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Complexity Theory is an eclectic collection of theoretical approaches to a wide variety of nonlinear problems that typically involve many degrees of freedom. Despite numerous claims, there does not appear to be a universal basis for the various approaches. Here we report on recent attempts to provide such a basis. Our approach is based on the partial order of Boltzmann states under majorization and thus is grounded in the Second Law of Thermodynamics. However, here we do not appeal to any energetic constraints. By majorizing the Boltzmann states we identify a new statistical mechanical entity, namely a multivalued function that maps Boltzmann entropy to the size or order of sets of incomparable Boltzmann entropy states. We call this thermodynamic complexity. This is a concave function of entropy, peaking near mid-entropy and falling to zero at maximum and minimum entropies. It remains to be seen if this approach can be rigorously applied to other areas, but heuristic arguments given here indicate broad applicability.
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- The Rise and Fall of Thermodynamic Complexity and the Arrow of Time
A. D. Kirwan Jr.
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