Roughness of sandpile surfaces.

J G Oliveira,J F F Mendes,G Tripathy

doi:10.1103/physreve.69.031105

Abstract

We study the surface roughness of prototype models displaying self-organized criticality (SOC) and their noncritical variants in one dimension. For SOC systems, we find that two seemingly equivalent definitions of surface roughness yield different asymptotic scaling exponents. Using approximate analytical arguments and extensive numerical studies we conclude that this ambiguity is due to the special scaling properties of the nonlinear steady state surface. We also find that there is no such ambiguity for non-SOC models, although there may be intermediate crossovers to different roughness values. Such crossovers need to be distinguished from the true asymptotic behavior, as in the case of a noncritical disordered sandpile model studied by Barker and Mehta [Phys. Rev. E 61, 6765 (2000)].

Highlights

We study the surface roughness of prototype models displaying self-organized criticalitySOCand their noncritical variants in one dimension
We find that there is no such ambiguity for non-SOC models, there may be intermediate crossovers to different roughness values. Such crossovers need to be distinguished from the true asymptotic behavior, as in the case of a noncritical disordered sandpile model studied by Barker and MehtaPhys
Since the original proposal by Bak, Tang, and Wisenfeld1͔ there has been a large body of work directed towards understanding the ubiquity of scale invariance in externally driven open dissipative systems using the concept of selforganized criticalitySOC

Summary

Roughness of sandpile surfaces

We study modifications of the LLS which have nonlinear steady state profiles but which do not display SOC and find that there is no such ambiguity in the asymptotic roughness exponent. To study the interfacial fluctuations in these models, we start with an initially flat pilehi(0)ϭ0, ᭙ iand add grains till the pile reaches a steady state with the mean surface making a critical angle ␺ (tan ␺L→ρϭ3/2) with the horizontal. This nonlinearity ofhi(t)͘ is a consequence of the boundary effects and is present for the ILLS as well, we demonstrate that the presence of SOC in LLS results in special scaling properties which is responsible for the observed differences between asymptotic scalings of W1 and W2. Where WS(L) is the root-mean-squarermswandering of the steady state interface profilehi,

Ϫs iϭ

Ϫd x

From the normalization condition

The parameters chosen are the same as in