invariant scaling laws that describe their relationships within some limits on the
range of values.
In describing the functional size, radius of gyration, Rg, of a polymer such as a polypeptide, composed of N monomers, assume each of the amino acids to be the same and that they are in a “good” hydrophobic solvent that didn’t stick the polymer together in a fold. Flory (1971) found a scaling law for certain broad classes of polymers and solvents, Rg ~ aN” where the exponent, v = 3/5, was universal, N indicated the number of monomers in the chain and the value of “pre-factor” a depended upon the particular monomer and solvent chosen. Log Rg plotted against log N has a “power law” slope of 0.60. For an equally static but less physical example, there is the well known Zipf law of “vocabulary balance’(Zipf, 1949). First reported for the 260,450 words of James Joyce’s Ulysses, the slope of the log of the rank of the words found (ordered from most to least along x) plotted against the log of their frequency (along y) results in a power law that is (generally) true for other
collections of words and in other languages.
An accessible example of a dynamical scaling law arises in a two dimensional lattice model of a forest which is to be set on fire with probability p independent random single tree ignitions. At some critical p, pc, the fire sweeps through the entire forest (“percolates”) and the correlation length of the connected clusters grows as |p-p,|’ with a universal scaling exponent, y = 4/3, for all Monte
Carlo, two dimensional percolation problems (Stauffer, 1985; Grimmett, 1989).
Mandelbrot’s scheme for the power laws that compose his fractal geometry of dynamical objects is a measure made on the pattern of occupancy in the embedding space by the reconstructed orbits of an attractor. It is, generally, mass = length” in which Do (the subscript that of the “capacity dimension”) is not the whole number of Euclidian dimensions, d, of the space in which the orbits are embedded. After Hausdorffs “convergence of external and internal measures” (Hurewicz and Wallman, 1948), the (capacity) fractal dimension Do is also defined as being larger than its topological dimension and smaller than its Euclidian
embedding dimension. Graphing a time series on a plane one can think of its
226
HOUSE_OVERSIGHT_013726