Comparison of SDL and LMC measures of complexity: Atoms as a testbed

Abstract

The "simple" measure of complexity of Shiner, Davison and Landsberg (SDL) and the "statistical" one, according to Lopez-Ruiz, Mancini and Calbet (LMC), are compared in atoms as functions of the atomic number Z. Shell effects i.e. local minima at the closed shells atoms are observed, as well as certain qualitative trends of SDL and LMC measueres of complexity. If we impose the condition that SDL and LMC behave similarly as functions of Z, then we can conclude that complexity increases with Z and for atoms the strength of disorder is zero and order is four.Comment: 14 pages, 2 figures, revised version, accepted for publiaction in Physics Letters

Statistical measure of complexity and correlated behavior of Fermi systems

“…Figure 4: S cor , D cor and C of electron gas the correlation parameter r s . The lines correspond to the expressions (31), (32), (33), with the parameters derived by the least squares fit method. (40), (41), with the parameters derived by the least squares fit method.…”

“…e S and D [50]. The usefulness of the improved version has been shown in many fields [29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45]. Moreover, the specific measure is suitably tailored for quantum systems, described by their very nature probabilistically via density distributions in position and momentum spaces, which are necessary for and enable a relatively easy calculation of S and D, entering the formulas C LMC = SD or C = e S D. The motivation of the present work, in the spirit of the above statements, is to extend our previous study of uniform Fermi systems [16], beyond information entropy, in order to include the complexity measure proposed by López-Ruiz et al [22], using probability distributions in momentum space.…”

“…Information-theoretical methods play an important role, not just in the clarification of fundamental concepts of quantum mechanics, but also provide a series of results concerning the information content of systems, the presence of interactions, correlation with experimental measured quantities, extraction of universal relations etc [2,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]. Additionally, many complexity measures have been proposed as indicators of complex behavior found in different systems scattered in a broad spectrum of fields [22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48].…”

“…The first investigation of C LM C in quantum many-body systems was carried out in atoms, for continuous electron distributions [33] and discrete ones [34]. An alternative definition of complexity is the SDL measure Γ αβ [26] (Shiner, Davison, Landsberg), defined and calculated in an analogous way as the LMC one.…”

“…It has been applied in atoms as well, starting from [32]. Comments on the validity of SDL and LMC measures are given in detail in Section 4 of [33], where it is stated that a welcome property of a definition of complexity might be the following: If one complicates the system by varying some of its parameters, and this leads to an increase of the adopted measure of complexity, then one could argue that this measure describes the complexity of the system properly.…”