Abstract:
We show that most of the empirical or semi-empirical isotherms proposed to extend the Langmuir formula to sorption (adsorption, chimisorption and biosorption) on heterogeneous surfaces in the gaseous and liquid phase belong to the family and subfamily of the Burr_{XII} cumulative distribution functions. As a consequence they obey relatively simple differential equations which describe birth and death phenomena resulting from mesoscopic and microscopic physicochemical processes. Using the probability theory, it is thus possible to give a physical meaning to their empirical coefficients, to calculate well defined quantities and to compare the results obtained from different isotherms. Another interesting consequence of this finding is that it is possible to relate the shape of the isotherm to the distribution of sorption energies which we have calculated for each isotherm. In particular, we show that the energy distribution corresponding to the Brouers-Sotolongo (BS) isotherm [1] is the Gumbel extreme value distribution. We propose a generalized GBS isotherm, calculate its relevant statistical properties and recover all the previous results by giving well defined values to its coefficients. Finally we show that the Langmuir, the Hill-Sips, the BS and GBS isotherms satisfy the maximum Bolzmann-Shannon entropy principle and therefore should be favoured.

Abstract:
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.

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 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.

Abstract:
We show that most of the empirical or semi-empirical isotherms proposed to extend the Langmuir formula to sorption (adsorption, chimisorption and biosorption) on heterogeneous surfaces in the gaseous and liquid phase belong to the family and subfamily of the Burr_{XII} cumulative distribution functions. As a consequence they obey relatively simple differential equations which describe birth and death phenomena resulting from mesoscopic and microscopic physicochemical processes. Using the probability theory, it is thus possible to give a physical meaning to their empirical coefficients, to calculate well defined quantities and to compare the results obtained from different isotherms. Another interesting consequence of this finding is that it is possible to relate the shape of the isotherm to the distribution of sorption energies which we have calculated for each isotherm. In particular, we show that the energy distribution corresponding to the Brouers-Sotolongo (BS) isotherm [1] is the Gumbel extreme value distribution Finally we propose a generalized GBS isotherm, calculate its relevant statistical properties and recover all the previous results by giving well defined values to its coefficients. In the course of the discussion we make contact with the Tsallis nonextensive theory [2] and the noninteger order reaction and fractal kinetics theory [3]. In the spirits of the present and previous publications, we propose an alternative formula to include fractality in the Michealis-Menten enzyme catalysis theory. Finally we suggest that the stochastic cluster model introduced by K.Weron [4] to account for the universal character of relaxation in disordered systems should be relevant for other phenomena in particular for heterogeneous sorption

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.

Abstract:
We show that if one uses the invariant form of the Boltzmann-Shannon continuous entropy, it is possible to obtain the generalized Pareto-Tsallis density function, using an appropriate "prior" measure m_{q}(x) and a "Boltzman constraint" which formally is equivalent to the Tsallis q-average constraint on the random variable X. We derive the Tsallis prior function and study its scaling asymptotic behavior. When the entropic index q tends to 1, m_{q}(x) tends to 1 for all values of x as this should be.

Abstract:
We have derived the dipolar relaxation function for a cluster model whose volume distribution was obtained from the generalized maximum Tsallis nonextensive entropy principle. The power law exponents of the relaxation function are simply related to a global fractal parameter $\alpha$ and for large time to the entropy nonextensivity parameter $q$. For intermediate times the relaxation follows a stretched exponential behavior. The asymptotic power law behaviors both in the time and the frequency domains coincide with those of the Weron generalized dielectric function derived from an extension of the Levy central limit theorem. They are in full agreement with the Jonscher universality principle. Moreover our model gives a physical interpretation of the mathematical parameters of the Weron stochastic theory and opens new paths to understand the ubiquity of self-similarity and power laws in the relaxation of large classes of materials in terms of their fractal and nonextensive properties.

Abstract:
We derive a universal function for the kinetics of complex systems. This kinetic function unifies and generalizes previous theoretical attempts to describe what has been called "fractal kinetic".The concentration evolutionary equation is formally similar to the relaxation function obtained in the stochastic theory of relaxation, with two exponents a and n. The first one is due to memory effects and short-range correlations and the second one finds its origin in the long-range correlations and geometrical frustrations which give rise to ageing behavior. These effects can be formally handled by introducing adequate probability distributions for the rate coefficient. We show that the distribution of rate coefficients is the consequence of local variations of the free energy (energy landscape) appearing in the exponent of the Arrhenius formula. We discuss briefly the relation of the (n,a) kinetic formalism with the Tsallis theory of nonextensive systems.

Abstract:
The purpose of this short paper dedicated to the 60th anniversary of Prof.Constantin Tsallis is to show how the use of mathematical tools and physical concepts introduced by Burr, L\.{e}vy and Tsallis open a new line of analysis of the old problem of non-Debye decay and universality of relaxation. We also show how a finite characteristic time scale can be expressed in terms of a $q$-expectation using the concept of $q$- escort probability.The comparison with the Weron et al. probabilistic theory of relaxation leads to a better understanding of the stochastic properties underlying the Tsallis entropy concept.

Abstract:
We Study in this paper the scaling and statistical properties of the $ac$ conductivity of thin metal-dielectric films in different regions of the loss in the metallic components and particularly in the limit of vanishing loss. We model the system by a 2D $RL-C$ network and calculate the effective conductivity by using a real space renormalization group method. It is found that the real conductivity strongly fluctuates for very small losses. The correlation length, which seems to be equivalent to the localization length, diverges for vanishing losses confirming our previous results for the decay of the real conductivity with the loss. We found also that the distribution of the real conductivity becomes log-normal below a certain critical loss $R_{c}$ which is size dependent for finite systems. For infinite systems this critical loss vanishes and corresponds to the phase transition between localized modes for finite losses and the extended ones at zero loss.