Possible dynamical paths towards the constrained optimization, and other fundamental aspects, of the LMC family of statistical measures of complexity

The celebrated LMC measure of complexity, advanced by Lopez-Ruiz, Mancini and Calbet thirty years ago, is based on the idea that scenarios ´ exhibiting large amounts of order, or large amounts of disorder, are characterized by low or vanishing amounts of complexity. According to this idea, complexity adopts its maximum value at some intermediate regime between extreme order and extreme disorder. Following on the LMC steps, researchers have introduced several other statistical measures of complexity, akin to the original LMC one, that also comply with the aforementioned requirements. These measures, which we collectively refer to as “LMC-measures”, are defined as products of information or entropic-like quantities. The LMC measures have been applied by scientists to the study of diverse systems or processes in physics, chemistry, and other fields, leading to a research literature of respectable size. In spite of the intriguing results yielded by those investigations, various fundamental issues concerning the LMC measures remain unaddressed. It seems timely, thirty years after the original LMC proposal, to reconsider its foundations. We shall discuss various basic aspects of the LMC measures, including some exploratory steps regarding possible dynamical mechanisms leading to probability densities optimizing the measures under suitable constraints.