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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-Singh-Maddala (BurrXII) double power law probability distribution function and its approximations (Pareto, loglogistic.) and extension (GB2…) 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 <i>et al</i>. Applied to thermodynamics, the entropy <i>nonextensivity</i> 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.

In this paper, we want to show how the BurrrXII-Singh-Maddala (BSM) [

This paper has been written for several reasons.

The BSM distribution has been used in a variety apparently independent fields: econometrics [

There is a widespread belief in physics that the phenomena characterized by power laws behaviour, ubiquitous in nature, should require extensions of the Boltzmann-Shannon (BS) entropy and that the BS entropy should be restricted to the family of Gaussian and exponential laws i.e. the distributions which obey the classical central limit theorem. Among the more than 20 proposed generalized entropies, the most famous are the Reyni [

We show in details, as this has been noticed by previous authors before us, that this is not necessarily true. It is possible quite naturally to obtain power law distributions by generalizing the definition of moments or using constraints determined by the data observation and keeping the BS entropy as a starting point for the maximization procedure [

The BSM cumulative distribution functions are characterized by three parameters, two form factors a and c and one scale parameter b. The power law exponents corresponding to, respectively the asymptotic behaviour of small and large values of its argument are a and μ = a/c. The Tsallis distribution, derived from the maximization of Tsallis entropy is the survival part of the cumulative BSM distribution when a is equal to one. This density function has been used in the literature to fit many experimental data. These data can be fitted with the same degree of precision using both density distributions. This has created confusion on the value of the so-called entropic index q which is different in both cases.

It must be stressed that the BSM density function includes an exponent a which is absent from the Tsallis density function. The experimental observations, which dates back to the beginning of the last century [

However, we will show in Section 8 that nonextensivity of the entropy may arise from the scale dependence of the characteristic exponents.

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.

The BSM cumulative density function is written as:

where a and c are form factors and b is a scaling factor.

Its density function is easily obtained by differentiation

It is solution of a differential equation

The function

The function

The differential equation describes a birth and death process modulated by a quasi hyperbolic function which can be modified to accommodate more complex situations.

The cumulative distribution function

The survival part of the cumulative function is given by

which has the same form as the Tsallis density function if

index. By contrast the power of the BSM density function is

the so called “escort probability” in Tsallis formalism. The Lévy power law exponent

and this has to be considered in interpreting experimental results.

As the mathematician Vladimir Arnold once said “Differential equations are the source of the development of modern sciences”, one can consider Equation (3) as the natural starting point to study complex systems in the same way the exponential is solution of a differential equation and is the natural starting point for the description of simple systems. For instance, it has been used in epidemiology [

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 (ad-, chemi-, bio-) in gaseous and aqueous phase [

The work of K. Weron and its collaborators has given a deep physical understanding of this distribution. Indeed they have derived the BSM survival function in the theory of relaxation by means of stochastic arguments linking the observed macroscopic properties to the mesooscopic and microscopic energetic and geometric organisation (fractal scaling, clustering, self-organiszation) of complex heterogeneous systems. We will deal with this interpretation in the next paragraph.

The analytical form of the BSM distribution function can be justified and the parameters

The quantities

where

Two situations have to be considered: the expecting value of

where

It is the Weibull survival function quite frequent in physical, chemical and biological phenomena. The parameter a arises from the stable scaling properties at the micro and mesoscopic levels. When

The BSM distribution function can be obtained by considering a more complex situation where due to complex frustrations, the number of reacting element is not fixed and is itself a random variable. In that case Equation (8) should be replaced by

As argued by Weron et al. the fluctuations of

Then the survival probability function of the entire system is

The solution of which has been obtained by Rodriguez [

This is the BSM survival distribution function in the time domain. The corresponding density function is

In this derivation of the BSM function, the parameter c appears to be a measure of aggregation with one or several characteristic lengths.

Starting from the Boltzmann-Shannon entropy

We determine

And generalizing the definition of the power of a variable in the same spirit as the definition of the deformed exponential

The third constraint can then be expressed as:

The constraint conditions can be understood as known prior information which can be used to achieve a least biased distribution.

Using the method of Lagrange optimization method and defining

The maximization of S(x) is obtained by solving the equation:

With the normalization constant determined by the partition function

The values of the

one gets finally, given the constraints

We have therefore

with

This finally gives the BSM density function (Equation (2)).

The expression for the k-th moment is given by [

The Tsallis density function (a = 1) is the survival function of

with the constraints

These results have already been obtained with other notations in the field of econometrics and meteorology [

Another tail distribution, the Cauchy distribution

The Weibull and the Pareto and Cauchy tail distribution entropies are well known and have been derived in the classical books on entropy maximization method [

For c = 0, one recovers the well-known results for the Weibull distribution

with

For c = 1, the log-logistic-Hill-Fisk constraints

with

Finally it must be noted that if

Making use of the results of the previous section, one can now write the expressions of the entropy corresponding to the various approximations.

From the general definition

We get for the BSM distribution

for the Weibull distribution

For the log-logistic-Hill-Fisk distribution

and

The general form of the entropy is therefore

K(a, c) is a constant that we can consider as the origin of the entropy for a couple of values a and c. The previous results are therefore compatible with the extensivity of the BS entropy. If we have two subsystems 1 and 2 added to form a larger system with have:

This is only true if

The aim of this paper is to discuss the probabilistic and stochastic foundation of the use of the BSM distribution to describe physical and chemical complex systems characterized by power-law, Levy and extreme value distributions. We will nevertheless in the last section touch the problem of the application of this formalism to thermodynamics.

In the case (a = 1, c = 0, q = 1), we have

The corresponding canonical entropy is

In the case a = 1, which is the case used in extensions of the classical thermodynamics, one has

Therefore if

The usual canonical thermodynamics expression for the entropy (Equation (47)) can be recovered if one introduces a c-temperature assuming that

which gives a c-depending temperature depending on the evolution of

Nonextensivity observed in systems with long range interactions can be accounted for if the parameter c (and q) are scale dependent. One can use an argument used by C. Beck [

Moreover, the system is no longer in an equilibrium state since

This interpretation of nonextensivity is in agreement with the signification of the parameter c which is related to the cluster organization (possibly multifractal) of the heterogeneous system as discussed in Section 4.

In conclusion, 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. We think that many phenomena exhibiting power-law behavior are not necessarily the consequence of a nonextensivity of the entropy as this has been assumed by a number of authors, myself [

F.Brouers, (2015) The Burr XII Distribution Family and the Maximum Entropy Principle: Power-Law Phenomena Are Not Necessarily “Nonextensive”. Open Journal of Statistics,05,730-741. doi: 10.4236/ojs.2015.57073