I discussed self-organized criticality (SOC) in Part 82. In a system that is in a state of SOC, big avalanches are rare, and small ones frequent. And all sizes are possible. There is power-law behaviour.
What is happening is that the average frequency of occurrence, N(s), of any particular size, s, of an avalanche is inversely proportional to some power τ of its size:
N(s) = s-τ
This is in contrast to the bell-shaped curve (Gaussian-law behaviour) observed in a different class of systems.
A log-log plot of the power-law equation gives a straight line (because log N(s) = -τs), with a negative slope determined by the value of the exponent τ. The system is scale-invariant: Usually the same straight line holds for all values of s. Large catastrophic events (corresponding to large values of s) are consequences of the same dynamics which causes small events. This scale invariance is also reminiscent of what happens in fractal structures: The same mechanism and dynamics is operative at all length scales.
Here are a couple of examples from real life:
Any country has a few cities with very large populations, and many cities with small populations. The figure below presents data for Germany. The y-axis shows the number of cities having a population equal to or greater than a given population size.
Similarly, the number of earthquakes of magnitude M is found to be proportional to 10-τM (the Gutenberg-Richter law). The figure below shows a graph of all the earthquakes recorded in 1995. The straight line is a plot of the Gutenberg-Richter equation with τ = 1. The value of τ varies a little from area to area, but worldwide it is τ ≈ 1. This is known as Zipf's law.
Power-law behaviour spans a very wide variety of complex phenomena in Nature, each exhibiting a characteristic value of τ. According to Bak (1996), large avalanches, not gradual variation, can lead to qualitative changes of behaviour, and may form the basis for emergent phenomena and complexity. Bak gave several examples to make the point that Nature operates at the SOC state.
Thermodynamics of small systems
To further illustrate the ubiquity of power-law behaviour in complex systems, I consider briefly the thermodynamics of 'small' systems.
The conventional (Boltzmann-Gibbs) formulation of thermodynamics is based on the following three assumptions:
- The microscopic interactions are short-ranged (compared to the size of the system).
- The time range of the microscopic 'memory' of the system is short in comparison to the observation time.
- The system evolves with time in a Euclidean-like space-time.
A system with these properties is commonly described as a large system. It has thermodynamic additivity or extensivity. In particular, the entropy S is an 'extensive' state parameter: Its magnitude is proportional to the 'size' of the system (i.e. the number of molecules in it); and for two independent systems the total entropy is simply the sum of the entropies of the two systems taken separately.
Entropy is a measure of the maximum energy available for doing useful work. It is also a measure of order and disorder, as expressed by the Boltzmann-Gibbs equation (cf. Part 22). So defined, entropy is an extensive state parameter; i.e. its value is proportional to the size of the system.
This formulation for entropy has had several notable failures, and has therefore been generalized by Tsallis. Tsallis highlighted the three basic assumptions (stated above) on which Boltzmann-Gibbs thermodynamics is based. This formulation fails whenever any of the assumptions is unjustified, i.e. whenever the system we are dealing with in not a large system.
Tsallis (1988) generalized this formulation of thermodynamics by introducing two postulates, the first generalizing the definition of entropy, and the second generalizing that of internal energy. An entropic index q was introduced, and the generalized entropy and the generalized internal energy were defined in terms of it.
The 'Tsallis entropy' so defined is nonextensive. Its limiting value for q = 1 is the standard (extensive) entropy. And (1-q) is a measure of the nonextensivity of the system. Moreover, the entropy is greater than the sum of entropies for two or more systems for q<1, and less than the sum for q>1. The combined system is said to be extensive for q = 1, superextensive for q<1, and subextensive for q>1.
I am skipping the technical details, but the key result for our discussion here is that the generalized version of the conventional Boltzmann weight factor, which governs the Maxwell-Boltzmann distribution of velocities in a gas, is no longer an exponential function; it is a power law. This is a very important result:
The Boltzmann factor need not be an exponential factor always. It can be a power law as well.
Thus in nonextensive systems the correlations among individual constituents do not decay exponentially with distance, but rather obey a power-law dependence. This has important implications for the occurrence of complex behaviour in many systems. Any number less than unity raised to a power less than unity becomes larger. For example, 0.40.3 = 0.76. Suppose a certain process or event in a system with q<1 is somewhat rare, with p equal to 0.4. If q = 0.3, the effective probability of the occurrence of that event becomes larger (0.76). A tornado or a cyclone is an example of how low-probability events can grow in intensity for nonextensive systems. Unlike the air molecules in normal conditions, the movements of air molecules in a tornado are highly correlated. Trillions and trillions of molecules are turning around in a correlated manner in a tornado. A vortex is a very rare (low-probability) occurrence, but when it is there, it controls everything because it is a nonextensive system.
It should be fully appreciated that the concept of a power-law distribution is counterintuitive, because it may lack any characteristic scale. The property prevented the use of power-law distributions in the natural sciences until the recent emergence of new paradigms (i) in probability theory, thanks to the work of Lévy and thanks to the application of power-law distributions in several problems pursued by Mandelbrot; and (ii) in the study of phase transitions, which introduced the concept of scaling for thermodynamic functions and correlation functions (Montegna and Stanley 2000).