Christian Schulzky

Modelling porous structures by repeated Sierpinski carpets

Porous materials such as sedimentary rocks often show a fractal character at certain length scales. Deterministic fractal generators, iterated upto several stages and then repeated periodically, provide a realistic model for such systems. On the fractal, diffusion is anomalous, and obeys the law 〈r2〉∼t2/dw, where 〈r2〉 is the mean square distance covered in time t and dw>2 is the random walk dimension. The question is: How is the macroscopic diffusivity related to the characteristics of the small scale fractal structure, which is hidden in the large-scale homogeneous material? In particular, do structures with same dw necessarily lead to the same diffusion coefficient at the same iteration stage? The present paper tries to shed some light on these questions.

Fractional diffusion, irreversibility and entropy

Abstract Three types of equations linking the diffusion equation and the wave equation are studied: the time fractional diffusion equation, the space fractional diffusion equation and the telegraphers equation. For each type, the entropy production is calculated and compared. It is found that the two fractional diffusions, considered as linking bridges between reversible and irreversible processes, possess counter-intuitive properties: as the equation becomes more reversible, the entropy production increases. The telegraphers equation does not have the same counter-intuitive behavior. It is suggested that the different behaviors of these equations might be related to the velocities of the corresponding random walkers.

Fractional Diffusion and Entropy Production

The entropy production rate for fractional diffusion processes is calculated and shows an apparently counter-intuitive increase with the transition from dissipative diffusion behaviour to reversible wave propagation. This is deduced directly from invariant and non-invariant factors of the (probability) density function, arising from a group method applied to the fractional differential equation which exists between the pure wave and diffusion equations. However, the counter-intuitive increase of the entropy production rate within the transition turns out to be a consequence of increasing quickness of processes as the wave case is approached. When this aspect is removed the entropy shows the expected decrease with the approach to the reversible wave limit.

Tsallis and Rényi entropies in fractional diffusion and entropy production

The entropy production rate for fractional diffusion processes using Shannon entropy was calculated previously, which showed an apparently counter intuitive increase with the transition from dissipative diffusion behaviour to reversible wave propagation. Renyi and Tsallis entropies, which have an additional parameter q generalizing the Shannon case (q=1), are shown here to have similar counter intuitive behaviours. However, the issue can be successfully treated in exactly the same manner as with Shannon entropy for q being not too large (i.e., generalizations near the Shannon case), whereas for larger q the Renyi and Tsallis entropies behave in a different way.

The similarity group and anomalous diffusion equations

A number of distinct differential equations, known as generalized diffusion equations, have been proposed to describe the phenomenon of anomalous diffusion on fractal objects. Although all are constructed to correctly reproduce the basic subdiffusive property of this phenomenon, using similarity methods it becomes very clear that this is far from sufficient to confirm their validity. The similarity group that they all have in common is the natural basis for making comparisons between these otherwise different equations, and a practical basis for comparisons between the very different modelling assumptions that their solutions each represent. Similarity induces a natural space in which to compare these solutions both with one another and with data from numerical experiments on fractals. It also reduces the differential equations to (extra-) ordinary ones, which are presented here for the first time. It becomes clear here from this approach that the proposed equations cannot agree even qualitatively with either each other or the data, suggesting that a new approach is needed.

Random walks on finitely ramified Sierpinski carpets

A new algorithm is presented that allows an efficient computer simulation of random walks on finitely ramified Sierpinski carpets. Instead of using a bitmap of the nth iteration of the carpet to determine allowed neighbor sites, neighbourhood relations are stored in small lookup tables and a hierarchical coordinate notation is used to give the random walker position. The resulting algorithm has low memory requirements, shows no surface effects even for extremely long walks and is well suited for modern computer architectures.

Clouds, fibres and echoes: a new approach to studying random walks on fractals

Up to now the general approach of constructing evolution differential equations to describe random walks on fractals has not succeeded. Is this because the true probability density function is inherently fractal? When plotted in the appropriate similarity variable, we find a cloud which is not too smooth. Further investigation shows that this cloud has a structure that might be overlooked if one is looking for the usual single-valued probability density function. The cloud is composed of an infinite family of smooth fibres, each of which describes the behaviour of the walk on an infinite echo point class. The fibres are individually smooth and so are naturally amenable to analysis with differential equations.

The pore structure of Sierpinski carpets

In this paper, a new method is developed to investigate the pore structure of finitely and even infinitely ramified Sierpinski carpets. The holes in every iteration stage of the carpet are described by a hole-counting polynomial. This polynomial can be computed iteratively for all carpet stages and contains information about the distribution of holes with different areas and perimeters, from which dimensions governing the scaling of these quantities can be determined. Whereas the hole area is known to be two dimensional, the dimension of the hole perimeter may be related to the random walk dimension.

AN EFFICIENT IMPLEMENTATION OF THE EXACT ENUMERATION METHOD FOR RANDOM WALKS ON SIERPINSKI CARPETS

In the following, we present a highly efficient algorithm to iterate the master equation for random walks on effectively infinite Sierpinski carpets, i.e. without surface effects. The resulting probability distribution can, for instance, be used to get an estimate for the random walk dimension, which is determined by the scaling exponent of the mean square displacement versus time. The advantage of this algorithm is a dynamic data structure for storing the fractal. It covers only a little bit more than the points of the fractal with positive probability and is enlarged when needed. Thus the size of the considered part of the Sierpinski carpet need not be fixed at the beginning of the algorithm. It is restricted only by the amount of available computer RAM. Furthermore, all the information which is needed in every step to update the probability distribution is stored in tables. The lookup of this information is much faster compared to a repeated calculation. Hence, every time step is speeded up and the total computation time for a given number of time steps is decreased.

Resistance scaling and random walk dimensions for finitely ramified Sierpinski carpets

We present a new algorithm to calculate the random walk dimensionof finitely ramified Sierpinski carpets. The fractal structure isinterpreted as a resistor network for which the resistance scalingexponent is calculated using Mathematica. A fractal form of theEinstein relation, which connects diffusion with conductivity, isused to give a numerical value for the random walk dimension.