Information geometric methods for complexity.
Research on the use of information geometry (IG), the application of differential geometric techniques to probability calculus, in modern physics has witnessed significant advances recently. In this review article, we report on the employment of IG methods to characterize the concepts of complexity in both classical and, whenever available, quantum physical settings. A paradigmatic example of a major change in complexity is given by the phase transition (PT). Hence we review both global and local aspects of PTs described in terms of the scalar curvature of the parameter manifold and the components of the metric tensor, respectively. We also report on the behavior of geodesic paths on the parameter manifold used to gain insight into dynamics of PTs. Going further on we survey measures of complexity arising in the geometric framework. Actually we quantify complexity of networks in terms of the Riemannian volume of parameter space of a statistical manifold associated with a given network. We deal with complexity measures that accounts for the interactions of a given number of parts of a system that cannot be described in terms of a smaller number of parts of the system itself and complexity measures of entropic motion on curved statistical manifolds that arise from a probabilistic description of physical systems in the presence of limited information. The Kullback-Leibler divergence, the distance to an exponential family and volumes of curved parameter manifolds, are examples of essential IG notions exploited in our discussion on complexity. We conclude discussing strengths, limits, and possible future applications of IG methods to the physics of complexity.
Publisher URL: http://arxiv.org/abs/1801.03026
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