Kinetic description of avalanching systems.
Avalanching systems are treated analytically using the renormalization group (in the self-organized-criticality regime) or mean-field approximation, respectively. The latter describes the state in terms of the mean number of active and passive sites, without addressing the inhomogeneity in their distribution. This paper goes one step further by proposing a kinetic description of avalanching systems making use of the distribution function for clusters of active sites. We illustrate an application of the kinetic formalism to a model proposed for the description of the avalanching processes in the reconnecting current sheet of the Earth's magnetosphere. A description of avalanching systems is proposed that makes use of the distribution function for clusters of active sites. A general kinetic equation is derived that describes the temporal evolution of the distribution function, in terms of growth and shrinking probabilities. The distribution of clusters is derived for the stationary regime, for a quite general class of avalanching systems or arbitrary dimensionality. The approach, including the probability calculation, is illustrated by an application of the kinetic description to the recently proposed burning model.