Power Laws in Nature

An example of a statistical macrosopic relation is the distribution of the magnitude of earthquakes. If $N(E)$ is the annual mean number of earthquakes (in a zone or worldwide) of size $E$ ( $\sim$ energy released), then empirically one finds over a wide range,

$$N(E) =E^{-b}$$ (7.1)

with the constant $b \sim 1$. The relation (7.1) is called the Gutenberg-Richter law and is obviously a statistical relation for observables - it does not specify when an earthquake of some magnitude will occur but only what the mean distribution in their magnitude is. The Gutenberg-Ricter law is a power-law and is therefore scale-invariant - a change of scale in $M$ can be absorbed in a normalisation constant, leaving the form of the law invariant. The scale-invariance of the law implies a scale-invariance in the phenomena itself: earthquakes happen on all scales and there is no typical or mean magnitude! There are many other natural phenomena which exhibit power laws over a wide range of the parameters: Volcanic activity, solar-flares, charge released during lightning events, length of streams in river networks, forest fires, and even the extinction rate of biological species! Some of these power laws refer to spatial scale-free structures, or fractals, while some others refer to temporal events and are examples of the ubiquitous "one-over-f " phenomena (see chapter 2). Can the frequent appearance of such power laws in complex systems be explained in a simple way? Note that the systems mentioned above are examples of dissipative structures, with a slow but constant inflow of energy and its eventual dissipation. The systems are clearly out of equilibrium, since we know that equilibrium systems tend towards uniformity rather than complexity. On the other hand the abovementioned systems display scale-free behaviour similar to that exhibited by equilibrium systems near a critical point of a second-order phase transition. However while the critical point in equilibrium systems is reached only for some specific value of an external parameter, such as temperature, for the dissipative structures above the scale free behaviour appears to be robust and does not seem to require any fine-tuning. Bak and collaborators proposed that many dissipative complex systems naturally self-organise to a critical state, with the consequent scale-free fluctuations giving rise to power laws. In short, the proposal is that self-organised criticality is the natural state of large complex dissipative systems, relatively independent of initial conditions. It is important to note that while the critical state in an equilibrium second-order phase transition is unstable (slight perturbations move the system away from it), the critical state of self-organised systems is stable: systems are continually attracted to it! The idea that many complex systems are in a self-organised critical state is intuitively appealing because it is natural to associate complexity with a state that is balanced at the edge between total order and total disorder (sometimes loosely referred to as the "edge of chaos"). Far from the critical point, one typically has a very ordered phase on one side and a greatly disordered phase on the other side. It is only at the critical point that one has large correlations among the different parts of a large system, thus making it possible to have novel emergent properties, and in particular scale-free phenomena. In addition to the examples mentioned above, self-organised criticality has also been proposed to apply to economics, traffic jams, forest fires and even the brain!