The nature of the critical state is described by the response of a system to external perturbation. For systems exhibiting noncritical behavior, the reaction of the system is described by a characteristic response time and characteristic length scale over which the perturbation is felt spatially. Although the response of a noncritical system may differ in detail as the system is perturbed at different positions and at different times, the distribution of responses is narrow and is well described by the average response. For a critical system, the same perturbation applied at different positions or at the same position at different times can lead to a response of any size. The average may not be a useful measure of the response; in fact, the average might not even exist.

Response Distributions

To illustrate the description of the critical state, consider the sandpile. We probe the state by adding one single grain of sand to a (randomly) chosen position on the slope. The extra grain will induce an avalanche characterized by such spatial and temporal measures as the total number s of sand grains involved in the avalanche and the lifetime t of the avalanche. We denote the statistical distributions describing the response by P(s) and P(t).