INTRODUCTION TO SELF-ORGANIZED CRITICALITY

Alexander Olemskoi (Personal webpage)

In the first part of lectures, an introduction to the phenomena of self-organized criticality, which offers considerable insight into a wide range of phenomena from earthquakes to traffic jams is given. In standard critical phenomena, there is a control parameter which an experimenter can vary to obtain the radical change in behaviour. Self-organized critical phenomena, by contrast, is exhibited by driven systems which reach a critical state by their intrinsic dynamics, independently of the value of any control parameter.

The archetype of a self-organized critical system is a sand pile. Sand is slowly dropped onto a surface, forming a pile. As the pile grows, avalanches occur which carry sand from the top to the bottom of the pile. The slope of the pile becomes independent of the rate at which the system is driven by dropping sand. This is the (self-organized) critical slope. In the second part of lectures we present the theory of a flux steady-state related to avalanche formation for the simplest model of a sand pile within the framework of the Lorenz approach. The stationary values of sand velocity and sand pile slope are derived as functions of a control parameter (driven sand pile slope). The additive noise of above values are introduced for building a phase diagram, where the noise intensities determine both avalanche and non-avalanche domains, as well as mixed one. Corresponding to the SOC regime, the last domain is crucial to affect of the noise intensity of the vertical component of sand velocity and especially sand pile slope.

To address to a self-similar behaviour, we use a fractional feedback as an efficient ingredient of the modified Lorenz system. In the spirit of Edwards paradigm, an effective thermodynamics is introduced to determine a distribution over an avalanche ensemble with negative temperature. Steady-state behavior of the moving grains number, as well as nonextensive values of entropy and energy is studied in detail. The power law distribution over the avalanche size is described within a fractional Lorenz scheme, where the energy noise plays a crucial role. This distribution is shown to be a solution of both fractional and nonlinear Fokker-Planck equation. As a result, we obtain new relations between the exponent of the size distribution, fractal dimension of phase space, characteristic exponent of multiplicative noise, number of governing equations, dynamical exponents and nonextensivity parameter.