Gibbs Paradox and the Concepts of Information, Symmetry, Similarity and Their Relationship

Abstract

We are publishing volume 10 of Entropy. When I was a chemistry student I was facinated by thermodynamic problems, particularly the Gibbs paradox. It has now been more than 10 years since I actively published on this topic [1][2][3][4]. During this decade, the globalized Information Society has been developing very quickly based on the Internet and the term "information" is widely used, but what is information? What is its relationship with entropy and other concepts like symmetry, distinguishability and stability? What is the situation of entropy research in general? As the Editor-in-Chief of Entropy, I feel it is time to offer some comments, present my own opinions in this matter and point out a major flaw in related studies. Definition of InformationWe are interested in the definition of information in the context of information theory. It is a surprise that a clear definition of the concept of "information" cannot be found in information theory textbooks. "Entropy as a measure of information" is confusing. I would like to propose a simple definition of information:Information ( I ) is the amount of the data after data compression.If the total amount of data is L, entropy ( S ) in information theory is defined as information loss, L S I = + . Let us consider a 100GB hard disk as an example: 100GB L =. A formatted hard disk will have 100GB S = and I = 0. Similar examples for defining information as the amount of data after compression are given in [5]. Based on this definition of information and the definition that (information theory) entropy is expressed as information loss, S L I = − , or in certain cases when the absolute values are unknown, S L I Δ = Δ − Δ , I was able to propose three laws of information theory [5]:

Distinguishability in Entropy Calculations: Chemical Reactions, Conformational and Residual Entropy

“…Pauli noticed this incompleteness and showed what additional condition must be imposed in order to define an extensive entropy, suggesting that entropy, as defined by Clausius, is not intrinsically an extensive property [4,9]. In any case, the extensivity and the indistinguishability are concepts closely connected to each other in the context of the Gibbs arguments, which have been supported and rejected more than once in an ongoing debate [3,4,7,10,11].…”

“…The solution of the so-called Gibbs paradox [3][4][5][6][7][8], is probably the most famous example where the same kind of correction has been applied. Gibbs proposed an ad hoc reduction in the entropy of an N -particle system by the amount −k B ln N !, where k B is Boltzmann's constant.…”

Study on Entropy and Similarity Measure for Fuzzy Set

“…In contrast, similarity measures based on the distance measure are applicable to general fuzzy membership functions, including nonconvex fuzzy membership functions [13], [14]. The correlation between entropy and similarity for fuzzy sets has been presented from various viewpoints [15]. Liu also proposed a relation between distance and similarity measures; in his paper, the sum of distance and similarity constitutes the total information [4].…”

Symmetry – An International and Interdisciplinary Scientific Open Access Journal

“…Symmetry related topics have been always my research interest [2][3][4]. In the journal Entropy [5], several symmetry related papers and several book reviews on symmetry related books [6][7][8] have been published and a special issue "Symmetry and Entropy" was published [9] with Professor Joe Rosen [10,11] serving as the Guest Editor.…”