Research Division Seminar
The LMC Measures of Complexity: A 30 Years Story with (Some) Roots at IAC and Many Facets, Including Astronomical Ones

Abstract

The LMC measure, proposed in 1995 by López-Ruiz, Mancini and Calbet, assigns a quantitative amount of complexity to a discrete probability distribution, or to a continuous probability density. The main idea motivating the LMC measure is the basic intuition, shared by most researchers studying complexity in natural phenomena, according to which systems exhibiting a large amount of order, or a large amount of disorder, have low or vanishing complexity. Consequently, the LMC measure is constructed in such a way that it vanishes in these two extreme cases, and adopts its maximum value for some intermediate regime. The above mentioned “boundary” restrictions, however, are not enough to determine a unique measure of statistical complexity. In fact, researchers have introduced several statistical measures of complexity, akin to the original LMC one, defined as products of information or entropic-like quantities, that also comply with the aforementioned requirements. These measures, which we collectively refer to as “LMC- measures”, have been applied by scientists to the study of diverse problems in Physics and other fields, leading to a research literature of respectable size. In spite of the intriguing results yielded by those investigations, various fundamental issues regarding the LMC measures remain unexplored. In particular, relatively little attention has been paid to identify, and investigate, the kind of dynamical processes governing time-dependent probability densities, that evolve towards densities optimizing the LMC measures, under suitable constraints. Given the remarkable amount of phenomenological applications of the LMC measures to diverse fields, including Astronomy, that have been developed by researchers over the years, it seems timely, thirty years after the LMC proposal, to re-consider some of its conceptual foundations. Our aim is to re-visit these issues, paying special attention to the possible dynamical origins of densities optimizing the LMC measures. In particular, we shall explore the main properties of a family of evolution equations satisfying an H-like theorem based on the LMC measures. We shall present these considerations within the context of other fundamental open questions concerning the LMC measures.