Mixed-Up-Ness or 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Mixed-up-ness can be traced to unpublished notes by Josiah Gibbs.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.

Testing the Minimum System Entropy and the Quantum of Entropy

“…The second and third laws of thermodynamics concern the state variable entropy directly. The concept of entropy is best defined and thought of as disordered energy [155] or as the mixing of states [162]. The quantum of entropy includes and implies the definition of entropy.…”

“…Simultaneously, the quantum of entropy includes the particle structure of matter and radiation. Using the arguments summarized in references [155,162], the quantum of entropy implies the second law.…”