Entropic and complexity measures in atomic and molecular systems

Abstract

Both entropy and complexity are central concepts for the understanding and development of Information Theory, playing an essential role in the increasingly numerous applications in a huge diversity of fields. Adequate methods of quantifying the entropy of physical systems and processes have been the object of design of numerous researchers for more than a century, usually based on a probabilistic description using discrete or continuous variables depending on the analysis framework considered. In this sense, Shannon entropy deserves a special mention as a pioneering measure; this has also given rise to the subsequent birth of, e.g., the Rényi and Tsallis entropies, or the relative entropy between distributions in close connection with the ‘mutual information’ among descriptive variables. The quantitative handle of the notion ‘complexity of a system’ (or process) arose much more recently, in the mid-1990s. The pioneering measure of complexity (LMC, in accordance with the names of its authors) includes the Shannon entropy, together with the distance to equiprobability or ‘disequilibrium’. Among the representative properties for any appropriate measure of complexity, it reaches its minimal value in the limits of extremal order and disorder. Complementary to LMC complexity, additional product-like complexity measures have been subsequently defined. Let us emphasize recent generalizations in terms of Rényi entropies, as well as those enclosing the content of gradient of the distribution as a local property to determine the searched complexity.