Chapter 13 - Davidson Physics

CHAPTER 14. FRACTALS AND KINETIC GROWTH MODELS 552The sum in (14.20) is over all the boxes and involves the probabilities raised to the qth power. Forq = 0, we haveln N(l)D 0 = − lim . (14.21)l→0 ln lIf we compare the form of (14.21) with (14.17), we can identify D 0 with the box dimension. Forq = 1, we need to take the limit of (14.20) as q → 1. Letu(q) = ln ∑ ip i q , (14.22)and do a Taylor-series expansion of u(q) about q = 1. We haveu(q) = u(1) + (q − 1) dudq + . . . (14.23)The quantity u(1) = 0 because ∑ i p i = 1. The first derivative of u(q) is given by∑dudq = i p i q ln p∑ ii p i q = ∑ p i ln p i , (14.24)iwhere the last equality follows by setting q = 1. If we use the above relations, we find that D 1 isgiven by∑iD 1 = limp i ln p i. (information dimension) (14.25)l→0 ln lD 1 is called the information dimension because of the similarity of the p ln p term in the numeratorof (14.24) to the information form of the entropy.It is possible to show that D 2 as defined by (14.20) is the same as the mass dimension defined in(14.1) and the correlation dimension D c . That is, box counting gives D 0 and correlation functionsgive D 2 (cf. Sander et al.).There are many objects in nature that differ in appearance but have similar fractal dimension.An example is the different visual appearance in three dimensions of diffusion-limited aggregationclusters and the percolation clusters at the percolation threshold. (Both objects have a fractaldimension of approximately 2.5.) In some cases this difference can be accounted for by the multifractalproperties of an object. For multifractals the various D q are different, in contrast tomonofractals for which the different measures are the same. Percolation clusters are an exampleof a monofractal, because p i ∼ l D0 , the number of boxes N(l) ∼ l −D0 , and from (14.20), D q = D 0for all q. Multifractals occur when the growth quantities are not the same throughout the object,as frequently happens for the strange attractors produced by chaotic dynamics. Diffusion-limitedaggregation is an example of a multifractal.14.6 ProjectsAlthough the kinetic growth models we have considered yield beautiful pictures, there is much wedo not understand. For example, the fractal dimension of DLA clusters can be calculated only byapproximate theories whose accuracy is unknown. Why do the fractal dimensions have the values