Fractals simple random walks

Figure 1 Comparison between a simple random walk model of particle deposition and an electrochemically deposited copper fractal. (A) 2000 random walks on a square grid, (B) the digitized image (256 x 256 pixels) of copper electrodeposited from 0.75 mol I copper sulfate and 1 moll sulfuric acid in an 11 cm Whatman 541 filter paper at 5V. Both fractals are displayed using Lotus for Windows.

The fractal exponent is thus significantly smaller than that of a simple random walk, and the structure of the coil is considerably more open. This result is in good agreement with experiment, as shown by the example of polystyrene dissolved in benzene, at sufficiently low concentrations that the polymers cannot penetrate each other (which would totally change the type of solution). 

Naturally, chains in solvents close to the theta state must become very long in order to attain the asymptotic self-avoiding behavior. The chain size required for self-avoidance to become significant is called the thermal blob length [8.3]. Chains smaller than a thermal blob size are approximately random-walk chains. Even chains much larger than this size behave as simple random walk fractals with D = 2 on length scales r < ft- This goes to infinity as theta-solvent conditions are approached. 

Abstract. The square and cubic lattice percolation problem and the selfavoiding random walk model were simulated by Monte Carlo method in order to obtain new understanding of the fractal properties of branched and hnear polymer molecules. The central point of this work refers to the comparison between the cluster properties as they emerge from the percolation problem on one hand and the random walk properties on the other hand. It is shown that in both models there is a drastic difference between two and three dimensional systems. In three dimensions it is possible to find a regime where the properties converge towards simple non-avoided random walk, while in two dimensions the topological reasons prevent a smooth transition of the properties pertaining to avoided and non-avoided random walks. 

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