Abstract
In this paper we recall for physicists how it is possible, using the principle of maximization of the Boltzmann-Shannon entropy, to derive the Burr-Bingh-Maddala (burr12) double power law probability distribution function and its approximations (Pareto, loglogistic ..) and extension first used in econometrics. this is possible using a deformation of the power function, as this has been done in complex systems for the exponential function. We give to that distribution a deep stochastic interpretation using the theory of Weron et al. applied to thermodynamics the entropy nonextensivity can be accounted for by assuming that the asymptotic exponents are scale dependent. Therefore functions which describe phenomena presenting power-law asymptotic behaviour can be obtained without introducing exotic forms of the entropy.
Highlights
We want to show how the BurrrXII-Singh-Maddala (BSM) [1] [2] distribution function, known as the q-Weibull distribution can be naturally derived from the maximum entropy principle using the Boltzmann-Shannon entropy with well-defined constraints including a generalization of the definition of the moment [3]-[5] similar to the deformation of the exponential
The BSM function has been used to establish a three parameters fractal kinetic equation which has been employed with some success to characterize macroscopically the sorption in gaseous and aqueous phase [11] [12] as well as in the theory of relaxation [22] [23] to justify the two asymptotic behaviours of the Havriliak-Negami formula [55] in the frequency range
We think that physicists have to learn a lot from the progress made the last decades by mathematicians and ex-physicists in the field of statistics in econometrics
Summary
We want to show how the BurrrXII-Singh-Maddala (BSM) [1] [2] distribution function, known as the q-Weibull distribution can be naturally derived from the maximum entropy principle using the Boltzmann-Shannon entropy with well-defined constraints including a generalization of the definition of the moment [3]-[5] similar to the deformation of the exponential As a consequence of the previous points we will show that phenomena described by function with one or two tails asymptotic power laws have not necessarily to be obtained by the maximization of an extension of the BS entropy This is has been known for a long time in fields outside of physics such as hydrology and econometrics
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