Donald L.Turcotte*^{†} and John B.Rundle^{‡}
*Department of Earth and Atmospheric Sciences, Cornell University, Ithaca, NY 14853; and ^{‡}Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder, CO 80309
The National Academy of Sciences convened an Arthur M. Sackler Colloquium on “Self-organized complexity in the physical, biological, and social sciences” at the NAS Beckman Center, Irvine, CA, on March 23–24, 2001. The organizers were D.L.T. (Cornell), J.B.R. (Colorado), and Hans Frauenfelder (Los Alamos National Laboratory, Los Alamos, NM). The organizers had no difficulty in finding many examples of complexity in subjects ranging from fluid turbulence to social networks. However, an acceptable definition for self-organizing complexity is much more elusive. Symptoms of systems that exhibit self-organizing complexity include fractal statistics and chaotic behavior. Some examples of such systems are completely deterministic (i.e., fluid turbulence), whereas others have a large stochastic component (i.e., exchange rates). The governing equations (if they exist) are generally nonlinear and may also have a stochastic driver. Many of the concepts that have evolved in statistical physics are applicable (i.e., renormalization group theory and self-organized criticality). As a brief introduction, we consider a few of the symptoms that are associated with self-organizing complexity.
The classic example of self-organizing complexity is the frequency-size distribution of earthquakes. Earthquakes are certainly complex yet they universally satisfy the relation (to a good approximation)
N ~ A^{−D}^{/2},
[1]
where N is the number of earthquakes in a specified time interval and region with their rupture area greater than A. This is the well known Guttenberg-Richter relation (1). The scaling exponent D is the fractal dimension introduced by B.Mandelbrot (2). In his book, Mandelbrot (3) pointed out the wide range of validity of the fractal scaling relation
[2]
where N_{i} is the number of objects of size r_{i}. Power-law scaling is second only to the Gaussian distribution in terms of applicability. Its relatively recent acceptance, in terms of the fractal paradigm, can be attributed to the fact that it cannot be used as a continuous probability distribution function without a cutoff. The integral of Eq. 2 diverges to infinity either at r=0 (for D > 1) or as r → ∞ (D < 1).
Another example of the applicability of power-law frequency-size scaling is in fragmentation. Certainly not all frequency-mass distributions of fragments are power law, but many are (4). An example is the frequency-mass distribution of asteroids and meteorites. One consequence is that number-area distribution of planetary craters satisfies Eq. 1 in many cases.
There are so many examples of the applicability of power-law scaling in biology that the term “allometry” was introduced to describe them. The classic example of allometric scaling in biology is the power-law scaling of a species’ metabolic rate with the species’ mass (5). It is applicable from ants to elephants.
A complex phenomenon is said to exhibit self-organizing complexity only if it has some form of power-law (fractal) scaling. It should be emphasized, however, that the power-law scaling may be applicable only over a limited range of scales.
Another classic example of self-organizing complexity is drainage networks. These networks are characterized by the concept of stream order. The smallest streams are first-order streams— two first-order streams merge to form a second-order stream, two second-order streams merge to form a third-order stream, and so forth. Drainage networks satisfy the fractal relation Eq. 2 with N_{i} the number of ith-order streams and r_{i} the mean length of these streams (6). This fractal scaling was recognized and generally accepted some 20 years before Mandelbrot’s introduction of the fractal concept.
There are many branching networks in biology that exhibit fractal scaling to a good approximation (7). Actual trees and plants, root systems, the vein structure of leaves, cardiovascular systems, and bronchial systems are examples. The concept of branch order for both actual trees and drainage networks can be traced back to Leonardo da Vinci.
Many time series are examples of self-organizing complexity. Examples include:
A velocity component at a point in a turbulent flow.
Global mean temperatures.
River flows.
Economic time series such as a stock market index or an exchange rate.
Intervals between heartbeats.
Time series are characterized by the probability distribution function of the values (usually a Gaussian) and correlations between adjacent values. A time series in which adjacent values are positively correlated is said to be persistent. The standard approach to quantifying persistence is to carry out a Fourier analysis. If the Fourier coefficients A_{n} have a power-law dependence on the wavelengths λ_{n}
[3]
a time series is said to be a self-affine fractal (8, 9). For a white noise β=0, if β=2 the time series is a Brownian (random) walk, and β=1 defines an 1/f or red noise. For a time series to exhibit self-organizing complexity it must satisfy Eq. 3, at least over
^{†} |
To whom reprint requests should be addressed. E-mail: turcotte@geology.cornell.edu. |