The
published below study is the characteristic one, where some issues concerned
with the fractality claims, those “should be” present
everywhere. To this extend it is appropriate to point out here, among others;
the interesting discussion appeared in #279 of “Science”:
a)
Biham, O., Malcai, O., Lidar, D.A., and Avnir, D., “Is
Nature Fractal? (Reply to Letters by B.Mandelbrot and
P.Pfeifer), Science, V. 279, p. 785, (1998)
b)
Biham, O., Malcai, O., Lidar, D.A., and Avnir, D., “Fractality in Nature (Reply to a Letter by A.A.Tsonis), Science, V. 279, p. 1615, (1998)
Example of
Inadequacy of Available Diffusion Theories to Describe the Transport Properties
of PointLike Objects in Multi Scale Heterogeneous Substructure
Y.G. Gordienko,
V.S.Travkin*
Institute
for Metal Physics,
*HSPT,
Abstract
Simulation of diffusion on the basis of fractional
diffusion equation was performed for investigation of anomalous transport
properties in plastically deformed AlMg single crystals. Digitized 2D surface
images were used for morphological analysis, computation of the spatial
correlation functions (by calculation of “fractal” dimensions) and simulation
for different scales of AlMg samples. On the basis of calculated fractal
dimension the fractional diffusion equation was constructed and diffusion
problem was numerically solved (with distribution of the given initial
heterogeneous concentration of "diffusing particles" in 2D domain
with periodically boundary conditions). These calculations were verified by
computer simulation of diffusion in 2D rectangular lattice for different types
of simulated random heterogeneous media with the same level "defect
substructure content" as it was observed in AlMg samples. These types of
media were created by random distribution of numerous parts of material
substance in 2D rectangular lattice ("defect substructure content"),
which have different sizes (1, 2, 4, 8, 16, 32). As a
result the different types of dynamics of diffusion processes (nonGaussian
shape of average particle density or "deformed Gaussian") were
observed for different media with the same "defect substructure
content". The conclusion is that measures of volume fraction and uniform
spatial correlation are not enough for characterization of diffusion transport
properties in multiscale heterogeneous media.
Motivation. The problem was motivated
by interest in diffusion processes of pointlike defects in metals under
fatigue loading among bandlike substructure regions, which are assumed to be
impenetrable for pointlike defects (Fig. 1).
The main aim is to show inadequacy of
the current approaches to determine and characterize universal characteristics
for anomalous diffusion. To determine universal characteristics for diffusion
and investigate the physical reasons for anomalous diffusion.
Problem Formulation

Fig. 1. Initial bandlike
pattern on AlMg surface <001>{100} For simulation of
diffusion processes on this heterogeneous
and anisotropic terrain we simplified the pattern
with roughening the whole picture to 1bit blackandwhite pattern shown in
Fig. 2. 

Fig 2. Highcontrast
snapshot used in simulations 
Then,
two different diffusion problems were considered: with perpendicular
(Fig. 3) and parallel (Fig. 4) mutual orientation of bands and initial
source of diffusing particles.
a) whole pattern b) initial source c) final distribution of
particles 
a) 
b) 
c) 


Fig. 3. Perpendicular
orientation of 
Fig. 4. Parallel
orientation of 

Analytical Solutions
In general, this simulation
with symmetry along one of the axes for each case can be described by equations
_{} 
_{} 
a) homogeneous case,
standard diffusion equation 
b) heterogeneous case,
fractal diffusion equation [1] 
where D_{i} = D_{^}_{ }for perpendicular and D_{i} = D_{} for parallel orientation,
_{}, d_{f}_{
}— fractal dimension of medium, d_{w} —anomalous diffusion exponent (d_{w}= 2
for standard diffusion and d_{w}= 1.333 for selfavoiding walks [2]) and by
definition fractional derivative is: _{}.
Periodic boundary conditions
for torus topology are:_{}, a>>1
and initial condition is as follows:
_{}
As a result in a limit x_{2}x_{1}®0 one can obtain:
_{} 
_{} 
a) homogeneous case, standard
diffusion equation 
b) heterogeneous case,
fractal diffusion equation, _{}, _{} 
The numerical solutions are
shown below for the x_{2}x_{1}=1, y_{0}=1, a=5, t =
[0,4], D_{i}=1.

d_{f}_{ }=2, d_{w}=1.333 

a) standard equation 
b) fractional equation 
Fig. 5. Numerical solutions
Simulation Results on
Perpendicular Orientation
Below simulation results on
heterogeneous cases are shown in comparison with a homogeneous case (empty 2D
embedding space).


a) homogeneous case 



b) heterogeneous case 

Fig. 6. Typical
distributions of particles after time T 



a) case homogeneous 
b) heterogeneous case 
Fig. 7. Typical
distributions fitted by Gaussian _{}with
parameters shown in the left upper corners 
Fractal Dimension 
Homogeneous 
Heterogeneous 
Capacity 
2 
1.91±.03 
Information 
2 
1.89±.015 
Correlation 
2 
1.87±.02 
NOTE. As one can easily note all Gaussian parameters
are nearly equal for this two qualitatively different
morphologies and they cannot be resolved, nevertheless different fractal
dimensions. The reason is availability of pronounced anisotropy of elongated
substructure regions ("bands"), which cannot impede diffusion of
particles along bands.
Simulation Results on Parallel
Orientation
Below simulation results on
heterogeneous cases are shown in comparison with a homogeneous case (empty 2D
embedding space).


a) homogeneous case 
b) heterogeneous case 
Fig. 8. Typical
distributions of particles after time T/4 



a) case homogeneous 
b) heterogeneous case 
Fig. 9. Typical
distributions fitted by Gaussian 
NOTE. In this case two qualitatively different
morphologies can be noted easily, because some Gaussian parameters are
substantially different. The reason is availability of "screening"
effect, which is caused by merging bands.
Continuous Transition from
Heterogeneous to Homogeneous Morphology
The typical heterogeneous
patterns with the same content of impenetrable phase and the same fractal dimensions
1.99±.01 are shown. They CANNOT be described
adequately by the standard diffusion and modern theories of diffusion on the
basis of fractional calculus.


b) smallest object 4´4 

c) smallest object 8´8 

d) smallest object 32´32 

e) empty space (smallest
object ~300´300) 

Fig. 10. Different diffusion
scenarios with increasing width of peak (w)
Conclusions
— diffusion of pointlike defects in anisotropic
heterogeneous substructure tend to be localized along direction of
"bandlike" defect substructure (see Fig. 6 and 8); it can improve localizaion of evolution processes in defect substructure
during further fatigue loading (which was observed in experiments [3]);
— comparison of simulated concentration
profiles (see Fig. 7 and 9) with results of analytical calculations on the
basis of fractional equation (Fig. 5) allows us to find INADEQUACY of the
current theories of fractional calculus for description of the transport properties
of pointlike objects in heterogeneous substructure;
— however, for quantitative conclusion simulated domain
should be extended (and it easily can be done) in the further investigation;
— the new anomalous feature
was found by simulations for heterogeneous patterns with the same content of
impenetrable phase, the same fractal dimension (which is »2), but with different sizes of randomly
distributed impenetrable obstacles (Fig. 10); it consists in impeded diffusion
for the patterns with the smaller obstacles. We assume that in the case the
finer impenetrable substructure contains multiscale structure, for example lagoons of linked
obstacles ("traps") and particles have to visit the same places (in
traps) much frequently than other ones
(outside traps).
— that is why we conclude
about necessity to improve the wellknown fractional diffusion equations by
taking into account the scales of the heterogeneous substructure.
References
1. B.B.Mandelbrot, C.J.G.Evertsz,
and Y.Hayakawa, Phys. Rev. A42 (1990) 4528.
2. L.Peliti, Random Walks with Memory,
in Fractals in Physics (Russian translation), (Moscow: Mir: 1988).
3. E.E.Zasimchuk and Yu.G.Gordienko,
to be published.
4. B.B.Mandelbrot, J.Fluid.Mechanics., 62
(1974) 331.
5. A.Bunde and S.Havlin,
(eds.) Fractals and Disordered Systems, (SpringerVerlag:
Berlin: 1991).