On an extension of Krivovichev's complexity measures

Abstract

<jats:p>An extension is proposed of the Shannon entropy-based structural complexity measure introduced by Krivovichev, taking into account the geometric coordinational degrees of freedom a crystal structure has. This allows a discrimination to be made between crystal structures which share the same number of atoms in their reduced cells, yet differ in the number of their free parameters with respect to their fractional atomic coordinates. The strong additivity property of the Shannon entropy is used to shed light on the complexity measure of Krivovichev and how it gains complexity contributions due to single Wyckoff positions. Using the same property allows for combining the proposed coordinational complexity measure with Krivovichev's combinatorial one to give a unique quantitative descriptor of a crystal structure's configurational complexity. An additional contribution of chemical degrees of freedom is discussed, yielding an even more refined scheme of complexity measures which can be obtained from a crystal structure's description: the six C's of complexity.</jats:p>

crystIT: complexity and configurational entropy of crystal structures via information theory

“…This measure of information essentially contains the information on coordinates that must be defined for the complete description of a crystallographic orbit when its Wyckoff position is known. In order to maintain consistency, Hornfeck subsequently renamed Krivovichev's information content I G as 'combinatorial' complexity, I comb :¼ I G , and defined configurational complexity I conf as the strong additive sum of coordinational and combinatorial complexities [see Hornfeck (2020) for the mathematical background]. Using our updated measure for combinatorial complexity (i.e.…”

“…Subsequently, he proposed a concise concept that has the potential to capture the full multifaceted challenges of defining the complexity of a crystal structure on the basis of the information content. Recently, Hornfeck (2020) has suggested a few improvements of the concept, emphasizing the importance of theory development and the current state of research in this relatively young area. Importantly, comparisons between Shannon entropy, crystal structure complexity and configurational entropy can be drawn, opening intriguing opportunities for the assessment of the configurational entropy of crystal structures and its change during phase transitions (Krivovichev, 2016).…”

crystIT: Complexity and Configurational Entropy of Crystal Structures via Information Theory

“…In order to maintain consistency, W. Hornfeck subsequently renamed S. Krivovichev's information content I G as combinatorial complexity, I G = I comb and defined configurational complexity I con f as the strong additive sum of coordinational and combinatorial complexities, see (Hornfeck, 2020) for the mathematical background. Using our updated measure for combinatorial complexity (i.e.…”

“…Subsequently he proposed a concise concept that has the potential to capture the full multifaceted challenges of defining the complexity of a crystal structure based on the information content. Recently W. Hornfeck (2020) has suggested a few improvements of the concept, emphasizing the importance of theory development and the current state of research in this relatively young area. Importantly, comparisons between Shannon entropy, crystal structure complexity and configurational entropy can be drawn, opening intriguing opportunities for the assessment of configurational entropy of crystal structures and its change during phase transitions (Krivovichev, 2016).…”

Embedding parallelohedra into primitive cubic networks and structural automata description

“…The SA approach provides a relatively easy and transparent tool to model the growth of orthogonal nets and to estimate their complexities by counting the number of states in the corresponding DFA. This measure may be viewed as reflecting an algorithmic (dynamic) complexity in contrast to informational (static) complexity, which is based on the application of the Shannon information theory (Krivovichev, 2012(Krivovichev, , 2014bHornfeck, 2020).…”