Chapter 13 - Davidson Physics

CHAPTER 14. FRACTALS AND KINETIC GROWTH MODELS 520Figure 14.1: Example of a spanning percolation cluster generated at p = 0.5927 on a L = 124square lattice. The other occupied sites are not shown.If the dimension of the object, D, and the dimension of the Euclidean space in which theobject is embedded, d, are identical, then the mass density ρ = M/R d scales asρ(R) ∝ M(R)/R d ∼ R 0 , (14.2)that is, its density is constant. An example of a two-dimensional object is shown in Figure 14.2.An object whose mass-length relation satisfies (14.1) with D = d is said to be compact.Equation (14.1) can be generalized to define the fractal dimension. We denote objects asfractals if they satisfy (14.1) with a value of D different from the spatial dimension d. If an objectsatisfies (14.1) with D < d, its density is not the same for all R, but scales asρ(R) ∝ M/R d ∼ R D−d . (14.3)Because D < d, we see that a fractal object becomes less dense at larger length scales. The scaledependence of the density is a quantitative measure of the ramified or stringy nature of fractalobjects. In addition, another characteristic of fractal objects is that they have holes of all sizes.This property follows from (14.3) because if we replace R by Rb, where b is some constant, weobtain the same power law dependence for ρ(R). Thus, it does not matter what scale of length isused, and thus all hole sizes must be present.Another important characteristic of fractal objects is that they look the same over a rangeof length scales. This property of self-similarity or scale invariance means that if we take part ofa fractal object and magnify it by the same magnification factor in all directions, the magnifiedpicture is similar to the original. This property follows from the scaling argument given for ρ(R).