We study the Abelian sandpile model on decorated one dimensional chains. We show that there are two types of avalanches, and determine the effects of finite, though large, system size L on the asymptotic form of distributions of avalanche sizes, and show that these differ qualitatively from the behavior on a simple linear chain. For large L, we find that the probability distribution of the total number of topplings s is not described by a simple finite-size scaling form, but by a linear combination of two simple scaling forms: ${\mathrm{Prob}}_{\mathit{L}}$(s)=(1/L)${\mathit{f}}_1$(s/L)+(1/${\mathit{L}}^2$)${\mathit{f}}_2$(s/${\mathit{L}}^2$), where ${\mathit{f}}_1$ and ${\mathit{f}}_2$ are nonuniversal scaling functions of one argument.

• Received 7 June 1995

DOI:https://doi.org/10.1103/PhysRevE.52.4804