The mixing distance

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 theorem 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.