Mixed-Up-Ness or Entropy?

SIMPLE SUMMARY: The second law of thermodynamics has a mystical appeal in disciplines with tenuous connections to its origins. We hypothesize that many of these appeals instead should be to another principle heretofore unrecognized: the law of mixed-up-ness (LOM). Instead of using a number such as entropy to characterize randomness, non-thermodynamic systems can be arranged in simple diagrams according to their degree of mixed-up-ness. Curiously, the evolution of such systems from degrees of low to high mixed-up-ness is both consistent with and richer than the principle of increasing entropy. ABSTRACT: Mixed-up-ness can be traced to unpublished notes by Josiah Gibbs. Subsequently, the concept was developed independently, and under somewhat different names, by other investigators. The central idea of mixed-up-ness is that systems states can be organized in a hierarchy by their degree of mixed-up-ness. In its purest form, the organizing principle is independent of thermodynamic and statistical mechanics principles, nor does it imply irreversibility. Yet, Gibbs and subsequent investigators kept entropy as the essential concept in determining system evolution, thus retaining the notion that systems evolve from states of perfect “order” to states of total “disorder”. Nevertheless, increasing mixed-up-ness is consistent with increasing entropy; however, there is no unique one-to-one connection between the two. We illustrate the notion of mixed-up-ness with an application to the permutation function of integer partitions and then formalize the notion of mixed-up-ness as a fundamental hierarchal principle, the law of mixed-up-ness (LOM), for non-thermodynamic systems.