Theory of the one-dimensional forest-fire model

Turbulent cascade processes are studied in terms of a one-dimensional forest-fire model. A hier- archy of steady-state equations for the forests and the holes between them is constructed and solved within a mean-field closure scheme. The exact hole distribution function is found to be ${\mathit{N}}_{\mathit{H}}$(s)=4N/[s(s+1)(s+2)], where N is the number of forests.