Forest Fires

Look at the figures which show the number of fires as a function of the area burnt, in different regions of the United States of America and Australia. On a log-log plot one sees that the data is well approximated by straight lines, meaning that the number of fires as a function of area is a power law:

$$N \propto A^{-\alpha}$$ (1.1)

with $\alpha \sim 1.3-1.5$. It is important to note that the straight line fits are for a wide range of the data (one can always fit a straight line to a small range), and for different geographical regions. This suggests a universality in the phenomena that requires a explanation. Power laws are observed in many other natural phenomena such as earthquakes and solar flare activity. It has been suggested that these phenomena are examples of self-organised criticality, that is, the systems are attracted to a state which is between that of total order and total disorder. The word "critical" is borrowed from well-studied thermal equilibrium systems that undergo second-order phase transitions at critical points and display power laws. However in the case of forest fires (or earthquakes etc.), the systems are far from equilibrium and the power-law behaviour, that is criticality, does not require fine-tuning -- it is self-organised. Given the complexity fo the actual systems, it is impossible at present to study those systems from first principles. Rather one studies simplified models to check whether power-laws emerge naturally. Power laws imply a self-similarity at different scales, so it is natural to suppose that self-organised criticality might be the common dynamical mechanism behind the wide occurence of fractal structures in nature. Power laws also appear in social contexts, for example the frequency of occurence of words in a literary text, and the magnitude of wars. We will come back to these examples later in the course.