The stretched exponential behavior and its underlying dynamics. The phenomenological approach
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
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 ()
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
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 ()
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.
, and for . We remark that the Laplace transform of the function is defined as follows INSneddon74 ():
for , whereas the inverse Laplace transform is given by
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
which for rational is equal to
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,
which is the inverse Fourier transform. In analogy to Eq. (4) we get the unique scaling relation
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
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
whereas (ii) for the two-sided Lévy stable distribution of two variables we get
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
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
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
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
where in Eq. (III.1) we employed the asymptotics of the square bracket, namely .
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
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
where an operator is independent on . Its formal solution for can be given by
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
The relation between the propagator and the Green function is given by
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
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
Thus, the Green function can be obtained by taking twice the inverse Laplace transform of Eq. (4):
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)
where . After calculating the inverse Laplace transform and then the inverse Fourier transform of , we get
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