To characterize strongly interacting statistical systems within a thermodynamical frameworkcomplex systems in particular -it might be necessary to introduce generalized entropies, Sg. A series of such entropies have been proposed in the past, mainly to accommodate important empirical distribution functions to a maximum ignorance principle. Until now the understanding of the fundamental origin of these entropies and its deeper relations to complex systems is limited. Here we explore this questions from first principles. We start by observing that the 4th Khinchin axiom (separability axiom) is violated by strongly interacting systems in general and ask about the consequences of violating the 4th axiom while assuming the first three Khinchin axioms (K1-K3) to hold and Sg = i g(pi). We prove by simple scaling arguments that under these requirements each statistical system is uniquely characterized by a distinct pair of scaling exponents (c, d) in the large size limit. The exponents define equivalence classes for all interacting and non interacting systems. This allows to derive a unique entropy, S c,d ∝ i Γ(d + 1, 1 − c ln pi), which covers all entropies which respect K1-K3 and can be written as Sg = i g(pi). Known entropies can now be classified within these equivalence classes. 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 is, to our knowledge, the first ab initio justification for the existence of generalized entropies. Even though here we assume Sg = i g(pi), we show that more general entropic forms can be classified along the same lines.Weakly interacting statistical systems can be perfectly described by thermodynamics -provided the number of states W in the system is large. Complex systems in contrast, characterized by long-range and strong interactions, can fundamentally change their macroscopic qualitative properties as a function of the number of states or the degrees of freedom. This leads to the extremely rich behavior of complex systems when compared to simple ones, such as gases. The need for understanding the macroscopic properties of such interacting systems on the basis of a few measurable quantities only, is reflected in the hope that a thermodynamic approach can also be established for interacting systems. In particular it is hoped that appropriate entropic forms can be found for specific systems at hand, which under the assumption of maximum ignorance, could explain sufficiently stationary macro states of these systems. In this context a series of entropies have been suggested over the past decades, [1-6] and Table 1. So far the origin of such entropies has not been fully understood within a general framework .Here we propose a general classification scheme of both interacting (complex) and non-or weakly-interacting statistical systems in terms of their asymptotic behavior under changes of the number of degrees of freedom of ...

This book is a comprehensive introduction to quantitative approaches to complex adaptive systems. Practically all areas of life on this planet are constantly confronted with complex systems, be it ecosystems, societies, traffic, financial markets, opinion formation, epidemic spreading, or the internet and social media. Complex systems are systems composed of many elements that interact with each other, which makes them extremely rich dynamical systems showing a huge range of phenomena. Properties of complex systems that are of particular importance are their efficiency, robustness, resilience, and proneness to collapse. The quantitative tools and concepts needed to understand the co-evolutionary nature of networked systems and their properties are challenging. The intention of the book is to give a self-contained introduction to these concepts so that the reader will be equipped with a conceptual and mathematical toolset that allows her to engage in the science of complex systems. Topics covered include random processes of path-dependent processes, co-evolutionary dynamics, the statistics of driven nonequilibrium systems, dynamics of networks, the theory of scaling, and approaches from statistical mechanics and information theory. The book extends well beyond the early classical literature in the field of complex systems and summarizes the methodological progress over the past twenty years in a clear, structured, and comprehensive way. The book is intended for natural scientists and graduate students.

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During the passage of Voyager 1 through the Saturn system, the infrared instrument acquired spectral and radiometric data on Saturn, the rings, and Titan and other satellites. Infrared spectra of Saturn indicate the presence of H(2), CH(4), NH(3), PH(3), C(2)H(2), C(2)H(6), and possibly C(3)H(4) and C(3)H(8). A hydrogen mole fraction of 0.94 is inferred with an uncertainty of a few percent, implying a depletion of helium in the atmosphere of Saturn relative to that of Jupiter. The atmospheric thermal structure of Saturn shows hemisphere asymmetries that are consistent with a response to the seasonally varying insolation. Extensive small-scale latitudinal structure is also observed. On Titan, positive identifications of infrared spectral features are made for CH(4), C(2)H(2), C(2)H(4), C(2)H(6), and HCN; tentative identifications are made for C(3)H(4) and C(3)H(8). The infrared continuum opacity on Titan appears to be quite small between 500 and 600 cm(-1), implying that the solid surface is a major contributor to the observed emission over this spectral range; between 500 and 200 cm(-1) theopacity increases with decreasing wave number, attaining an optical thickness in excess of 2 at 200 cm(-1). Temperatures near the 1-millibar level are independent of longitude and local time but show a decrease of approximately 20 K between the equator and north pole, which suggests a seasonally dependent cyclostrophic zonal flow in the stratosphere of approximately 100 meters per second. Measurements of the C ring of Saturn yield a temperature of 85 +/- 1 K and an infrared optical depth of 0.09 +/- 0.01. Radiometer observations of sunlight transmitted through the ring system indicate an optical depth of 10(-1.3 +/-0.3) for the Cassini division. A phase integral of 1.02 +/- 0.06 is inferred for Rhea, which agrees with values for other icy bodies in the solar system. Rhea eclipse observations indicate the presence of surface materials with both high and low thermal inertias, the former most likely a blocky component and the latter a frost.

We show how the dependence of phase space volume Ω(N ) of a classical system on its size N uniquely determines its extensive entropy. We give a concise criterion when this entropy is not of Boltzmann-Gibbs type but has to assume a generalized (non-additive) form. We show that generalized entropies can only exist when the dynamically (statistically) relevant fraction of degrees of freedom in the system vanishes in the thermodynamic limit. These are systems where the bulk of the degrees of freedom is frozen and is practically statistically inactive. Systems governed by generalized entropies are therefore systems whose phase space volume effectively collapses to a lower-dimensional 'surface'. We explicitly illustrate the situation for binomial processes and argue that generalized entropies could 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.