Constructal Theory Of Parabolic Scaling
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The parabolic scaling of rank-size distributions is a very common phenomenon in natural or societal systems such as city sizes, petroleum reservoirs, or galactic intensities. Often, the divergence from a log-log linear power-law is explained by the finite size of the system studied. Several distribution functions were also proposed to best fit this kind of distribution, such as the parabolic fractal or the stretched exponential. In order to explain the emergence of this rank-size distribution pattern, we propose instead to consider a generic mechanism of constructal tree-shaped invasion of a territory combined with the constructal rank-size distribution of a growing flow architecture as the mechanism potentially generating a wide variety of parabolically scaling distributions. Simulations of this mechanism were conducted and we showed that it generated shorter and curved distribution tails similar to a parabolic scaling. In conclusion, we propose to consider the constructal law as the first principle behind the generation of parabolic scaling of rank-size distributions in natural, societal, and engineered systems. Keywords: constructal law, flow architecture, parabolic scaling, power-law, rank-size distribution, S-curve, Zipf’s law. 1 Introduction Power-laws and scale-free system architectures are very common in natural or social systems. These phenomena have attracted a strong interested coming from the scientific community, in order to understand and explain the emergence of the scale-free properties of systems. A frequently encountered property concerns the scale free distribution of the size of the constitutive elements of the system studied, when sorted by considering the rank of these elements relative to their size. This subject is of uttermost interest and some researchers  consider that
constructal law, flow architecture, parabolic scaling, power-law, rank-size distribution, S-curve, Zipf’s law.