Complexity by Henrik Jensen

Abstract for a Centre for Complexity Seminar

Complexity Science can be considered as the systematic study of emergence [1]. We take a pragmatic viewpoint and consider emergence to be the occurrence of properties, or phenomena, at the aggregate level, which the individual constituents do not possess. This viewpoint stresses that new emergent properties can only appear when components are interacting.  

Statistical mechanics has developed methods to quantify and derive emergent properties from a knowledge of the interaction between components. Perhaps the most spectacular example of this consists in the equilibrium theory of order parameters describing the macroscopic change from one phase to another. We will briefly recall this in the context of how emergent vortices in the 2dXY model are crucial for the understanding of the phase diagram and other important physical properties.  

This leads naturally to think about situations where the number of states of the system, W, isn’t given by a Cartesian product of states available to the independent constitutes. In such cases the Shannon-Boltzmann-Gibbs entropy is not extensive. We will discuss how extensive group entropies [2,3] can be defined for sub-exponential, exponential and super exponential dependencies of W on the number of constituents N. A new measure of essential emergence can be derived from the group entropies.  


[1] Henrik Jeldtoft Jensen, Complexity Science. The study of emergence, Cambridge Univesity Press, 2022.  

[2] Henrik Jeldtoft Jensen and Piergiulio Tempesta, Group Structure as a Foundation for Entropies. Entropy, 26, 266, (2024). 

[3]  P. Tempesta and H.J. Jensen, Universality Classes and Information-Theoretic Measures of Complexity Via Group Entropies.  Scientific Reports, 10:5952 (2020).  

Getting here