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We discuss the Brouers-Sotolongo fractal (BSf) kinetics model. This formalism interpolates between the first and second order kinetics. But more importantly, it introduces not only a fractional order n but also a fractal time parameter a which characterizes the time variation of the rate constant. This exponent appears in non-exponential relaxation and complex reaction models as demonstrated by the extended use of the Weibull and Hill kinetics which are the two most popular approximations of the BSf ( n, a) kinetic equation as well in non-Debye relaxation formulas. We show that the use of nonlinear programs allows an easy and precise fitting of the data yielding the BSf parameters which have simple physical interpretations.

These last years the Brouers-Sotolongo fractal kinetics model [

On the other hand many of the works (we quote some of the most recent) dealing with pharmacokinetics [

In this paper we want to show that BSf model contains all of these formulas as approximations, each of them corresponding to a particular choice of the parameters. More elaborate methods have been introduced using the notion of fractional derivative [

In this section we derive the

whose solution is

If we use the deformed exponential [

and if we define

we can write

with a bit of algebra one obtains a first order differential equation

with a time dependent reaction rate

For

and for

These two behaviors expressing memory and aging effects appearing when

For increasing populations, we have

For n = 1, one recovers the first order memoryless exponential decrease or increase behavior with

These results do not exhibit the

with a characteristic time

The form

Equation (15) is solution of a “fractal” differential equations

The effective reaction rate has two asymptotic behaviors:

For

The two asymptotic behaviors of the population evolutionary law Equation (15) are:

independent on n for

It is important to notice that in BSf kinetics the exponent for large t is given by the ratio involving the two exponent parameters:

For special values of the parameters n and

1)

which is the first order kinetics.

2)

which is a “Weibull kinetics”. If

3)

This is the “Tsallis” kinetics.

4)

this is the second order kinetics.

5)

this the fractal second order or Hill kinetics.

It is important to note that as soon as

For increasing population, one has

solution of

In the Weibull case

The Hill equation is obtained in the case

It is of interest to recall that the solutions (30) an (32) can be obtained from exponential kinetics by assuming a distribution of the rate constant K due to fluctuations of the exponent of the Arrhenius law:

In the application of

When n and a are ≠1, one can no longer define a time independent rate constant and the relevant quantity characterizing the time evolution of the process is the characteristic time

The quantity

which gives

In the case of Weibull kinetics and Hill kinetics, this reduces respectively to

Although the BSf model has been invoked in a number of papers, it has been used correctly in a very few. We can mention the work of Harissa et al. [

As an illustration we have applied the ^{c}” software which allows a much better precision. As can be seen from the results, the

These results do not agree with the linearized calculations of [^{2} (case 3) to demonstrate the quality of the fitting. A

better fitting. These two simple facts are not always recognized by experimentalists and this can lead to erroneous results and the use of empirical not physical formulas.