The mixing distance

Abstract

Mixing character denotes a property concerning the composition of sets. It was first used in order to present criteria for ligand systems in molecules that assure the existence of chiral isomers. Since then the concept of mixing character has been found to have further fields of applications in physics and pure mathematics. Various theorems have been given in the meantime that are reviewed here mostly in the form of equivalent definitions. The mixing distance is another. still more general concept suggested by a study of the time dependent behaviour of nonisolated thermodynamical systems. It describes a dissimilarity of probability distributions, the decrease of which characterizes irreversible phenomena. In the present paper a variety of equivalent definitions and theorems concerning the mixing distance are presented that are of use for further applications. Moreover, a theorem is deduced which is a generalization of a fundamental theprem by Hardy, Littlewood, and Polya on inequalities. Applications of the results given in the present paper will be explored in a subsequent publication that is mostly concerned with the concept of statistical information.

Beyond heat baths II: framework for generalized thermodynamic resource theories

“…The curves have been introduced into Helmholtz theories [9], the entropy theory [13], and grand-potential theories [15]. These resource-theory applications have roots in works by Ruch, Schranner, Seligman, and others (e.g., [44]). Extant results are generalized concisely below.…”

Beyond heat baths: Generalized resource theories for small-scale thermodynamics

“…Having defined the f divergences and the rescaled Lorenz curve, we state their relationship with equimajorization. Ruch, Schranner, and Seligman first proved this relationship for continuous systems [43], using tools from measure theory. Uhlmann proved the relationship more directly, for discrete systems, which we address [45].…”

Generalizations of the Entropy Concept: Principles, Limitations, and Prospects for Applications

“…The question of the validity of the 17-theorem, and some further applications of it, have been discussed by Schlogl and the a~t h o r .~ I Some properties of the mixing distance, particularly for the case of continuous probability distributions, have been considered by Ruch, Schranner, and Seligman. 32 The relevance of mixing character and distance to information theory has been further studied by Ruch and L e~c h e .~~ Regarding some of their conclusions, the author34 has expressed some misgivings, whch, however, are not shared by Ruch and L e~c h e .~~ Despite the existence of other measures of disorder, the entropy still possesses some properties that are unique,36 and indeed this fact plays an essential role in the application of mixing character itself to actual system^.^' All of this will be elaborated on in the chapters that follow.…”