Fluctuations at the self-organized critical state

Abstract

Avalanche dynamics is defined as coupled internal motion at two separate time scales. The state parameter $S(t)$ measures the amount of tension or potential energy stored in the system. Fluctuations of ${S(t)=F}_{d}^{\ensuremath{\uparrow}}\ensuremath{-}{F}_{a}^{\ensuremath{\downarrow}}$ are caused by the action ${F}_{d}$ on $S$ of the driving force, giving rise to slow, continuous motion, and the antagonistic action ${F}_{a}$ of a relaxation force causing rapid, discrete events (avalanches). The arrows indicate the directions of the forces (increase or decrease). A state parameter may be chosen such that both forces act as repellers in state space. Self-organized criticality (SOC) emerges when ${F}_{a}$ is an internal force that exists if, and only if, ${F}_{d}$ is present. When this contingency condition is fulfilled, the two antagonistic forces trap the system trajectory inside a SOC attractor, which is a state-space region of overlapping basins for the two types of motion. The conclusions are based primarily on the case of river meandering dynamics, described in the paper.