# Generalized fractal kinetics in complex systems (application to biophysics and biotechnology)

**ABSTRACT** We derive a universal function for the kinetics of complex systems characterized by stretched exponential and/or power-law behaviors. 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 α 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.The fractal (n,α) kinetic has been applied to a few problems of fundamental and practical importance in particular the sorption of dissolved contaminants in liquid phase. Contrary to the usual practice in that field, we found that the exponent α, which is implicitly equal to 1 in the traditional analysis of kinetic data in terms of first- or second-order reactions, is a relevant and useful parameter to characterize the kinetics of complex systems. It is formally related to the system energy landscape which depends on physical, chemical and biological internal and external factors.We discuss briefly the relation of the (n,α) kinetic formalism with the Tsallis theory of non-extensive systems.

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**ABSTRACT:**In the article, a new integrated kinetic Langmuir equation (IKL) is derived. The IKL equation is a simple and easy to analyze but complete analytical solution of the kinetic Langmuir model. The IKL is compared with the nth-order, mixed 1,2-order, and multiexponential kinetic equations. The impact of both equilibrium coverage θ(eq) and relative equilibrium uptake u(eq) on kinetics is explained. A newly introduced Langmuir batch equilibrium factor f(eq) that is the product of both parameters θ(eq)u(eq) is used to determine the general kinetic behavior. The analysis of the IKL equation allows us to understand fully the Langmuir kinetics and explains its relation with respect to the empirical pseudo-first-order (PFO, i.e., Lagergren), pseudo-second-order (PSO), and mixed 1,2-order kinetic equations, and it shows the conditions of their possible application based on the Langmuir model. The dependence of the initial adsorption rate on the system properties is analyzed and compared to the earlier published approximate equations.Langmuir 10/2010; 26(19):15229-38. · 4.38 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The influence of lateral non-specific and specific interactions on the kinetics in dilute solutions is analyzed within the framework of the Langmuir model. Regular solution theory is used to derive kinetic equations for dilute solutions (RSK model). RSK equations are modified to include simple Kiselev associative interactions and deviations from the regular solution theory (mRSK model) and LF-type energetic heterogeneity (LF-mRSK). Derived models lead to significantly different kinetic behavior than the commonly used FG model or the SRT approach. The influence of the equilibrium uptake u(eq) and coverage θ(eq) on the observed effects of lateral interactions is discussed. A new kind of kinetic plot for data analysis is also presented. The mixed LF-mRSK model is applied to analysis of solute adsorption on mesoporous carbon.Journal of Colloid and Interface Science 06/2011; 361(2):603-11. · 3.55 Impact Factor - SourceAvailable from: scielo.brRevista Brasileira De Ensino De Fisica - REV BRAS ENSINO FIS. 01/2008; 30(1).

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arXiv:cond-mat/0508449v1 [cond-mat.dis-nn] 19 Aug 2005

Generalized Fractal Kinetics in Complex

Systems ( Application to Biophysics and

Biotechnology)

F.Brouers and O.Sotolongo-Costa

Department of Physics, University of Lige, 4000, Belgium

Faculty of Physics, Chair of Complex Systems H.Poincar, University of Havana,

Cuba

fbrouers@ulg.ac.be, oscarso@fisica.uh.cu

Abstract

We derive a universal function for the kinetics of complex systems characterized by

stretched exponential and/or power-law behaviors.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 func-

tion obtained in the stochastic theory of relaxation, with two exponents α 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 distri-

bution of rate coefficients is the consequence of local variations of the free energy

(energy landscape) appearing in the exponent of the Arrhenius formula.

The fractal (n,α) kinetic is the applied to a few problems of fundamental and

practical importance in particular the sorption of dissolved contaminants in liquid

phase. Contrary to the usual practice in that field, we found that the exponent α,

which is implicitly equal to 1 in the traditional analysis of kinetic data in terms of

first or -second order reactions, is a relevant and useful parameter to characterize the

kinetics of complex systems. It is formally related to the system energy landscape

which depends on physical, chemical and biological internal and external factors.

We discuss briefly the relation of the (n,α) kinetic formalism with the Tsallis

theory of nonextensive systems.

Key words: Fractal kinetics, complex systems, nonextensive systems, energy

landscape, Levy distributions, sorption in aqueous solutions.

PACS: 05.20.Dd, 89.75.-k, 82.39.Rt, 82.40.Qt.

Preprint submitted to Elsevier Science2 February 2008

Page 2

Introduction

The physical origin of anomalous kinetics in complex systems like glass, poly-

mers, solutions, proteins, enzymes... has received much attention in recent

years ([1][2][3][4][5]).

This is due to the observation that in many instances, the kinetics cannot

be accounted for without introducing a time-dependent reaction rate coeffi-

cient to describe properly the experimental data. As in many other related

problems an accurate relation between microscopic properties and the global

macroscopic observables is lacking due to many-body interactions, the in-

evitable homogenization and coarse-graining resulting from complexity and

experimental techniques. What has been done in practice to deal with this

situation is to generalize formula used in simple reaction kinetics (first and

second order reaction) and introduce one or several supplementary empirical

parameters to fit experimental data. The aim of these works is to establish

correlations between macroscopic observables and external parameters (tem-

perature, concentration, pH, ...). Another method widely used is to consider a

few reaction steps. The total rate equation is written as the sum of elementary

rate equations (first or second order). Their respective weight and the rate co-

efficients are fitted to the data. This procedure can lead to ambiguous results

and conclusions. In particular it does not reproduce power-law behaviors often

encountered in these systems. In this spirit, Frauenfelder and collaborators [3]

have used intuitive arguments to fit experimental biomolecular reaction data

in protein materials with empirical stretched exponential or power-law func-

tions. They trace the ’anomalous” kinetics to the distribution of the reaction

rate in the Arrehnius formula. In 1988, to account for power law behavior,

Kopelman [1] proposed a phenomenological fractal like kinetics to account for

reactions in materials prepared as fractals. This lead more recently Savageau

[2] to introduce a model where instead of introducing a time-dependence to the

rate coefficient, the reactant concentrations are raised to non-integer powers.

More recently, Weron [4] and collaborators using results from their stochas-

tic theory of relaxation in dipolar systems [6], introduced a time dependent

power-law reaction rate coefficient to generalize first and second order kinetic

equations in order to apply them to biomolecular reactions. Simultaneously,

Mendes and collaborators [7], using results of the Tsallis nonextensive entropy

theory [8] to solve non-linear differential equations, introduced the concept of

a n-order kinetic equation whose solution has a formal expression similar to

the Tsallis generalized Pareto distribution.

The purpose of the present paper is to use some results of two recent papers on

non-Debye relaxation [9][10] to incorporate the ideas developed in the previous

quoted works [1][3][4][7], in one unified formalism in order to the set the basis

of a general theory of reaction kinetics in complex systems.

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The challenge which has to be faced, in this important problem, is to give a

physical or statistical meaning to the empirical parameters and, when this is

possible, to relate the ”anomalous” kinetics to universality, i.e. universal scal-

ing functions independent on the microscopic or mesoscopic detailed properties

of the system.

1 The (n,α) kinetic equation

The most general chemical kinetic equation for one given species (A) in a

complex system composed of A,B,C,.. reacting atoms and molecules can be

written formally as

dcA

dt

where K is the rate coefficient and α,β,γ...refer to the concentrations of chem-

ical species A,B,C,...present in the reaction and the sum α + β + γ is the

overall order of the reaction. In some cases, the concentration cB,cC... can be

considered as constant, thus the above equation reduces (for instance for the

reactant A) to the form

= Kcα

Acβ

Bcγ

C...(1)

−dcn(t)

dt

= Kncn(t)n

(2)

In that way, the parameter n becomes the overall order of the reaction. The

solution of this differential equation for one of the reactants is given by [7]

cn(t) = cn(0)[(1 + (n − 1)cn(0)n−1Knt]−

1

n−1

(3)

which has the form of a generalized Pareto function, solution of the Tsallis

entropy maximization [8] and has an asymptotic power law behavior cn(t) ∝

t−1/(n−1).

If we use the deformed n-exponential and n-logarithm introduced by Tsallis

and collaborators in the context of nonextensive systems [8]:

expn(x) = (1 − (n − 1)x)−

1

n−1

if1 − (n − 1)x > 0, 0 otherwise

andlnn(x) =x1−n− 1

1 − n

(4)

with

expn( lnn(x)) =lnn( expn(x)) = 1

we can write the solution (3) in a more compact form:

cn(t) = cn(0)expn(−t/τn) (5)

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Page 4

with a characteristic time:

τn= (cn(0)n−1Kn)−1

(6)

This will allow the definition of an effective time-dependent rate coefficient:

−dcn(t)

dt

= c(0)1

τn(expn(−(t/τn))n

= c(0)1

τn(1 + (n − 1)(t/τn))−

= Kn(t)cn(t)

(7)

1

n−1−1

with

Kn(t) =1

τn(1 + (n − 1)(t/τn))−1

(8)

For t << τn, one has a slowing down of the effective rate :

Kn(t) =1

τn(1 − (n − 1)(t/τn)) t << τn

(9)

and for n ?= 1, t >> τn

Kn(t) ∝ (1/(n − 1))t−1

(10)

This behavior is a manifestation of what has been call aging [11] which appears

as soon as n ?= 1. For n = 1, one recovers the exponential behavior with

K(t) = 1/τ.

These results do not exhibit the t < τ power law time dependence of the reac-

tion rate which describes adequately the experimental data of many complex

systems [4][12][13][14][15]. This behavior can appear quite naturally if we in-

troduce in (6), instead of the n-exponential, a n−Weibull function introduced

by Mendes [16] and used also in the theory of relaxation [9][10]

cn,α(t) = c(0)expn(−(t/τn,α)α) = c(0)[(1 + (n − 1) (t/τn,α)α]−

1

n−1

(11)

with a characteristic time:

τn,α= [Kn,αc(0)n−1]−1/α

(12)

The effective time-dependent rate coefficient Kn,α(t) now reads

Kn,α(t) = αtα−1

τα

n,α

(1 + (n − 1)(t/τn,α)α)−1

(13)

Equation (11) is solution of a fractional differential equation :

−dcn,α(t)

dtα

= Kn,αcn,α(t)n

(14)

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Page 5

by introducing a fractional time index α and a non-integer reaction order

n. Fractional derivation and fractal time concepts have been introduced in

physics (diffusion in disordered and porous media, random walks ...[17]) in

the theory of dielectric response [20][18][10] and in economy [19].

The effective rate coefficient Kn,α(t) has the two asymptotic behaviors

for

for

t → 0

t → ∞

Kn,α(t) ∝ tα−1

Kn,α(t) ∝ t−1

(15)

For t → 0, we get the same power-law variation of the rate coefficient as in the

work of Weron et al. [4] as well as in the fractal phenomenological description

of non-homogeneous reaction dynamics called fractal-like kinetics [1], if we

identify the Kopelman fractal parameter h < 1 with 1 − α. As noted in [5],

the concept of effective time-dependent rate constant breaks down for t → 0,

α < 1, since in that limit Kn,α(t) diverges. The general solution (11) of the

fractional differential equation (14) does not suffer from such difficulty and is

well defined in the positive time domain. In any case, as for real geometric

fractals, for physical reasons, there is in each case a natural small time cut-off.

The two asymptotic behaviors of the concentration evolutionary law equation

(11) are:

cn,α(t) = c(0)[(1 − (t/τn,α)α+ ...]

independent of n for t << τn,α, while for n ?= 1 and t >> τn,α

(16)

cn,α(t) = c(0)(n − 1) (t/τn,α)−α/(n−1)

(17)

The ratio of the the two asymptotic exponents α and α/(n − 1) yields the

value of the apparent order of the reaction n.

For special values of the two parameters n and α, some other typical solutions

are recovered

a. If n = 1,α = 1, we have

−dc(t)

dt

= K1c(t)

→

c(t) = c(0)exp(−K1t) (18)

which is a first order kinetic

b. If n = 1,α ?= 1, we have

−dcα(t)

dtα

= Kαcα(t) →

cα(t) = c(0)exp(−Kαtα) (19)

which is a ”Weibull kinetics”. If 0 < α < 1, it is a ”stretched exponential

kinetic”.

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c. If n ?= 1, α = 1, equations (11) gives

cq(t) = cn(0)[(1 + (n − 1)cn(0)n−1Knt]−

1

n−1≡ expn(−c(0)n−1Knt) (20)

which is solution of (2).

d. If n = 2, α = 1, we have

−dc(t)

dt

= K2c2(t)

→

1/c(t) − 1/c(0) = K2t (21)

This is the second order kinetic.

e. If n = 2, α ?= 1, we have

−dcα(t)

dtα

= K2,αcα(t)2

→ cα(t) = c(0)[(1 + c(0)(K2,αt)α]−1

(22)

This is a generalized second order kinetic.

Cases (b) and (e) have been discussed in [4].

It is important to note that as soon as n ?= 1, the time dependence of the

kinetics depends on the initial concentration.

We will call the kinetic giving rise to the concentration evolutionary law (11),

the (n,α) kinetic:

cn,α(t) = c(0)[(1 + (n − 1) (t/τn,α)α]−

1

n−1

(23)

τn,α= [c(0)n−1Kn,α]−1/α

(24)

is the characteristic time of the complex kinetic. It depends on the initial

concentration and the two exponents n and α. For n → 1, cn,α(t) tends to a

Weibull exponential with τ1,α= [K1,α]−1/α. One can define a ”half-reaction

time” τ1/2which is the time necessary to transform half of the relevant quantity

by solving the equation

(1 + (n − 1)(τ1/2/τn,α)α)−1/(n−1)= 1/2 (25)

which gives using the definition of lnn(x) (4) :

τ1/2= τn,α(lnn2)1/α

(26)

Kinetics are ”memoryless” only when n = α = 1. If n = 1, kinetics are

”memoryless” in the fractal time tf = tαsince with the change of variable,

the Weibull function reduces to a memoryless exponential.

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Page 7

One can introduce in this problem a ”response function” as it is done in Weron

et al.[4] for n = 1 and n = 2. We have more generallyfor any real n:

f(t) = −

1

c(0)

dcn,α(t)

dt

= αtα−1

τn,α(1 + (n − 1)(

t

τn,α)α)−

1

n−1−1

(27)

this function has the two asymptotic behaviors:

fort → 0f(t) ∝ (t/τn,α)α−1

f(t) ∝ (t/τn,α)−(α/n−1)−1n ?= 1

(28)

fort → ∞

For n = 1 and n = 2, they coincide with those of [4].

2 Arrhenius law and exponential conspiracy

The results of the previous section can be understood physically as a con-

sequence of what is has been called ”exponential conspiracy”, an expression

coined by Boucheau [21] and proposed as exercise in textbooks on probabil-

ity theory(for example [22]). It is generally accepted that the temperature

dependence of the reaction rate K has an Arrhenius form which we will write:

K = K0exp(±E/kT) (29)

K0is the pre-exponential factor and E the relevant energy (in thermodynam-

ics systems this energy is the Gibbs free energy which depends on the enthalpy

(heat of reaction) and the entropy : G = H −TS). The sign + corresponds to

an ”exothermic” reaction (i.e. the energy corresponds to an attraction energy

and the rate decreases with the temperature). This is the case for instance in

physisorption, when the overall adsorption enthalpy resulting from adsorption

and desorption is positive. The sign - corresponds to an ”endothermic” reac-

tion and E is an activation energy barrier to be overcome. In that case the

rate increases with temperature. We have written the two terms ”exothermic”

and ”endothermic”, because due to the variation of entropy with T, paradox-

ically in some complex systems, an endothermic reaction can occur without

activation energy (see for instance [23]).

In disordered systems frozen out of equilibrium, the exponent factor E/kT

varies due to fluctuations of local energies and local temperatures. The dis-

tribution of energies depends on what has been called by Frauenfelder [3] the

”energy landscape”, a concept taken from the theory of glasses, and variations

of the inverse of local temperature (1/T) have been used to introduce, what

has been called ”super-statistics” by Beck and Cohen [24].

Here we will assume, as in the theory of heterogeneous catalysis, in the theory

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of glass and in the theory of adsorption [25], that the probability distribution

of the energy E varies for large values as exp(−E/E0). This means that the

large energy values are statistically exponentially small. The reference energy

E0is linked to the width of the energy density distribution fE(E) [25]. With

this assumption, using the basic probability theory relation

fE(E)dE = fK(K)dK (30)

it is straightforward to show, that the distribution of the rate coefficient K,

has the asymptotic form

fK(K) = µ(K/K0)−1±µ

withµ = kT/E0

(31)

Therefore, in the ”exothermic” case (sign + in (18)),

K → ∞

fK(K) ∼ K−1−µ

(32)

while in the ”endothermic” case (sign - in (18))

K → 0fK(K) ∼ K−1+µ

(33)

In the first case, the distribution fK(K) is a Pareto distribution and is the

simplest density distribution belonging to the domain of attraction of the sta-

ble L´ evy distributions [26][27]. If we assume that the variations of K induced

by the fluctuations of E are represented by a L´ evy distribution Lµ(λ) , one

can, using the well-known relation (the Laplace transform of a one-sided Levy

distribution is a stretched exponential) :

?∞

0

exp(−λKt)Lµ(λ)dλ = exp(−(Kt)µ) (34)

obtain the generalized first order (Weibull) kinetic (case 2 with µ = α) as a

compounded exponential first order kinetic . In the second case, if we use the

Gamma density distribution which has the power law asymptotic behavior

(µλµ−1) for small values of λ,

gµ(λ) =

µ

Γ(µ)(µλ)µ−1exp(−µλ) (35)

we obtain equation (3) with µ = 1/(n − 1)

?∞

0

exp(−λ(Kt))gµ(λ)dλ = (1 +1

µ(Kt))−µ

(36)

If we use the Weibull distribution, we can then write [28]

?∞

0

exp(−λ(Kt)α)gµ(λ)dλ = (1 +1

µ(Kt)α)−µ

(37)

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Page 9

or using a result established by Weron and collaborators in the stochastic

theory of relaxation [6] :

?∞

0

exp(−λKt) MLα,µ(λ)dλ = (1 + (1/µ)(Kt)α)−µ

(38)

where MLα,µ(λ) is a generalized Mittag-Lefller distribution:

MLα,µ(λ) =

∞

?

k=0

(−1)kΓ(µ + k)

k!Γ(µ)Γ[(α(µ + k)](λ)α(µ+k)−1

(39)

This last result is more difficult to interpret. It can be understood [6][9], as

the result of the random character of the number of active centers, geometric

frustrations and dynamic constraints or as a consequence of the interplay of

”exothermic” and ”endothermic” processes in the kinetics of complex mate-

rials. In conclusion, the (n,α) kinetic equation, can be obtained formally by

introducing an adequate distribution for the exponent of the Arrehnius law as

conjectured by Fraunfelder [3].

3Probabilistic interpretation of the (n,α) kinetic equation

A comparison with the stochastic theory of relaxation [6][4][9][10] is of interest

to understand the physical meaning of equation (23). We first note that

cn,α(t) = c(0)[(1 + (n − 1) (t/τn,α)α]−

1

n−1

(23)

is related to the BurrXII distribution function (Ba,b,c(x) = 1 − (1 + axb)−c

,x > 0) [28], named by reference to the number it occupies in the main Table

of Burr’s original paper [29]). If we introduce an effective random reaction

waiting time˜θ , the quantity cn,α(t)/c(0) can be viewed as the probability

that the reactant has not yet reacted at time t:

cn,α(t)/c(0) = Pr(˜θ > t) = 1 −

?t

0fn,α(˜θ)d˜θ (40)

where

fn,α(˜θ) = α

˜θα−1

τn,α(1 + (n − 1)(

˜θ

τn,α)α)−

1

n−1−1

(41)

This distribution belongs to the domain of attraction of the Levy distribution

with a tail exponent µ = α/(n − 1) and therefore generalizes the Pareto or

Zipf-Mandelbrot distributions used in fractal reactions kinetics of previous

works [5]. If µ < 1, an expectation value of˜θ cannot be defined and an escort

probability function [30] has to be used to determine τn,αfrom the knowledge

of fn,α(˜θ).

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Page 10

If α = 1, the probability density function (41) reduces to the Tsallis general-

ized Pareto distribution [8] if (n − 1) = (q − 1)/(2 − q) i.e. n = 1/(2 − q)

fq(˜θ) =

1

τn,α(1 + (q − 1)/(2 − q)

˜θ

τn,α)−

1

q−1

which maximizes the Tsallis entropy of the random variable˜θ.

The relation between the (n,α) kinetics and the nonextensivity of the en-

tropy and the formal relation of the reaction order n with the Tsallis entropy

index q is worth further investigations. The characteristics of the complex

systems studied in the present work (see the introduction) are similar to the

ones (frozen non-equilibrium states with memory effects and long range cor-

relations) of what has been called nonextensive systems by the Tsallis school

[8].

4 Application to biotechnology and biophysics

In this last section, we give some examples of problems in the field of biothech-

nology and biophysics, where we think the application of the (n,α) kinetics

can open new paths to understand anomalous kinetics from the point of view

of the theory of complex systems.

4.1 Sorption of dissolved contaminants in liquid phase

The sorption (adsorption, chemisorption, biosorption) of pollutants from aque-

ous solutions plays a significant role in water pollution control. It is therefore

important to be able to predict the rate at which contamination is removed

from aqueous solutions and how this rate depend on physical, chemical, bio-

logical and environment variables in order to design an appropriate treatment

plant. Sorption of dissolved contaminant is a complex phenomena caused by

several mechanisms including London-van der Waals forces, Coulomb forces,

hydrogen bonding, ligand exchange fluctuations, chemisorption, dipole-dipole

forces and hydrophobic forces and biosorption for biological materials. There-

fore these systems can be considered to belong to the class of complex systems

[31]. The quantity adsorbed at time t, qtis defined as

qt=(c0− ct)V

W

(42)

where c0is the initial concentration of the solution, ct, the concentration at

time t, V, the volume of the solution and W, the weight of the adsorbent.

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In this context, the kinetic equations are determined with reference to the

quantity of dissolved contaminant necessary to reach equilibrium(for ex. [23]).

We have then

qt= qe(1 − exp(−K1t)

where qeis the mass of solute adsorbed at equilibrium, qtis the mass of solute

adsorbed at time t and K1is the rate coefficient. Equation (43) is called pseudo

first- order equation by contrast to the simple exponential first-order equation

(18).

(43)

In the same way, one can define a pseudo second-order reaction:

1

qe− qt

= K2t +1

qe

(44)

In agreement with the ideas developed in sections 1 and 2, we can introduce

the pseudo-(n,α) equation

qt(α,n) = qe[1 − (1 + qn−1

e

(n − 1)Kn,αtα)−

1

n−1](45)

which reduces to (43) for n = 1,α = 1 and to (44) for n = 2, α = 1. We can

write (45) more compactly using the definition (4) of the deformed exponential

expn(x),

rn,α(t) = qt(α,n)/qe= 1−expn((t/τn,α)α) withτn,α= (qn−1

e

Kn,α)−1/α

(46)

The definition of the deformed logarithm (4) associated with expn(x) allows

us to write the following relation

R(t) = Log((1 − rn,α(t))1−n− 1

n − 1

) = αLog(t) − αLog(τn,α)(47)

which can be used to make a linear fit of the data (rn,α(t) = qt(α,n)/qe) and

obtain the values of α and τn,α. The value of n to be chosen is the one which

can give the better fit in the Log-Log plot.

In two different collaborations we have analyzed the kinetics of various pol-

lutants (phenol, tannic acid, gallic acid, melano¨ ıdine on activated carbon [32]

and various dyes pollutants from the textile industry on biological materials

(algaes and agaves) [33]. It appears that, quite generally the data can be fitted

quite well to pseudo-(n,α) kinetic. For t << τ, the concentration qe(n,α) does

not depend on n (cf eq.16) and can be fitted to a n-independent power law

qe(n,α) ∝ tα. The value of n (i.e. the overall order of the reaction) has to

be determined from the large time (near saturation) behavior of the kinetics.

Contrary to the usual practice in that field, we found that the exponent α,

which is implicitly equal to 1 in the traditional analysis of kinetic data [23]

in terms of a pseudo-first or -second order reaction, is more appropriate to

characterize the kinetics of sorption of dissolved contaminants in liquid phase.

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Page 12

It yields a better fit and moreover it is related to the system energy land-

scape which varies with external parameters. In Fig.1, we show an example

(adsorption kinetic on activated carbon of m´ elano¨ ıdine, a dye formed in the

crystallization process of sacharose [32]) where the best fit is obtained with

a pseudo-(1.5,α) kinetic equation. One cannot fit properly the data with a

quasi-first or second-order kinetic where α = 1. This is understandable since

one observes that for small t , qe(n,α) ∝ t0.56. The dependence of the two

quantities α and n on the physical, chemical environmental and biological pa-

rameters of the couple adsorbent-pollutant (pH, T, clustering, ligand field and

architecture of large biomolecules...) is the subject of current studies [32][33].

4.2 Kinetics in photosynthesis processes

In this subsection, we want to show how the fractal (n,α) rate equation (41)

can be used in situation where a two-steps first order equation has been used

to fit the kinetics of photosynthesis processes. The example chosen is the kinet-

ics of the conversion of protochlorophyllide into chlorophyllide. The method

used to follows the kinetics in that problem is the observation of spectral

changes recorded by the technique of spectrofluorometry under short-time il-

lumination. For instance in [34] the authors have observed the transformation

of a 647 nm pigment (species a) by 630 nm photons. Contrary to the trans-

formation induced by 647 nm photons, where the kinetics is first order, the

transformation under 630 nm irradiation follows an unusual kinetic. It has

been assumed that this particular kinetics is due to the presence of an other

protochlorophyllide species (named b), i.e. a pigment with another associa-

tion with the lipoproteins. We refer to the specialized literature for details

[34][35][34]. The two-steps model, often used when the rate equation cannot

be fitted to a first or second-order kinetics, lead Boardman, in this particular

problem [35], to the following rate equation (percentage of phototransformed

quantity),

T(%) = 100 − Aexp(−K1t) − (100 − A)exp(−K2t) (48)

A is the proportion in % of the complex which is transformed. This formula

can be fitted to the experimental data up to 85% [34]. What differs in 647

pigment and 630 nm pigment is the link with the lipoproteins. The pigment-

protein links are most probably fluctuating locally due to the complexity of

the organization of the molecules inside the prolamellar body. If instead of

two, a distribution of ”species” is present, it is more appropriate to use, a

rate equation deriving from a distribution of exponentials as the (n,α) rate

equation. We have verified that in this particular case, (42) can fit perfectly

the experimental curve, also for transformation larger than 85% (Fig.1). The

two asymptotic behaviors appears to be power-law, a behavior a simple two-

step mechanism cannot account for. The best fit obtained with a nonlinear

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method is given by

T(%) = 100(1 − (1 + (1.9 − 1) (t/0.07)0..96)−1/(1.9−1))) (49)

This correspond to a fractal-time exponent α = 0.96 and a reaction order pa-

rameter n = 1.9. The characteristic time is τα= 0.07 and from the inspection

of the experimental curve (Fig.1), one can see that it is very close to τ1/2. The

linear fit (r(t) = T/100 ) gives the same result (Fig.2):

R(t) = Log((1 − r(t))1−1.9− 1

1.9 − 1

) = 0.96Log(t) − 0.96Log(0.07) (50)

with a regression coefficient of 0.9997.

The temperature dependence of the transformation rate under 633 nm photons

[36] indicates that both endo- and exothermic effects are competing. As we

have suggested at the end of section 2, in this particular situation, fluctuations

of the rate coefficient in the exponent of the Arrhenius law (29) can give rise

to (n,α) kinetics.

4.3 Complexity of DNA

One method widely used to study the complexity of DNA is the so-called Cot

method. The method splits the double strands of DNA into single strands by

raising the temperature or by other denaturing process. One then studies the

kinetics of the reassociation of dissociated single strands. Since it involves two

single strands, the renaturing into the original form is assumed to follow a

(2,1) kinetics f = c (t)/c (0) = [(1 + c(0)(Kt)]−1. This equation is the basis

of the Cot analysis of the rate of renaturation of sequence heterogeneity (or

complexity) of DNA, The quantity c(0) is the initial concentration of DNA,

f= c (t)/c (0) the fraction of single-stranded molecules which decreases with

time and K the rate constant for the reassociation of complementary strands.

The value of c(0)t when f = 0.5 is known as c(0)t1/2.

The rate coefficient K is characteristic of a particular DNA and is related to

its complexity in terms of sequence composition. The quantity c(0)t1/2is the

reciprocal of K and can therefore be used as a measure of sequence complexity.

The higher the value of c(0)t1/2, the more complex is the DNA.

The Cot method which was developed in the 1960’s and widely used in the

70’ was then nearly abandoned. It made a comeback recently [37] as a much

cheaper method because of its ability to concentrate on the low copy sequences,

the highly repeated sequences being irrelevant as far as the genetic information

is concerned.

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Due to the complexity of the DNA structure, it would be surprising if there

would be no short time memory effects in the reassociation process, since

it involves quite complex biomolecules. Indeed, fingerprints of fractality and

nonextensivity in DNA fragment distribution has been reported recently [38].

The ideas and formalism developed in this paper might be of interest also in

that field of primary importance.

5 Discussion

Using ideas and theoretical tools borrowed from recent works on the theory of

relaxation, we have derived a universal function for the kinetics of complex sys-

tems characterized by stretched exponential and/or power-law behaviors This

kinetic function unifies and generalizes previous theoretical attempts to de-

scribe what has been called ”fractal kinetic”. The concentration evolutionary

equation (12) is formally similar to the BurrXIIrelaxation function obtained

in the theory of relaxation, with two exponents α 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. As in the theory of relaxation, these effects can be

formally handled by introducing adequate probability distributions for the

rate coefficient. We have shown that the distribution of rate coefficients is the

consequence of local variations of the free energy appearing in the exponent

of the Arrhenius formula. The scaling (power-law) behavior of the kinetic is

therefore an other example of what has been called ”exponential conspiracy”

[21]. The two macroscopic observables n and α are formally related to the

energy landscape of the complex system which varies if physical, chemical or

biological external factors are modified.

The fractal (n,α) kinetic has been applied to a few problems of fundamental

and practical importance [32][33][39], examples of which have been presented

in section 4.

In references [9] we have shown how a universal relaxation function can be

derived if we use distributions of macroscopic waiting times maximizing the

nonextensive Tsallis entropy. Similar conclusions can be drawn in the present

problem, if we introduce local reaction waiting time in a probabilistic deriva-

tion of the universal kinetic function.

The relation between the (n,α) kinetic and nonextensive thermostatistics will

be the subject of further studies.

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6 Captions of Figures

Fig.1 (n,α) kinetic of adsorption of melano¨ ıdine in aqueous solution on acti-

vated carbon with n = 1.5 and α = 0.56.

Fig.2 (n,α) kinetic (eq.49) of the conversion of protochlorophyllide into chloro-

phyllide (transformation of 647 nm pigment by 630 nm photons). The smaller

points for t>0.4 sec. are results of the two first order model (eq.48).

Fig.3 Log-Log plot (eq.47) applied to data of Fig.2.

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