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3 ∗ 1 A comparison study of slow–subdiffusion and subdiffusion 0 2 K.D. Lewandowska n a J Department of Radiological Informatics and Statistics, Medical University of Gdan´sk, 0 2 ul. Tuwima 15, 80-210 Gdan´sk, Poland. and ] h c Tadeusz Koszto lowicz e m Institute of Physics, Jan KochanowskiUniversity, - ul. S´wi¸etokrzyska 15, 25-406Kielce, Poland. t a t s . t a m We study slow–subdiffusionincomparisonto subdiffusion. Bothofthe processes are treated as random walks and can be described within con- - d tinuous time random walk formalism. However, the probability density of n the waiting time of a random walker to take its next step ω(t) is assumed o over a long time limit in the form ω(t) 1/tα+1 for subdiffusion, and c in the form ω(t) h(t)/t for slow–subdiff∼usion [h(t) is a slowly varying [ ∼ function]. We show that Green functions for slow–subdiffusionand subdif- 1 fusioncanbe verysimilarwhenthe subdiffusioncoefficientDα depends on v time. This createsthe possibilityof describingslow–subdiffusionby means 4 ofsubdiffusion witha smallvalue ofthe subdiffusion parameterα, andthe 0 subdiffusion coefficient Dα varying over time. 7 4 . PACS numbers: 05.40.-a,05.60.-k, 05.40.Fb 1 0 3 1. Introduction 1 : v Anomalous diffusion is usually described within a framework of the con- i X tinuous time random walk in which the probability densiy of finding a random walker at time t at point x, P(x,t), depends on the probability den- r a sity of the waiting time of the random walker to take its next step ω(t) and the probability density of jump length λ(x). When ω(t) and λ(x) are independent then the probability density P(x,t) reads in terms of the ∗ Presented at 25th Marian Smoluchowski Symposium on Statistical Physics (1) 2 lewandowska printed on January 22, 2013 Laplace transform ω(t) ωˆ(s) ∞e−stf(t)dt and the Fourier trans- L{ } ≡ ≡ 0 form λ(x) λˆ(k) ∞ eikxf(x)dx [1] F{ } ≡ ≡ −∞ R R 1 ωˆ(s) 1 Pˆ(k,s) = − . (1) s 1 ωˆ(s)λˆ(k) − As far as we know, an inverse transform of the above equation in the most general casehasnotbeenfoundyet, withtheexpection of afewvery special cases. For this reason, Eq.(1)is usually considered within thelimit of small values of s and k. For subdiffusion, it is assumed that the first moment of ω(t) (the average value) is infinite and that the Laplace transfom ωˆ(s) has the following form for small values of s ωˆ(s) = 1 ταs , (2) ∼ α − where subdiffusion parameter α obeys 0 < α < 1 and τ is a positive α parameter, whereas all the moments of the natural order of λ(x) are finite. Assuming that λ(t) is a symmetric function, the Fourier transform λˆ(k) has the following form for small values of k k2 λˆ(k) = 1 σ2 , (3) ∼ − 2 whereσ2 is thesecond momentof λ(x). Computingtheinverse Laplace and Fourier transforms leads to ω(t) being proportional to 1/t1+α (therefore ω(t) is refered to as a heavy–tailed distribution) and λ(x) in the form of Gauss distribution. A subdiffusion equation which is obtained as a result of continuous time random walk formalism is a linear differential equation with a fractional time derivative. A random walk in which ω(t) is a superheavy–tailed distribution has been recently studied [2, 3]. The superheavy–tailed distribution means that h(t) ω(t) , (4) ∼ t where h(t) is a slowly varying function, i.e. h(χt)/h(t) 1 when t + → → ∞ for all positive values of χ. All the fractional moments of ω(t) ( ωc(t) = h i xcω(t)dt, c > 0) are infinite. Such a process qualitatively differs from subdiffusion and is usually called slow–subdiffusion. It is rather difficult to R find an equation with fractional derivatives which describes such a process since an order of the derivative would be very close to zero. InthispaperwewillcompareGreenfunctionsforsubdiffusionandslow– subdiffusionandwewillalsoshowthatslow–subdiffusioncouldbedescribed by subdiffusion with small values of subdiffusion parameter α and with subdiffusion coefficient D which depends on time. α lewandowska printed on January 22, 2013 3 2. Green functions Green functions for subdiffusive transport in an unrestricted system is well known. They can be obtained after substituting Eqs. (2) and (3) into Eq. (1) and then computing the inverse Laplace and Fourier transforms. After calculation we obtain 1 x P(x,t) = f t; | | , (5) 2√D α/2−1,α/2 √D α (cid:18) α(cid:19) where [4] 1 ∞ 1 a k f (t;a) = . (6) ν,β t1+ν k!Γ( kβ ν) −tβ kX=0 − − (cid:18) (cid:19) The Green function for slow–subdiffusive transport in an unrestriced system was derived in [2] and reads over a long time limit P (x,t) = V(t)e−|x|√2V(t)/σ2 , (7) s s 2σ2 whereV(t)isthecomplementarycumulativedistributionfunctionofwaiting times V(t) = 1 tω(t)dt = ∞ω(t)dt, V(0) = 1 and V(+ ) = 0. In the − 0 t ∞ further consideration we take ω(t) in the following form R R (r 1)lnr−1η ω(t)= − , (8) (η+t)lnr(η+t) where r > 1 and η > 1, which provides [2] lnη r−1 V(t)= . (9) ln(η+t) (cid:20) (cid:21) 3. Comparision between the Green functions for slow–subdiffusion and subdiffusion In Figs. 1–3 we present the comparision between Green functions for slow–subdiffusion and subdiffusion calculates from the formulae given by Eqs. (7) and (5), respectively. We are interested in testing if we can choose the parameters in such a way that functions (7) and (8) cover function (5). We arbitrarily chose thevalues of theparameters r, σ2, η andα. Inorder to match Green functions for slow–subdiffusion and subdiffusion for all times, the subdiffusion coefficient D has to depend on time. Therefore, subdiffu- α sion coefficient D was treated as a fitting parameter. All calculations were α performed for r = 2.34, σ2 = 0.1 and α = 0.01 and the values of the rest 4 lewandowska printed on January 22, 2013 of the parameters were given in the legend and the figures’ captions. It is interesting to find the dependance of the subdiffusion coefficient on the rest of parameters for the given value of t. The preliminary studies showed that the subdiffusioncoefficent most strongly dependson η. In Fig. 4 we present the dependence between the fitting parameter D and the parameter η. α 0,015 t=105, D =1125 α t=1010, D =2520 α t=1015, D =3860 α t=1020, D =5120 α 0,01 t=1025, D =6190 α t=1030, D =7050 α ) x,t ( P 0,005 0 -500 -250 0 250 500 x Fig.1. The comparision between Green functions for subdiffusion [Eq. (5)] and slow–subdiffusion[Eq.(7)]. The latter Greenfunction was calculatedfor η =1.01. The values of time and subdiffusion coefficient are given in the legend. The lines represents the Green functions for slow–subdiffusion, the symbols — for subdiffusion. All quantities are given in arbitrarily chosen units. As can be seen in Figs. 1–3, Green functions for subdiffusion and slow– subdiffusion are in quite good agreement for all values of η for the subdiffusion coefficient D depending on time. Moreover, we can observe that α the values of the subdiffusion coefficient depend strongly on the values of parameter η. In Fig. 4 we can observe that for higher values of η we have obtained lower values of D . Therefore, we conclude that it is possible to α describe slow–subdiffusion as subdiffusion with small values of subdiffusion parameter and a subdiffusion coefficient which depends on time. A more detailed description of the subject presented in this paper will be present elsewhere. lewandowska printed on January 22, 2013 5 0,4 t=105, D =1.7 α t=1010, D =3.8 α t=1015, D =5.9 α 0,3 t=1020, D =7.7 α t=1025, D =9.3 α t=1030, D =10.6 α ) x,t0,2 ( P 0,1 0 -10 -5 0 5 10 x Fig.2. The same as in Fig. 1, but for η =3.54. 4. Final reamrks In this paper we have compared Green functions obtained for subdiffusion and slow–subdiffusion. It seems that it is possible to descibe slow– subdiffusion by means of subdiffusionwith a small value of subdiffusion parameter and the subdiffusion coefficient depending on time although these two processes seem to be qualitatively different from each other. We have encountered a similar situation when we have approximated the solution to nonlinearsundiffusionequationsbythesolutiontothesubdiffusionequation with a fractional time derivative with the subdiffusioncoefficient depending on time. It would be interesting to study the dependence of the subdiffusion coefficient on time or the dependence of the agreement between the Green functions for slow–subdiffusion and subdiffusionon the subdiffusion parameter α. Moreover, investigation like this could allow one to find a more transparent physical interpretation of slow–subdiffusion. Acknowledgments This paper was partially supported by the Polish National Science Cen- tre under grant No. N N202 19 56 40 (1956/B/H03/2011/40). 6 lewandowska printed on January 22, 2013 t=105, D =0.71 α 0,5 t=1010, D =1.6 α t=1015, D =2.5 α t=1020, D =3.25 α 0,4 t=1025, D =3.9 α t=1030, D =4.45 α ) x,t0,3 ( P 0,2 0,1 0 -10 -5 0 5 10 x Fig.3. The same as in Fig. 1, but for η =11.23. REFERENCES [1] R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000); J. Phys. A 37, R161 (2004). [2] S.I. Denisov and H. Kantz, Phys. Rev. E 83, 041132 (2011). [3] S.I. Denisov, S.B. Yuste, S. Bystrik Yu, H. Kantz, and K. Lindenberg Phys. Rev. E 84, 061143 (2011). [4] T. Kosztol owicz, J. Phys. A 37, 10779 (2004). lewandowska printed on January 22, 2013 7 30 25 20 Dα15 10 5 0 0 2 4 6 8 10 12 η Fig.4. The dependence of the subdiffusion coefficient Dα on the parameter η for t=105.