Entropy and Geometric Objects

Abstract

Different notions of entropy can be identified in different scientific communities: (i) the thermodynamic sense; (ii) the information sense; (iii) the statistical sense; (iv) the disorder sense; and (v) the homogeneity sense. Especially the "disorder sense" and the "homogeneity sense" relate to and require the notion of space and time. One of the few prominent examples relating entropy to both geometry and space is the Bekenstein-Hawking entropy of a Black Hole. Although this was developed for describing a physical object-a black hole-having a mass, a momentum, a temperature, an electrical charge, etc., absolutely no information about this object's attributes can ultimately be found in the final formulation. In contrast, the Bekenstein-Hawking entropy in its dimensionless form is a positive quantity only comprising geometric attributes such as an area A-the area of the event horizon of the black hole, a length L P-the Planck length, and a factor 1/4. A purely geometric approach to this formulation will be presented here. The approach is based on a continuous 3D extension of the Heaviside function which draws on the phase-field concept of diffuse interfaces. Entropy enters into the local and statistical description of contrast or gradient distributions in the transition region of the extended Heaviside function definition. The structure of the Bekenstein-Hawking formulation is ultimately derived for a geometric sphere based solely on geometric-statistical considerations.

Geometric Entropy of Plant Leaves: A Measure of Morphological Complexity

“…Like any physical object, a Plant leaf has a mass, momentum, temperature, and electric charge. We attempt to describe the geometric entropy of an individual plant leaf by the mathematical information of physical objects gathered from the formulation of the Bekenstein-Hawking entropy (Schmitz 2018). The formulation is purely geometrical and is devoid of any of the attributes of physical objects.…”

“…Information and entropy unify the notion of geometry and energy in all biological systems (Crofts 2007;Jost 2022). The notion of entropy in the 'disorder' sense (Altieri et al 2019) relates to the perception of entropy in space and time until the dimensionless form of Bekenstein -Hawking entropy is constructed using a phase-field approach (Schmitz 2018). In this paper, we consider plant leaves to be made up of 2-D circular elements (cells) and derive their geometric entropy through mathematical formulations based on Schmitz's geometric approach (Schmitz 2018).…”

Thermodynamic Characterization of Planar Shapes and Curves, and the Query of Temperature in Black Holes

“…The Kossel-Jackson-Temkin models of crystal growth are quite often referred to in the literature. The Kossel-Temkin entropy introduces a length scale into evolving thermodynamics of diffuse interfaces [9] [28] [29] [30]. The entropy of the crystal boundary can be simply calculated from the already formed surface where every step corresponds to a new state.…”

“…The shape entropy becomes important when the intrinsic interactions start to dominate at moderate densities [8]. The entropy of shapes revealed as the interface between two phases can be determined by using discrete formulation but it can be extended to continuous formulation [9]. The fragmentation entropy of spiral tilings can also be calculated by the same method [10].…”

A Phase-Field Perspective on Mereotopology

“…Nothing is a priori known about the exact "shape" of the phase field function in the transition region. Reasoning towards a specification of this shape is based on statistical distributions of gradients in the interface and is described in [68] and [69]. In spite of not knowing this exact shape, a number of terms/expressions can already be qualitatively identified, Table 1, which all allow the identification and description of the transition region (expressions (5), (6), (7 :"overlap"), (8), ( 13) and ( 14) in table 1), Figure 4.…”