy for the perfectly random system is subtracted from the actual entropy calculated for the real particle configuration has been considered [3]. The different idea of using a linear transformation of configurational entropy as a measure of the degree of inhomogeneity of the spatial distribution, mentioned in [4], was applied to point [5] and finite-sized [6] objects.To investigate how it reveals an information concerning the changes of spatial disorder in the thermodynamic limit first we briefly recall the two formulas of [6]. When a binary image of size…”