Simplest possible self-organized critical system

In order to pinpoint the nature of self-organized criticality, a simplest possible system exhibiting the phenomenon is introduced and analyzed. Its phase space is fully parametrized by two integer variables, one describing the state of a medium (sandpile), the other describing the state of a disturbance (avalanche) propagating in the medium, modifying it in the process. For asymptotically large systems, a scaling limit is obtained in which the system's state and dynamics is given by two real numbers and a simple partial differential equation. These results provide a full and transparent description of the dynamics that drives this system critical and keeps it in that state.