R. Kanno

National Institute for Fusion Science (NIFS-Repository)

§29. Diffusion on a Self-similar Structure Kanno,Il. \Vc analyze diffusion phenon1ena on a self-sirnilar structure with fractal in the tirne-axis. The behavior of this rnotion shows the non-Gaussian-type. This behavior can be seen nu-rnerically, for exan1ple, for the field-line trans-port in stochastic n1agnetic fields between mag-netic islands1). Previous analytical studies at-ternpt to treat the rnotion in real space-tirne with integral dimensions. To clarify the rela-tion between the non-Gaussian-type behavior and the fractal nature, "Yve choose a different starting point. We notice that the randorn par-ticle is able to n1ove on only fractal mediun1 with fractal in the tirne-axis. This means that frorn the particle's viewpoint, space-tin1e seen1s to have non-integral dirnensions, i.e. we can sa~ that this n1otion is the randon1 walk in frac-tal space-tirne. It is well known that in real space-tirne, a fractal structure has infinite or zero length . To rneasure a finite and non-zero length of any self-sirnilar set X, usually the Hausdorff length He is used; He(X) = lim Hpe(_X) =constant for onlv £, p---+0 J (1) where e is a real-nurnber and describes the Hausdorff dirnension of the set X, and H~( _X) is a length of X divided by JV parts {Xi} and is given by H!(X) = inf {~d/1 0 < d;::; p,X <;;; iQ1 X;}. (2) Here di is a diarneter of the i- th part _Xi and is rneasured in real-space ·with integral dirnen-sions. By using the Hausdorff length , arbitrary fractal-space and fractal-tirne can be n1easured. Now; we assurne that these Hausdorff lengths can be used to specify coordinates ( x, i), and derivatives and integrals of a function on these coordinates are acceptable. Fractal space-tirne 210 ( x, t) is defined by these Hausdorff lengths. i = Ha(x), t = H f3 (t). (3) Here, space-tirne with a = 1 and f3 = 1 n1eans real space-tirne. By following the path integralrnethod devel-oped by Fcynrnan2) , we carry out integrations and finally obtain a result as follows. Xb- Xa { 1 } 1/2 { ( - - )2 } f(b,a)= 4nD(tb-ta) exp -4D(tb-ta) . (4) In real space-tirne, the transition probabil-ity f can be seen to be proportional to exp {-z2a j4Drf3} under the expectation that important contributions to the integral off will occur only for srnall z and r, where z = i 1/a and r = T- 11f3 are n1easured in real space-tirne. Thus, by re-normalizing the probability in real-space, the equation ( 4) can be rewritten as fol-lows. 1 { 2 2a } f(z, r) = B exp - 4Drf3 , (5) where B is the nonnalizing factor. Using eq.(5), \ve can calculate R(r) - z2 and G = - -2 z4 /(3z2 ) -1; R(r) = ( 4D) 1far (.l...) ____ :..._2_a :__ (3 /a r (2~) r ' (6) r (ia) r (2:) _.:___.:_______.:.....__:__ - 1 ' 3 [ r (2:)] 2 (7) G = where r is the garnrna function. In real space-tirne, the randorn walk shows the non-Gaussian process, if G -=/: 0, i.e. a -=/: 1. On the other hand, for a case of f3 -=/: 1 and a = 1, the pro-cess represents the fractional Brownian process . Thus, we have the unified representation of the Brownian process (a = 1, f3 = 1), the frac-tional Brownian process (a = 1, (3 -=/: 1), and the non-Gaussian process with fractal nature (a-=/: 1). References 1) Zimbardo,G. and Veltri,P. , Phys. Rev. E 51 (1995) 1412. 2) Feynrnan,R.P., Rev. Mod. Phys. 20(1948)367. 

Diffusion on a Self-similar Structure

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