Entropies for complex systems: generalized-generalized entropies

Complex Systems are ubiquitous in nature and man-made systems. In natural sciences, in social and economic models and in mathematical constructions are studied and analyzed, are applied in practical problems but without a clear and universal definition of "complexity", let alone classification and quantification. Following the "three-level scheme" of physical theories, observations/experiments, phenomenology, microscopic interactions, we need, starting from the experience of observation to establish appropriate phenomenological parameters and concepts, and in conjunction with a possible knowledge of the nature of microscopic structures to deepen our understanding of a particular system which we "understand as complex". Information Geometry seems to be a useful phenomenological framework, which using generalized entropies, provides some classification and quantification tools. But we need the next level, microscopic structure and interactions of the parts of complex systems. A useful direction is the conceptual niche of hyper-networks and super graphs, where a strong involvement of algebra offers concrete techniques. We believe that appropriate algebraic structures may systematize our approach to microscopic structures of complex systems, and help associate the information geometric phenomenology with concrete properties. In this paper after a short discussion of the problem of "definition of complexity", we introduce our information geometric quantities derived from generalized entropies. Then we present our results of application of information geometry for classification of complex systems. Finally we present our ideas for an abstract algebraic approach which may offer a framework for the microscopic study of complex systems.

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THE ENTROPY OF NON-ERGODIC COMPLEX SYSTEMS — A DERIVATION FROM FIRST PRINCIPLES

International Journal of Modern Physics Conference Series ◽

10.1142/s2010194512007817 ◽

2012 ◽

Vol 16 ◽

pp. 105-115 ◽

Cited By ~ 3

Author(s):

STEFAN THURNER ◽

RUDOLF HANEL

Keyword(s):

Complex Systems ◽

Degrees Of Freedom ◽

Power Laws ◽

Distribution Functions ◽

Log P ◽

Critical Systems ◽

Scaling Exponents ◽

Special Cases ◽

Statistical Systems ◽

Generalized Entropies

In information theory the 4 Shannon-Khinchin1,2 (SK) axioms determine Boltzmann Gibbs entropy, S ~ -∑i pi log pi, as the unique entropy. Physics is different from information in the sense that physical systems can be non-ergodic or non-Markovian. To characterize such strongly interacting, statistical systems – complex systems in particular – within a thermodynamical framework it might be necessary to introduce generalized entropies. A series of such entropies have been proposed in the past decades. Until now the understanding of their fundamental origin and their deeper relations to complex systems remains unclear. To clarify the situation we note that non-ergodicity explicitly violates the fourth SK axiom. We show that by relaxing this axiom the entropy generalizes to, S ~∑i Γ(d + 1, 1 - c log pi), where Γ is the incomplete Gamma function, and c and d are scaling exponents. All recently proposed entropies compatible with the first 3 SK axioms appear to be special cases. We prove that each statistical system is uniquely characterized by the pair of the two scaling exponents (c, d), which defines equivalence classes for all systems. The corresponding distribution functions are special forms of Lambert-W exponentials containing, as special cases, Boltzmann, stretched exponential and Tsallis distributions (power-laws) – all widely abundant in nature. This derivation is the first ab initio justification for generalized entropies. We next show how the phasespace volume of a system is related to its generalized entropy, and provide a concise criterion when it is not of Boltzmann-Gibbs type but assumes a generalized form. We show that generalized entropies only become relevant when the dynamically (statistically) relevant fraction of degrees of freedom in a system vanishes in the thermodynamic limit. These are systems where the bulk of the degrees of freedom is frozen. Systems governed by generalized entropies are therefore systems whose phasespace volume effectively collapses to a lower-dimensional 'surface'. We explicitly illustrate the situation for accelerating random walks, and a spin system on a constant-conectancy network. We argue that generalized entropies should be relevant for self-organized critical systems such as sand piles, for spin systems which form meta-structures such as vortices, domains, instantons, etc., and for problems associated with anomalous diffusion.

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