The stretched exponential behavior and its underlying dynamics. The phenomenological approach

# The stretched exponential behavior and its underlying dynamics. The phenomenological approach

K. Górska H. Niewodniczański Institute of Nuclear Physics, Polish Academy of Sciences, Division of Theoretical Physics, ul. Eliasza-Radzikowskiego 152, PL 31-342 Kraków, Poland    A. Horzela H. Niewodniczański Institute of Nuclear Physics, Polish Academy of Sciences, Division of Theoretical Physics, ul. Eliasza-Radzikowskiego 152, PL 31-342 Kraków, Poland    K. A. Penson Sorbonne Universités, Université Pierre et Marie Curie (Paris 06), CNRS UMR 7600, Laboratoire de Physique Théorique de la Matière Condensée (LPTMC), Tour 13-5ième ét., B.C. 121, 4 pl. Jussieu, F 75252 Paris Cedex 05, France    G. Dattoli ENEA - Centro Ricerche Frascati, via E. Fermi, 45, IT 00044 Frascati (Roma), Italy    G. H. E. Duchamp Université Paris XIII, LIPN, Institut Galilée, CNRS UMR 7030, 99 Av. J.-B. Clement, F 93430 Villetaneuse, France
###### Abstract

We show that the anomalous diffusion equations with a fractional spacial derivative in the Caputo or Riesz sense are strictly related to the special convolution properties of the Lévy stable distributions which stem from the evolution properties of stretched or compressed exponential function. The formal solutions of these fractional differential equations are found by using the evolution operator method where the evolution is conceived as integral transform whose kernel is the Green function. Exact and explicit examples of the solutions are reported and studied for various fractional order of derivatives and for different initial conditions.

MSC 2010: Primary 35R11; Secondary 26A33, 60G18, 60G52, 49M20

Key Words and Phrases: fractional calculus, stretched or compressed exponential function, fractional ordinary and partial differential equations, Green functions

## I Introduction

In recent experiments of relaxation processes in a variety of complex materials and systems, such as supercooled liquids, spin glasses, amorphous solids, molecular systems, glass soft matter, porous and noncrystaline silicon, etc. WGotze92 (); JCPhillips96 (); CAAngell00 (); LCipelletti05 (); LPavesi96 (); IMihalcescu96 (), the relaxation properties have been successfully fitted by stretched exponentials RKohlrausch54 (); GWilliams70 (); RSAnderssen04 ()

 fα(tτ0)=e−(tτ0)α,fort>0and0<α<1,

called also the Kohlrausch-Williams-Watts functions (the KWW functions). Moreover, it has been shown, e.g. in KWeron96 (), that the KWW function is related to the Cole-Cole relaxation processes which appear in the systems of disordered or anomalous structures. The importance of the KWW function for the survival propability in relation to relaxation function has been also noted and dissed in KWeron96 (). The decay according to the KWW pattern is by no means restricted to common relaxation phenomena. For example, it appears in biological context, e.g. in protein folding JSabelko99 () and in -Helix formation in photo-switchable peptides JBredenbeck05 (); JAIhalainen07 (), etc. The stretched exponential behavior is also used to describe the local variations of transport speed YZhang07 (); BDybiec08 (). Exponential decay faster than the Debye law () is also observed. This case is called the compressed exponential, which in the theory of slow magnetization is named the Kolmogorov-Avrami-Fatuzzo relaxation HXi08 (); AAdanlete11 (); NZurauskiene14 ().

The function is related to Lévy stable distributions HPollard46 (); HBergstrom52 (): for these are one-sided and for these are two-sided cases. As it is known there exists an intimate relation between the Lévy stable distributions and the anomalous diffusion; for it is the sub-diffusion and for it is the super-diffusion. We are going to approach this connection in a novel way. The physical formulation is based on the Fokker-Planck equation with the spatial fractional derivative, usually of Riemann-Liouville, Caputo or Riesz types. The case of is matched with the standard diffusion equation. For the compressed exponential functions correspond to the partial differential equation with the space derivative of order KGorska13 (); EOrsinger12 (). The sub- and super-diffusion equations are usually derived via the probability approach like the continuous-time random walks JKlafter80 (), the master equation DBedeaux71 (), the generalized master equation VMKenkre73 (), and related methods ACompte96 (); MMagdziarz06 (); BDybiec10 (). The intense activity in this field has apparently been initiated by Saichev and Zaslavsky in AISaichev97 (). The relevant references can be traced back from two recent papers ECapelasdeOliverira14 (); RGarra14 (). A very complete and lucid exposition of theoretical aspects of diffusion phenomena can be found in MMeerschaert12 (). Nevertheless, the anomalous diffusion equations should stem from the natural evolution property which is satisfied by : namely a suitably defined product of two stretched (compressed) exponential functions should give another stretched (compressed) exponential function. From the mathematical point of view, the use of the complex analysis technique underlying this condition will allow us to obtain the fractional Fokker-Planck equations. This alternative approach to a derivation of the anomalous equations and the operational method of solving them are the main objectives of this work. We believe that this will open new possibilities of looking at the stretched and compressed exponentials which have in the natural way built-in the anomalous behavior.

The paper is organized as follows. In Sec. II we recall and comment on two relations: the first is between the stretched exponential and the one-sided Lévy stable distribution, and the second one is between the compressed exponential and two-sided Lévy stable distribution. We use the well-known unique relation between the one- and two-variables Lévy stable distributions whose evolution property will be studied in the paper. In Sec. II we derive two types of integral evolution equations related to two various behaviors of . The differential version of these equations will be found in Sec. III. In connection with the used Lévy stable distribution they will contain the fractional derivative in the Caputo or Riesz sense. The solution of these differential equations in terms of the Green function are given in Sec. IV. Sec. V is devoted to finding the solution of these equations with the help of operational methods based on the integral transforms whose kernels are given by the appropriate Lévy stable distributions. Sec. VI contains the explicit examples of solutions of the equations. We conclude the paper in Sec. VII.

## Ii Lévy stable distributions

In a discrete setting and from a phenomenological point of view the KWW functions for can be treated as a sum of weight function of exponential decays ( for ) GDattoli14 (), namely

 fα(tτ0)=∑je−tτ0ujgα(uj)Δuj, (1)

where is a probability density, i.e. and , which describes the structure of the sample where relaxation process is governed by the stretched exponential function. In the limit of large number of elements in the sum and for the infinitesimally small changes , Eq. (1) goes over to the Laplace transform GDattoli14 (); KWeron91 ()

 fα(tτ0)=L[gα(u);tτ0] =∫∞0e−tτ0ugα(u)du (2) =∫∞0e−xgα((tτ0)α,x)dx. (3)

In Eq. (2) the variable of integration is changed to and it gives the relation VMZolotarev83 (); KGorska12 (); KAPenson16 ():

 τ0tgα(τ0tx)≡gα((tτ0)α,x). (4)

Systematic treatment of relations of type Eq. (4) can be rephrased in terms of subordination laws, consequences of the properties of Mellin transform BDybiec10 (); KAPenson16 (); FMainardi03 (). Eq. (4) represents the one-to-one correspondence between the one- and two- variables Lévy stable distributions. It leads to the evolution equation with the fractional derivative which will be studied below. In this sense Eq. (4) is crucial for our considerations.

In order to limit the proliferation of notation we shall employ the same symbol to both sides of Eq. (4), and later on, to Eq. (10). Their meaning will be clear from the context and should not lead to confusion.

The probability density , via the inverse Laplace transform of Eq. (2) and Eq. (4), is uniquely given by one-sided Lévy stable distributions HPollard46 ():

 gα((tτ0% )α,x)=L−1{exp[−(tτ0ξ)α];x}=12πi∫Lexξe−(tτ0ξ)αdξ, (5)

, and for . We remark that the Laplace transform of the function is defined as follows INSneddon74 ():

 fL(p)=L[f(u);p]=∫∞0e−upf(u)du,

for , whereas the inverse Laplace transform is given by

 f(u)=L−1[fL(p);u]=12πi∫LeupfL(p)dp,

where inside the integration contour the function is analytic. Eq. (5) goes to the Dirac function, namely , in the limit of , see (ISGradsteyn07, , Eq. (17.13.94) on p. 1115). The function has the essential singularity at (JMikusinski59, , Eq. (4)) and the “heavy” long tail as (JMikusinski59, , Eq. (5)). Taking in Eq. (5) the integral contour defined in HPollard46 () and using Eq. (4), we get

 gα((tτ0)α,x)=Im{∫∞0e−xξe−(tτ0ξ)αe−iπαdξπ}, (6)

which for rational is equal to

 glk((tτ0)lk,x)=k−1∑j=1(−1)jx−1−jlkj!Γ(−lkj)(tτ0)jlk×l+1Fk(1,Δ(l,1+jlk)Δ(k,1+j)∣∣z), (7)

with , see KAPenson10 (). The generalized hypergeometric function is denoted as , where is the “upper” list of parameters and is the “lower” list APPrudnikov_v3 (). The symbol is equal to a list of elements . The basic example of is the so-called Lévy-Smirnov distribution . The other explicit and exact examples of can be obtained by using Eq. (4) and further examples are quoted in KAPenson10 ().

The compressed exponential behavior can be written as , which for is related via the Fourier transform INSneddon74 () to , the two-sided Lévy distribution,

 fα(tτ0)=12fα(|tτ0|) =F[12~gα(u);%$tτ0$] =∫∞−∞eitτ0u% 12~gα(u)du (8) =∫∞−∞eix12~gα((tτ0)α,x)dx, (9)

where, similarly to Eqs. (2) and (3), we use the new variable of integration . Let us recall that the Fourier transform of the function is given by

 gF(ω)=F[g(x);ω]=∫∞−∞e−iωxg(x)dx,

where

 g(x)=F−1[gF(ω);x]=12π∫∞−∞eiωxgF(ω),

which is the inverse Fourier transform. In analogy to Eq. (4) we get the unique scaling relation

 τ0t~gα(τ0tx)≡~gα((tτ0)α,x), (10)

whose evolution property will be studied in this work. The inversion of the Fourier transform given in Eq. (II) and the use of property (10) leads to

 ~gα((tτ0)α,x)=F−1{exp[−(% tτ0)α|ξ|α];x}=∫∞−∞e−ixξe−(tτ0)α|ξ|αdξ2π (11)

for and , which is the symmetric two-sided Lévy stable distribution, see Eq. (8) for of HBergstrom52 () applied for (10). Observe that Eq. (11) defines also the symmetric Lévy stable distribution for HBergstrom52 (); KGorska11 (). In the limit of vanishing the two-sided Lévy stable distribution goes to the Dirac function, i.e. for . Eq. (11) expressed as the finite sum of the generalized hypergeometric function is given in (KGorska11, , Eqs. (4) and (5) for , , and ), namely

 ~gpq((tτ0)pq,x) =M−1∑j=1(−1)1+jπj!Γ(1+mMj)(t/τ0)∓2pMjx1∓2pMj ×m+1FM(1,Δ(m,1+mMj)Δ(M,1+j)∣∣~z), (12)

where , , . The lower sign in the powers of and is for , whereas the upper sign is for . For we get the Gauss (normal) distribution, . For more examples see KGorska11 ().

From the very natural evolution assumption written for the stretched (compressed) exponential function for , it appears that an appropriately defined composition of two stretched (compressed) exponentials should give another stretched (compressed) exponential. It means that the appropriate convolution of the stable distributions of two variables gives another stable distribution of two variables: (i) for the one-sided Lévy stable distribution of two variables we have

 gα((t2−t0τ0)α,x)=∫x0gα((t2−t1τ0)α,y)gα((t1−t0τ0)α,x−y)dy, (13)

whereas (ii) for the two-sided Lévy stable distribution of two variables we get

 ~gα((t2−t0τ0)α,x)=∫∞−∞~gα((t2−t1τ0)α,y)~gα((t1−t0τ0)α,x−y)dy. (14)

The proofs of Eqs. (13) and (14) are given in Appendix A. Eqs. (13) and (14) are the new types of Laplace and Fourier convolutions, respectively, where both of the arguments of functions, i.e. and , vary. We point out that Eqs. (13) and (14) differ from the standard Laplace and Fourier convolutions of the one variable Lévy stable distributions which lead to the stable laws VMZolotarev83 (); WFeller71 (). For the stretched exponential the stable law has the form

 e−(b1p)αe−(b2p)α=e−(bp)αforbα1+bα2=bα.

Eqs. (13) and (14) are the integral forms of the evolution equations whose differential forms will be found in the next section.

The properties inherent in Eqs. (13) and (14) naturally illustrate the transitivity of evolution expressed in two-variable version of ’s and ’s. Apparently, these extensions were not employed in the current context before.

## Iii Fractional differential equation related to Lévy stable distributions

### iii.1 The differential form of Eq. (13)

We start by rephrasing Eq. (13) for and infinitesimally small . The r.h.s of Eq. (13) can be estimated by taking the asymptotics of for . Using Eq. (6), where instead of we take its first two terms of asymptotic expansion, i.e. , we obtain

 gα((ΔT)α,x−y)≃Im{∫∞0e−(x−y)ξdξπ} (1) −Im{(ΔT)αe−iπα∫∞0e−(x−y)ξξαdξπ} (2) =Im{1π(x−y)−(ΔT)αe−iπαΓ(1+α)π(x−y)1+α}. (3)

In Eqs. (1) and (2) we employed (APPrudnikov_v1, , Eq. (2.3.3.1) on p. 322). Substitution of Eq. (3) into Eq. (13) written for and gives

 RHS of Eq. (???)=Im{∫x0gα(Tα,y)x−ydyπ} (4) −Im{(ΔT)αeiπαΓ(1+α)∫x0gα(Tα,y)(x−y)1+αdyπ}. (5)

The integral in Eq. (4) can be calculated by inserting Eq. (6) into it, and then changing the order of integration. That leads to

 Im{∫x0gα(Tα,y)x−ydyπ}=Im{∫∞0e−(Tξ)αexp(−iαπ)} ×Im[∫x0e−yξx−ydy]dξπ2} (6) =Im{∫∞0e−xξe−(Tξ)αexp(−iαπ)dξπ} (7) =gα(Tα,x).

The second integral in Eq. (III.1) is calculated in Appendix B and it is equal to , see Eq. (5). In Eq. (7) we employ the definition given in Eq. (6).

Next, we calculate the integral in Eq. (5) as follows: we apply , integrate Eq. (5) by parts and use the definition of the one-sided Lévy stable distribution in Eq. (6). Thus,

 Im{Γ(1+α)eiπα∫x0gα(Tα,y)(x−y)1+αdyπ} =Γ(α)Im[limy→xgα(Tα,y)(y−x)α] (8) −Im[Γ(α)eiπα∫x0g′α(Tα,y)(x−y)αdyπ], (9)

where . From the theory of residues FWByron92 () the limit in Eq. (8) is equal to

 limy→xgα(Tα,y)(y−x)α=12πi∮Sδgα(Tα,z)(z−x)αdz. (10)

The contour is a circle around with the radius () in a counterclockwise manner. Inserting Eq. (6) into Eq. (10) and changing the order of integration we have

 Im[limy→xgα(Tα,y)(y−x)α] =Im{∫∞0e−(Tξ)αexp(−iπα) ×Im[12πi∮Sδe−zξ(z−x)αdz]dξπ}.

Setting and then taking the limit of small we obtain that the integral over closed contour has only the real value which is equal to . That leads to vanishing of the integral and consequently to vanishing Eq. (8). Consequently, Eq. (9) is equal to

 Im[Γ(α)eiπα∫x0g′α(Tαy)(x−y)αdyπ]=∫x0g′α(Tα,y)(x−y)αdyΓ(1−α), (11)

which follows from (ISGradsteyn07, , Eq. (8.334.3)) and the fact that the integral in Eq. (11) is real. That can be shown by calculating the derivative of given in Eq. (6), inserting it in Eq. (9), changing the order of integration, and using (ISGradsteyn07, , Eq. (3.381.1) on p. 346).

Expressing the l.h.s. of Eq. (13) written for and infinitesimally small into the Taylor series up to the second term we get

 gα((T+ΔT)α,x)≃gα(Tα,x)+∂Tαgα(Tα,x) ×[(T+ΔT)α−Tα] (12) ≃gα(Tα,x)+(ΔT)α∂Tαgα(Tα,x), (13)

where in Eq. (III.1) we employed the asymptotics of the square bracket, namely .

Collecting together all the contributions we obtain that the right hand side of Eq. (13) for and , where , has the form specified below:

 ∂Tαgα(Tα,x)=−C∂αxgα(Tα,x),0<α<1. (14)

In Eq. (14) denotes the so-called fractional derivative in the Caputo sense IPodlubny99 (); SGSamko93 (); Mainardi1 ()

 C∂αxf(x)=1Γ(1−α)∫x0f′(y)dy(x−y)α,

where and .

### iii.2 The differential form of Eq. (14)

Let us now find the differential form of Eq. (14) for and such that . Similarly to the approach presented in the previous subsection we will estimate the r.h.s. of Eq. (14) for infinitesimally small values of . Employing Eq. (11) adjusted for and taking the asymptotic expansion of up to the second term, we obtain

 ~gα((ΔT)α,x−y)≃∫∞−∞e−i(x−y)ξ[1−(ΔT)α|ξ|α]dξ2π =δ(x−y)−(ΔT)α2|x−y|−1−αcos(απ2)Γ(−α). (15)

In Eq. (III.2) we have used formulas (1.1) on p. 102 and (1.11) on p. 103 of YABrychkov77 (). Inserting Eq. (III.2) into the r.h.s. of Eq. (14) gives

 RHS of Eq. (???)=~gα(Tα,x)−(ΔT)α2Γ(−α)cos(πα2)×∫∞−∞~gα(Tα,y)|x−y|1+αdy. (16)

The integral in Eq. (16) is related to the fractional derivative in the Riesz sense RGorenflo98 (); SGSamko93 ():

 R∂α|x|f(x)=−12Γ(−α)cos(απ2)∫∞−∞f(y)dy|x−y|1+α (17)

for .

The first two terms of the Taylor series of the l.h.s. of Eq. (14) read

 ~gα((T+ΔT)α,x)≃~gα(Tα,x)+(ΔT)α∂Tα~gα(Tα,x), (18)

which is in analogy to Eq. (13). Substituting Eqs. (18), (16) and (17) into Eq. (14) with and infinitesimally small we obtain

 ∂Tα~gα(Tα,x)=R∂α|x|~gα(Tα,x) (19)

for , see also Mainardi1 (); Mainardi ().

## Iv The Green function method

We recall the fundamental facts about the propagator and the Green function, compare FWByron92 (). We consider the differential equation in the form

 ∂yh(y,x)=^Oh(y,x), (1)

where an operator is independent on . Its formal solution for can be given by

 h(y,x)=∫K(y,x;y′,x′)h(y′,x′)dx′.

The function is a propagator related to Eq. (1) which is an response of the system on the given initial (or boundary) conditions which are the Dirac functions. The propagator has the following properties: (i) ; (ii) ; and (iii) for the propagator satisfies , which is the Laplace convolution for the one-sided Lévy stable distribution and the Fourier convolution for the two-sided Lévy distribution.

can be used to define the Green function , which fulfills

 (∂y−^O)G(y,x;t′,x′)=δ(x−x′)δ(y−y′). (2)

The relation between the propagator and the Green function is given by

 G(y,x;y′,x′)=Θ(y−y′)K(y,x;y′,x′)+AK(y,x;y′,x′), (3)

where is the Heaviside step function and is a constant.

### iv.1 The Green function of Eq. (14)

In this subsection we will find the Green function relevant to Eq. (14) denoted by . For that purpose we solve Eq. (2) with and by taking twice the Laplace transform of this equation and applying (ISGradsteyn07, , Eq. (17.13.95) on p. 1115). The twofold use of Laplace transform, one in () and another one in () gives

 L[L[(∂Tα+C∂αx)G(Tα,x;~Tα,~x);τ];κ]=e−τ~Tαe−κ~x,

where . The Laplace transform of the first derivative with respect to is proportional to , i.e. . Similarly the Laplace transform of the fractional derivative in the Caputo sense of is given by , see (IPodlubny99, , Eq. (2.140) on p. 80 for ) or (VKiryakova13, , Eq. (6) for ). The symbols and denote the Laplace transforms with respect to time and space such that . Assuming that , we get

 GLκLτ(τα,κ;~Tα,~x)=e−τ~Tαe−κ~xτ+κα, (4)

Thus, the Green function can be obtained by taking twice the inverse Laplace transform of Eq. (4):

 G(Tα,x;~Tα,~x)=L−1[L−1[GLκLτ(τ,κ;~Tα,~x);Tα];x] =L−1[Θ(Tα−~Tα)e−κ~xe−(Tα−~Tα)κα;x] =Θ(x−~x)Θ(Tα−~Tα)gα(Tα−~Tα,x−~x). (5)

In Eq. (IV.1) we have applied (INSneddon74, , Eq. (3-3-11) on p. 146 of vol. 2) and the definition of the one-sided Lévy stable distribution. Comparing Eqs. (IV.1) and (3) we have and

 K(Tα,x;~Tα,~x)=Θ(x−~x)gα(Tα−~Tα,x−~x)

with given in Eq. (5). From the properties of the one-sided Lévy stable distribution it can be seen that the propagator satisfies properties (i), (ii) and (iii) which are listed at the beginning of this Section. The property (ii) can be easily shown to hold by demonstrating that the Laplace transform vanishes. It comes from (INSneddon74, , Eq. (3-3-11) on p. 146) and (IPodlubny99, , Eq. (2.140) on p. 80). The property (iii) is proven in Appendix A.

### iv.2 The Green function of Eq. (19)

The Green function for which is the solution of Eq. (19), i.e. Eq. (2) with and , can be obtained by taking the Laplace transform in for and Fourier transform in for :

 F[L[(∂Tα−R∂α|x|)~G(Tα,x;~Tα,~x);τ];κ]=e−τ~Tα. (6)

The Fourier transform of the fractional derivative in the Riesz sense is equal to , see (RGorenflo98, , Eq. (1.17)) and SGSamko93 (). Thus, Eq. (6) reads

 ~GFκLτ(τ,κ;~Tα,~x)=e−τ~Tατ+|κ|α,

where . After calculating the inverse Laplace transform and then the inverse Fourier transform of , we get

 ~G(Tα,x;~Tα,~x) =F−1[L−1[~GFκLτ(τ,κ;~Tα,~x);Tα];x] =F−1[Θ(Tα−~Tα)e−Tα|κ|α;x] =Θ(Tα−~Tα)~gα(Tα−~Tα,x−~x). (7)

The two-sided Lévy stable distribution is defined in Eq. (11). Comparison of Eq. (IV.2) with Eq. (3) indicates that and propagator is given by

 ~K(Tα,x;~Tα,~x)=~gα(Tα−~Tα,x−~x). (8)

We stress that the properties of imply that the requirements (i)-(iii) are satisfied for defined in Eq. (8). For example, the property (ii) follows from (INSneddon74, , Eq. (2-3-8) on p. 39 of vol. 1) and (RGorenflo98, , Eq. (1.17)).

## V The evolution operator method

### v.1 The case of Eq. (14)

From the mathematical point of view, Eq. (14) with the initial condition is the Cauchy-like problem. Its formal solution for and can be expressed via evolution operator in the following way

 gα(Tα,x)=^Uα(T)h(x)=exp(−TαC∂αx)h(x) (1)